Theorem dsupivthlem 7234

Description: Lemma for dsupivth 7235.

Hypotheses

Ref	Expression
isupivth.1	|- A e. RR
isupivth.2	|- B e. RR
isupivth.3	|- U e. RR
isupivth.4	|- A < B
isupivth.5	|- (A[,]B) (_ D
isupivth.6	|- D (_ CC
isupivth.7	|- F e. (D-cn->CC)
isupivth.8	|- (x e. (A[,]B) -> (F` x) e. RR)
isupivth.9	|- S = {x e. (A[,]B) | (F` x) = U}
dsupivth.10	|- ((F` B) < U /\ U < (F` A))
dsupivth.11	|- C = sup(S, RR, < )
dsupivthlem.12	|- G = {<.a, b>. | (a e. D /\ b = -u(F` a))}

Assertion

Ref	Expression
dsupivthlem	|- (C e. (A(,)B) /\ (F` C) = U)

Distinct variable groups:   A,a,b,x   B,a,b,x   C,a,b   D,a,b,x   F,a,b,x   x,G   x,U

Proof of Theorem dsupivthlem

Step	Hyp	Ref	Expression
1		isupivth.1	. . . 4 |- A e. RR
2		isupivth.2	. . . 4 |- B e. RR
3		isupivth.3	. . . . 5 |- U e. RR
4	3	renegcl 5396	. . . 4 |- -uU e. RR
5		isupivth.4	. . . 4 |- A < B
6		isupivth.5	. . . 4 |- (A[,]B) (_ D
7		isupivth.6	. . . 4 |- D (_ CC
8		isupivth.7	. . . . 5 |- F e. (D-cn->CC)
9		dsupivthlem.12	. . . . 5 |- G = {<.a, b>. | (a e. D /\ b = -u(F` a))}
10	7, 8, 9	negfcncf 7212	. . . 4 |- G e. (D-cn->CC)
11	6	sseli 2061	. . . . . 6 |- (x e. (A[,]B) -> x e. D)
12		fveq2 3715	. . . . . . . 8 |- (a = x -> (F` a) = (F` x))
13	12	negeqd 5341	. . . . . . 7 |- (a = x -> -u(F` a) = -u(F` x))
14		negex 5345	. . . . . . 7 |- -u(F` x) e. V
15	13, 9, 14	fvopab4 3771	. . . . . 6 |- (x e. D -> (G` x) = -u(F` x))
16	11, 15	syl 10	. . . . 5 |- (x e. (A[,]B) -> (G` x) = -u(F` x))
17		isupivth.8	. . . . . 6 |- (x e. (A[,]B) -> (F` x) e. RR)
18		renegclt 5417	. . . . . 6 |- ((F` x) e. RR -> -u(F` x) e. RR)
19	17, 18	syl 10	. . . . 5 |- (x e. (A[,]B) -> -u(F` x) e. RR)
20	16, 19	eqeltrd 1545	. . . 4 |- (x e. (A[,]B) -> (G` x) e. RR)
21		isupivth.9	. . . . 5 |- S = {x e. (A[,]B) | (F` x) = U}
22	16	eqeq1d 1480	. . . . . . 7 |- (x e. (A[,]B) -> ((G` x) = -uU <-> -u(F` x) = -uU))
23	17	recnd 5295	. . . . . . . 8 |- (x e. (A[,]B) -> (F` x) e. CC)
24	3	recn 5294	. . . . . . . . 9 |- U e. CC
25		neg11t 5389	. . . . . . . . 9 |- (((F` x) e. CC /\ U e. CC) -> (-u(F` x) = -uU <-> (F` x) = U))
26	24, 25	mpan2 695	. . . . . . . 8 |- ((F` x) e. CC -> (-u(F` x) = -uU <-> (F` x) = U))
27	23, 26	syl 10	. . . . . . 7 |- (x e. (A[,]B) -> (-u(F` x) = -uU <-> (F` x) = U))
28	22, 27	bitr2d 528	. . . . . 6 |- (x e. (A[,]B) -> ((F` x) = U <-> (G` x) = -uU))
29	28	rabbii 1801	. . . . 5 |- {x e. (A[,]B) | (F` x) = U} = {x e. (A[,]B) | (G` x) = -uU}
30	21, 29	eqtr 1492	. . . 4 |- S = {x e. (A[,]B) | (G` x) = -uU}
31	1, 2, 5	ltlei 5562	. . . . . . . . 9 |- A <_ B
32		lbicc2t 6345	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ A <_ B) -> A e. (A[,]B))
33	1, 2, 31, 32	mp3an 914	. . . . . . . 8 |- A e. (A[,]B)
34	6, 33	sselii 2062	. . . . . . 7 |- A e. D
35		fveq2 3715	. . . . . . . . 9 |- (a = A -> (F` a) = (F` A))
36	35	negeqd 5341	. . . . . . . 8 |- (a = A -> -u(F` a) = -u(F` A))
37		negex 5345	. . . . . . . 8 |- -u(F` A) e. V
38	36, 9, 37	fvopab4 3771	. . . . . . 7 |- (A e. D -> (G` A) = -u(F` A))
39	34, 38	ax-mp 7	. . . . . 6 |- (G` A) = -u(F` A)
40		dsupivth.10	. . . . . . . 8 |- ((F` B) < U /\ U < (F` A))
41	40	pm3.27i 324	. . . . . . 7 |- U < (F` A)
42	17	rgen 1695	. . . . . . . . 9 |- A.x e. (A[,]B)(F` x) e. RR
43		fveq2 3715	. . . . . . . . . . 11 |- (x = A -> (F` x) = (F` A))
44	43	eleq1d 1537	. . . . . . . . . 10 |- (x = A -> ((F` x) e. RR <-> (F` A) e. RR))
45	44	rcla4v 1869	. . . . . . . . 9 |- (A e. (A[,]B) -> (A.x e. (A[,]B)(F` x) e. RR -> (F` A) e. RR))
46	33, 42, 45	mp2 43	. . . . . . . 8 |- (F` A) e. RR
47	3, 46	ltneg 5585	. . . . . . 7 |- (U < (F` A) <-> -u(F` A) < -uU)
48	41, 47	mpbi 189	. . . . . 6 |- -u(F` A) < -uU
49	39, 48	eqbrtr 2629	. . . . 5 |- (G` A) < -uU
50	40	pm3.26i 320	. . . . . . 7 |- (F` B) < U
51		ubicc2t 6346	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A <_ B) -> B e. (A[,]B))
52	1, 2, 31, 51	mp3an 914	. . . . . . . . 9 |- B e. (A[,]B)
53		fveq2 3715	. . . . . . . . . . 11 |- (x = B -> (F` x) = (F` B))
54	53	eleq1d 1537	. . . . . . . . . 10 |- (x = B -> ((F` x) e. RR <-> (F` B) e. RR))
55	54	rcla4v 1869	. . . . . . . . 9 |- (B e. (A[,]B) -> (A.x e. (A[,]B)(F` x) e. RR -> (F` B) e. RR))
56	52, 42, 55	mp2 43	. . . . . . . 8 |- (F` B) e. RR
57	56, 3	ltneg 5585	. . . . . . 7 |- ((F` B) < U <-> -uU < -u(F` B))
58	50, 57	mpbi 189	. . . . . 6 |- -uU < -u(F` B)
59	6, 52	sselii 2062	. . . . . . 7 |- B e. D
60		fveq2 3715	. . . . . . . . 9 |- (a = B -> (F` a) = (F` B))
61	60	negeqd 5341	. . . . . . . 8 |- (a = B -> -u(F` a) = -u(F` B))
62		negex 5345	. . . . . . . 8 |- -u(F` B) e. V
63	61, 9, 62	fvopab4 3771	. . . . . . 7 |- (B e. D -> (G` B) = -u(F` B))
64	59, 63	ax-mp 7	. . . . . 6 |- (G` B) = -u(F` B)
65	58, 64	breqtrr 2635	. . . . 5 |- -uU < (G` B)
66	49, 65	pm3.2i 285	. . . 4 |- ((G` A) < -uU /\ -uU < (G` B))
67		dsupivth.11	. . . 4 |- C = sup(S, RR, < )
68	1, 2, 4, 5, 6, 7, 10, 20, 30, 66, 67	isupivth 7233	. . 3 |- (C e. (A(,)B) /\ (G` C) = -uU)
69	68	pm3.26i 320	. 2 |- C e. (A(,)B)
70		ioossicc 6338	. . . . . 6 |- (A(,)B) (_ (A[,]B)
71	70, 69	sselii 2062	. . . . 5 |- C e. (A[,]B)
72	6	sseli 2061	. . . . . 6 |- (C e. (A[,]B) -> C e. D)
73		fveq2 3715	. . . . . . . 8 |- (a = C -> (F` a) = (F` C))
74	73	negeqd 5341	. . . . . . 7 |- (a = C -> -u(F` a) = -u(F` C))
75		negex 5345	. . . . . . 7 |- -u(F` C) e. V
76	74, 9, 75	fvopab4 3771	. . . . . 6 |- (C e. D -> (G` C) = -u(F` C))
77	72, 76	syl 10	. . . . 5 |- (C e. (A[,]B) -> (G` C) = -u(F` C))
78	71, 77	ax-mp 7	. . . 4 |- (G` C) = -u(F` C)
79	68	pm3.27i 324	. . . 4 |- (G` C) = -uU
80	78, 79	eqtr3 1494	. . 3 |- -u(F` C) = -uU
81		ssid 2076	. . . . . 6 |- CC (_ CC
82		cncffvelrn 7211	. . . . . 6 |- ((D (_ CC /\ CC (_ CC /\ F e. (D-cn->CC)) -> (C e. D -> (F` C) e. CC))
83	7, 81, 8, 82	mp3an 914	. . . . 5 |- (C e. D -> (F` C) e. CC)
84		neg11t 5389	. . . . . 6 |- (((F` C) e. CC /\ U e. CC) -> (-u(F` C) = -uU <-> (F` C) = U))
85	24, 84	mpan2 695	. . . . 5 |- ((F` C) e. CC -> (-u(F` C) = -uU <-> (F` C) = U))
86	72, 83, 85	3syl 20	. . . 4 |- (C e. (A[,]B) -> (-u(F` C) = -uU <-> (F` C) = U))
87	71, 86	ax-mp 7	. . 3 |- (-u(F` C) = -uU <-> (F` C) = U)
88	80, 87	mpbi 189	. 2 |- (F` C) = U
89	69, 88	pm3.2i 285	1 |- (C e. (A(,)B) /\ (F` C) = U)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642  {crab 1645   (_ wss 2043   class class class wbr 2614  {copab 2661  ` cfv 3177  (class class class)co 3954  supcsup 4553  CCcc 5212  RRcr 5213  -ucneg 5273   <_ cle 5275   < clt 5466  (,)cioo 6302  [,]cicc 6305  -cn->ccncf 7205

This theorem is referenced by:  dsupivth 7235

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-sup 4554  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680