Theorem ruclem35 7545

Description: Lemma for ruc 7550. The supremum we have constructed lies between all values of the G and H functions. Compare ruclem29 7539, which states the opposite for the input function F.

Hypotheses

Ref	Expression
ruclem.0	|- F:NN-->RR
ruclem.1	|- C = ({<.1, <.((F` 1) + 1), ((F` 1) + 2)>.>.} u. (F |` (NN \ {1})))
ruclem.2	|- D = {<.<.x, y>., z>. | ((x e. (RR X. RR) /\ y e. RR) /\ z = if(((1st` x) < y /\ y < (2nd` x)), <.(((2 x. y) + (2nd` x)) / 3), ((y + (2 x. (2nd` x))) / 3)>., <.(((2 x. (1st` x)) + (2nd` x)) / 3), (((1st` x) + (2 x. (2nd` x))) / 3)>.))}
ruclem.3	|- G = (1st o. (D seq1 C))
ruclem.4	|- H = (2nd o. (D seq1 C))
ruclem.5	|- S = sup(ran G, RR, < )
ruclem.a	|- A e. NN

Assertion

Ref	Expression
ruclem35	|- ((G` A) < S /\ S < (H` A))

Distinct variable groups:   x,y,z   z,F

Proof of Theorem ruclem35

Step	Hyp	Ref	Expression
1		ruclem.0	. . . 4 |- F:NN-->RR
2		ruclem.1	. . . 4 |- C = ({<.1, <.((F` 1) + 1), ((F` 1) + 2)>.>.} u. (F |` (NN \ {1})))
3		ruclem.2	. . . 4 |- D = {<.<.x, y>., z>. | ((x e. (RR X. RR) /\ y e. RR) /\ z = if(((1st` x) < y /\ y < (2nd` x)), <.(((2 x. y) + (2nd` x)) / 3), ((y + (2 x. (2nd` x))) / 3)>., <.(((2 x. (1st` x)) + (2nd` x)) / 3), (((1st` x) + (2 x. (2nd` x))) / 3)>.))}
4		ruclem.3	. . . 4 |- G = (1st o. (D seq1 C))
5		ruclem.4	. . . 4 |- H = (2nd o. (D seq1 C))
6		ruclem.a	. . . 4 |- A e. NN
7	1, 2, 3, 4, 5, 6	ruclem26 7536	. . 3 |- (G` A) < (G` (A + 1))
8	1, 2, 3, 4, 5	ruclem17 7527	. . . . . . 7 |- G:NN-->RR
9		ffn 3633	. . . . . . 7 |- (G:NN-->RR -> G Fn NN)
10	8, 9	ax-mp 7	. . . . . 6 |- G Fn NN
11		peano2nn 5937	. . . . . . 7 |- (A e. NN -> (A + 1) e. NN)
12	6, 11	ax-mp 7	. . . . . 6 |- (A + 1) e. NN
13		fnfvelrn 3819	. . . . . 6 |- ((G Fn NN /\ (A + 1) e. NN) -> (G` (A + 1)) e. ran G)
14	10, 12, 13	mp2an 699	. . . . 5 |- (G` (A + 1)) e. ran G
15	1, 2, 3, 4, 5	ruclem33 7543	. . . . . 6 |- (ran G (_ RR /\ ran G =/= (/) /\ E.w e. RR A.v e. ran G v <_ w)
16	15	suprubi 6064	. . . . 5 |- ((G` (A + 1)) e. ran G -> (G` (A + 1)) <_ sup(ran G, RR, < ))
17	14, 16	ax-mp 7	. . . 4 |- (G` (A + 1)) <_ sup(ran G, RR, < )
18		ruclem.5	. . . 4 |- S = sup(ran G, RR, < )
19	17, 18	breqtrr 2645	. . 3 |- (G` (A + 1)) <_ S
20	1, 2, 3, 4, 5, 6	ruclem22 7532	. . . 4 |- (G` A) e. RR
21	1, 2, 3, 4, 5, 12	ruclem22 7532	. . . 4 |- (G` (A + 1)) e. RR
22	1, 2, 3, 4, 5, 18	ruclem34 7544	. . . 4 |- S e. RR
23	20, 21, 22	ltletr 5599	. . 3 |- (((G` A) < (G` (A + 1)) /\ (G` (A + 1)) <_ S) -> (G` A) < S)
24	7, 19, 23	mp2an 699	. 2 |- (G` A) < S
25	1, 2, 3, 4, 5, 12	ruclem23 7533	. . . . . 6 |- (H` (A + 1)) e. RR
26		fvelrnb 3766	. . . . . . . . 9 |- (G Fn NN -> (u e. ran G <-> E.w e. NN (G` w) = u))
27	10, 26	ax-mp 7	. . . . . . . 8 |- (u e. ran G <-> E.w e. NN (G` w) = u)
28		breq2 2628	. . . . . . . . . . 11 |- ((G` w) = u -> ((H` (A + 1)) < (G` w) <-> (H` (A + 1)) < u))
29	28	negbid 613	. . . . . . . . . 10 |- ((G` w) = u -> (-. (H` (A + 1)) < (G` w) <-> -. (H` (A + 1)) < u))
30		ltnsymt 5544	. . . . . . . . . . 11 |- (((G` w) e. RR /\ (H` (A + 1)) e. RR) -> ((G` w) < (H` (A + 1)) -> -. (H` (A + 1)) < (G` w)))
31		fveq2 3730	. . . . . . . . . . . . . 14 |- (w = if(w e. NN, w, 1) -> (G` w) = (G` if(w e. NN, w, 1)))
32	31	eleq1d 1543	. . . . . . . . . . . . 13 |- (w = if(w e. NN, w, 1) -> ((G` w) e. RR <-> (G` if(w e. NN, w, 1)) e. RR))
33		1nn 5936	. . . . . . . . . . . . . . 15 |- 1 e. NN
34	33	elimel 2398	. . . . . . . . . . . . . 14 |- if(w e. NN, w, 1) e. NN
35	1, 2, 3, 4, 5, 34	ruclem22 7532	. . . . . . . . . . . . 13 |- (G` if(w e. NN, w, 1)) e. RR
36	32, 35	dedth 2387	. . . . . . . . . . . 12 |- (w e. NN -> (G` w) e. RR)
37	36, 25	jctir 293	. . . . . . . . . . 11 |- (w e. NN -> ((G` w) e. RR /\ (H` (A + 1)) e. RR))
38	31	breq1d 2634	. . . . . . . . . . . 12 |- (w = if(w e. NN, w, 1) -> ((G` w) < (H` (A + 1)) <-> (G` if(w e. NN, w, 1)) < (H` (A + 1))))
39	1, 2, 3, 4, 5, 34, 12	ruclem32 7542	. . . . . . . . . . . 12 |- (G` if(w e. NN, w, 1)) < (H` (A + 1))
40	38, 39	dedth 2387	. . . . . . . . . . 11 |- (w e. NN -> (G` w) < (H` (A + 1)))
41	30, 37, 40	sylc 68	. . . . . . . . . 10 |- (w e. NN -> -. (H` (A + 1)) < (G` w))
42	29, 41	syl5cbi 209	. . . . . . . . 9 |- (w e. NN -> ((G` w) = u -> -. (H` (A + 1)) < u))
43	42	r19.23aiv 1746	. . . . . . . 8 |- (E.w e. NN (G` w) = u -> -. (H` (A + 1)) < u)
44	27, 43	sylbi 199	. . . . . . 7 |- (u e. ran G -> -. (H` (A + 1)) < u)
45	44	rgen 1701	. . . . . 6 |- A.u e. ran G -. (H` (A + 1)) < u
46	15	suprnubi 6066	. . . . . 6 |- (((H` (A + 1)) e. RR /\ A.u e. ran G -. (H` (A + 1)) < u) -> -. (H` (A + 1)) < sup(ran G, RR, < ))
47	25, 45, 46	mp2an 699	. . . . 5 |- -. (H` (A + 1)) < sup(ran G, RR, < )
48	18	breq2i 2632	. . . . 5 |- ((H` (A + 1)) < S <-> (H` (A + 1)) < sup(ran G, RR, < ))
49	47, 48	mtbir 192	. . . 4 |- -. (H` (A + 1)) < S
50	22, 25	lenlt 5590	. . . 4 |- (S <_ (H` (A + 1)) <-> -. (H` (A + 1)) < S)
51	49, 50	mpbir 190	. . 3 |- S <_ (H` (A + 1))
52	1, 2, 3, 4, 5, 6	ruclem27 7537	. . 3 |- (H` (A + 1)) < (H` A)
53	1, 2, 3, 4, 5, 6	ruclem23 7533	. . . 4 |- (H` A) e. RR
54	22, 25, 53	lelttr 5598	. . 3 |- ((S <_ (H` (A + 1)) /\ (H` (A + 1)) < (H` A)) -> S < (H` A))
55	51, 52, 54	mp2an 699	. 2 |- S < (H` A)
56	24, 55	pm3.2i 285	1 |- ((G` A) < S /\ S < (H` A))

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649   \ cdif 2047   u. cun 2048  ifcif 2365  {csn 2413  <.cop 2415   class class class wbr 2624   X. cxp 3174  ran crn 3177   |` cres 3178   o. ccom 3180   Fn wfn 3183  -->wf 3184  ` cfv 3188  (class class class)co 3969  {copab2 3970  1stc1st 4083  2ndc2nd 4084  supcsup 4582  RRcr 5245  1c1 5247   + caddc 5249   x. cmul 5251   / cdiv 5306   <_ cle 5307  NNcn 5308   < clt 5498  2c2 5963  3c3 5964   seq1 cseq1 6308

This theorem is referenced by:  ruclem36 7546

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-3 5973  df-n0 6102  df-z 6138  df-seq1 6309

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