Theorem isupivth 7225

Description: The intermediate value theorem, increasing case with supremum solution. (Contributed by Paul Chapman, 22-Jan-2008.)

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}
isupivth.10	|- ((F` A) < U /\ U < (F` B))
isupivth.11	|- C = sup(S, RR, < )

Assertion

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

Distinct variable groups:   x,A   x,B   x,F   x,U

Proof of Theorem isupivth

Step	Hyp	Ref	Expression
1		isupivth.11	. . . 4 |- C = sup(S, RR, < )
2		isupivth.1	. . . . 5 |- A e. RR
3		isupivth.2	. . . . 5 |- B e. RR
4		isupivth.3	. . . . 5 |- U e. RR
5		isupivth.4	. . . . 5 |- A < B
6		isupivth.10	. . . . . 6 |- ((F` A) < U /\ U < (F` B))
7	2, 3, 5	ltlei 5554	. . . . . . . . . 10 |- A <_ B
8		lbicc2t 6337	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A <_ B) -> A e. (A[,]B))
9	2, 3, 7, 8	mp3an 913	. . . . . . . . 9 |- A e. (A[,]B)
10		fvres 3719	. . . . . . . . 9 |- (A e. (A[,]B) -> ((F |` (A[,]B))` A) = (F` A))
11	9, 10	ax-mp 7	. . . . . . . 8 |- ((F |` (A[,]B))` A) = (F` A)
12	11	breq1i 2616	. . . . . . 7 |- (((F |` (A[,]B))` A) < U <-> (F` A) < U)
13		ubicc2t 6338	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A <_ B) -> B e. (A[,]B))
14	2, 3, 7, 13	mp3an 913	. . . . . . . . 9 |- B e. (A[,]B)
15		fvres 3719	. . . . . . . . 9 |- (B e. (A[,]B) -> ((F |` (A[,]B))` B) = (F` B))
16	14, 15	ax-mp 7	. . . . . . . 8 |- ((F |` (A[,]B))` B) = (F` B)
17	16	breq2i 2617	. . . . . . 7 |- (U < ((F |` (A[,]B))` B) <-> U < (F` B))
18	12, 17	anbi12i 481	. . . . . 6 |- ((((F |` (A[,]B))` A) < U /\ U < ((F |` (A[,]B))` B)) <-> ((F` A) < U /\ U < (F` B)))
19	6, 18	mpbir 190	. . . . 5 |- (((F |` (A[,]B))` A) < U /\ U < ((F |` (A[,]B))` B))
20		fvres 3719	. . . . . . . . 9 |- (c e. (A[,]B) -> ((F |` (A[,]B))` c) = (F` c))
21	20	breq1d 2619	. . . . . . . 8 |- (c e. (A[,]B) -> (((F |` (A[,]B))` c) <_ U <-> (F` c) <_ U))
22	21	rabbii 1796	. . . . . . 7 |- {c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U} = {c e. (A[,]B) | (F` c) <_ U}
23		supeq1 4549	. . . . . . 7 |- ({c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U} = {c e. (A[,]B) | (F` c) <_ U} -> sup({c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U}, RR, < ) = sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ))
24	22, 23	ax-mp 7	. . . . . 6 |- sup({c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U}, RR, < ) = sup({c e. (A[,]B) | (F` c) <_ U}, RR, < )
25	24	eqcomi 1471	. . . . 5 |- sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) = sup({c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U}, RR, < )
26		eqid 1468	. . . . 5 |- {c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U} = {c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U}
27		isupivth.7	. . . . . . 7 |- F e. (D-cn->CC)
28		isupivth.6	. . . . . . . 8 |- D (_ CC
29		ssid 2070	. . . . . . . 8 |- CC (_ CC
30		isupivth.5	. . . . . . . 8 |- (A[,]B) (_ D
31		rescncf 7207	. . . . . . . 8 |- ((D (_ CC /\ CC (_ CC /\ (A[,]B) (_ D) -> (F e. (D-cn->CC) -> (F |` (A[,]B)) e. ((A[,]B)-cn->CC)))
32	28, 29, 30, 31	mp3an 913	. . . . . . 7 |- (F e. (D-cn->CC) -> (F |` (A[,]B)) e. ((A[,]B)-cn->CC))
33	27, 32	ax-mp 7	. . . . . 6 |- (F |` (A[,]B)) e. ((A[,]B)-cn->CC)
34	30, 28	sstri 2063	. . . . . . . 8 |- (A[,]B) (_ CC
35		axresscn 5240	. . . . . . . 8 |- RR (_ CC
36	34, 29, 35	3pm3.2i 816	. . . . . . 7 |- ((A[,]B) (_ CC /\ CC (_ CC /\ RR (_ CC)
37		fvres 3719	. . . . . . . . 9 |- (x e. (A[,]B) -> ((F |` (A[,]B))` x) = (F` x))
38		isupivth.8	. . . . . . . . 9 |- (x e. (A[,]B) -> (F` x) e. RR)
39	37, 38	eqeltrd 1540	. . . . . . . 8 |- (x e. (A[,]B) -> ((F |` (A[,]B))` x) e. RR)
40	39	rgen 1690	. . . . . . 7 |- A.x e. (A[,]B)((F |` (A[,]B))` x) e. RR
41		cncffvrn 7208	. . . . . . 7 |- ((((A[,]B) (_ CC /\ CC (_ CC /\ RR (_ CC) /\ A.x e. (A[,]B)((F |` (A[,]B))` x) e. RR) -> ((F |` (A[,]B)) e. ((A[,]B)-cn->CC) -> (F |` (A[,]B)) e. ((A[,]B)-cn->RR)))
42	36, 40, 41	mp2an 695	. . . . . 6 |- ((F |` (A[,]B)) e. ((A[,]B)-cn->CC) -> (F |` (A[,]B)) e. ((A[,]B)-cn->RR))
43	33, 42	ax-mp 7	. . . . 5 |- (F |` (A[,]B)) e. ((A[,]B)-cn->RR)
44		isupivth.9	. . . . . 6 |- S = {x e. (A[,]B) | (F` x) = U}
45	37	eqeq1d 1475	. . . . . . 7 |- (x e. (A[,]B) -> (((F |` (A[,]B))` x) = U <-> (F` x) = U))
46	45	rabbii 1796	. . . . . 6 |- {x e. (A[,]B) | ((F |` (A[,]B))` x) = U} = {x e. (A[,]B) | (F` x) = U}
47	44, 46	eqtr4 1490	. . . . 5 |- S = {x e. (A[,]B) | ((F |` (A[,]B))` x) = U}
48	2, 3, 4, 5, 19, 25, 26, 43	ivthlem8 7223	. . . . . 6 |- (sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) e. (A(,)B) /\ ((F |` (A[,]B))` sup({c e. (A[,]B) | (F` c) <_ U}, RR, < )) = U)
49	48	pm3.27i 324	. . . . 5 |- ((F |` (A[,]B))` sup({c e. (A[,]B) | (F` c) <_ U}, RR, < )) = U
50	2, 3, 4, 5, 19, 25, 26, 43, 47, 49	ivthlem9 7224	. . . 4 |- sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) = sup(S, RR, < )
51	1, 50	eqtr4 1490	. . 3 |- C = sup({c e. (A[,]B) | (F` c) <_ U}, RR, < )
52	48	pm3.26i 320	. . 3 |- sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) e. (A(,)B)
53	51, 52	eqeltr 1536	. 2 |- C e. (A(,)B)
54	51	fveq2i 3712	. . 3 |- ((F |` (A[,]B))` C) = ((F |` (A[,]B))` sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ))
55		ioossicc 6330	. . . . 5 |- (A(,)B) (_ (A[,]B)
56	55, 53	sselii 2056	. . . 4 |- C e. (A[,]B)
57		fvres 3719	. . . 4 |- (C e. (A[,]B) -> ((F |` (A[,]B))` C) = (F` C))
58	56, 57	ax-mp 7	. . 3 |- ((F |` (A[,]B))` C) = (F` C)
59	54, 58, 49	3eqtr3 1495	. 2 |- (F` C) = U
60	53, 59	pm3.2i 285	1 |- (C e. (A(,)B) /\ (F` C) = U)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  A.wral 1637  {crab 1640   (_ wss 2037   class class class wbr 2609   |` cres 3162  ` cfv 3172  (class class class)co 3948  supcsup 4547  CCcc 5204  RRcr 5205   <_ cle 5267   < clt 5458  (,)cioo 6294  [,]cicc 6297  -cn->ccncf 7197

This theorem is referenced by:  dsupivthlem 7226  reeff1olem1 7364

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-n0 6047  df-z 6083  df-q 6194  df-rp 6219  df-seq1 6245  df-ioo 6298  df-icc 6301  df-exp 6501  df-sqr 6600  df-re 6682  df-im 6683  df-cj 6684  df-abs 6685  df-cncf 7198

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