Theorem ruclem35 7756

Description: Lemma for ruc 7761. The supremum we have constructed lies between all values of the G and H functions. Compare ruclem29 7750, 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 7747	. . 3 |- (G` A) < (G` (A + 1))
8	1, 2, 3, 4, 5	ruclem17 7738	. . . . . . 7 |- G:NN-->RR
9		ffn 3734	. . . . . . 7 |- (G:NN-->RR -> G Fn NN)
10	8, 9	ax-mp 7	. . . . . 6 |- G Fn NN
11		peano2nn 6080	. . . . . . 7 |- (A e. NN -> (A + 1) e. NN)
12	6, 11	ax-mp 7	. . . . . 6 |- (A + 1) e. NN
13		fnfvelrn 3927	. . . . . 6 |- ((G Fn NN /\ (A + 1) e. NN) -> (G` (A + 1)) e. ran G)
14	10, 12, 13	mp2an 701	. . . . 5 |- (G` (A + 1)) e. ran G
15	1, 2, 3, 4, 5	ruclem33 7754	. . . . . 6 |- (ran G (_ RR /\ ran G =/= (/) /\ E.w e. RR A.v e. ran G v <_ w)
16	15	suprubii 6230	. . . . 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	breqtrri 2713	. . 3 |- (G` (A + 1)) <_ S
20	1, 2, 3, 4, 5, 6	ruclem22 7743	. . . 4 |- (G` A) e. RR
21	1, 2, 3, 4, 5, 12	ruclem22 7743	. . . 4 |- (G` (A + 1)) e. RR
22	1, 2, 3, 4, 5, 18	ruclem34 7755	. . . 4 |- S e. RR
23	20, 21, 22	ltletri 5741	. . 3 |- (((G` A) < (G` (A + 1)) /\ (G` (A + 1)) <_ S) -> (G` A) < S)
24	7, 19, 23	mp2an 701	. 2 |- (G` A) < S
25	1, 2, 3, 4, 5, 12	ruclem23 7744	. . . . . 6 |- (H` (A + 1)) e. RR
26		fvelrnb 3871	. . . . . . . . 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 2696	. . . . . . . . . . 11 |- ((G` w) = u -> ((H` (A + 1)) < (G` w) <-> (H` (A + 1)) < u))
29	28	notbid 614	. . . . . . . . . 10 |- ((G` w) = u -> (-. (H` (A + 1)) < (G` w) <-> -. (H` (A + 1)) < u))
30		ltnsym 5686	. . . . . . . . . . 11 |- (((G` w) e. RR /\ (H` (A + 1)) e. RR) -> ((G` w) < (H` (A + 1)) -> -. (H` (A + 1)) < (G` w)))
31		fveq2 3835	. . . . . . . . . . . . . 14 |- (w = if(w e. NN, w, 1) -> (G` w) = (G` if(w e. NN, w, 1)))
32	31	eleq1d 1583	. . . . . . . . . . . . 13 |- (w = if(w e. NN, w, 1) -> ((G` w) e. RR <-> (G` if(w e. NN, w, 1)) e. RR))
33		1nn 6079	. . . . . . . . . . . . . . 15 |- 1 e. NN
34	33	elimel 2451	. . . . . . . . . . . . . 14 |- if(w e. NN, w, 1) e. NN
35	1, 2, 3, 4, 5, 34	ruclem22 7743	. . . . . . . . . . . . 13 |- (G` if(w e. NN, w, 1)) e. RR
36	32, 35	dedth 2437	. . . . . . . . . . . 12 |- (w e. NN -> (G` w) e. RR)
37	36, 25	jctir 291	. . . . . . . . . . 11 |- (w e. NN -> ((G` w) e. RR /\ (H` (A + 1)) e. RR))
38	31	breq1d 2702	. . . . . . . . . . . 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 7753	. . . . . . . . . . . 12 |- (G` if(w e. NN, w, 1)) < (H` (A + 1))
40	38, 39	dedth 2437	. . . . . . . . . . 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 207	. . . . . . . . 9 |- (w e. NN -> ((G` w) = u -> -. (H` (A + 1)) < u))
43	42	r19.23aiv 1789	. . . . . . . 8 |- (E.w e. NN (G` w) = u -> -. (H` (A + 1)) < u)
44	27, 43	sylbi 197	. . . . . . 7 |- (u e. ran G -> -. (H` (A + 1)) < u)
45	44	rgen 1744	. . . . . 6 |- A.u e. ran G -. (H` (A + 1)) < u
46	15	suprnubii 6232	. . . . . 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 701	. . . . 5 |- -. (H` (A + 1)) < sup(ran G, RR, < )
48	18	breq2i 2700	. . . . 5 |- ((H` (A + 1)) < S <-> (H` (A + 1)) < sup(ran G, RR, < ))
49	47, 48	mtbir 190	. . . 4 |- -. (H` (A + 1)) < S
50	22, 25	lenlti 5732	. . . 4 |- (S <_ (H` (A + 1)) <-> -. (H` (A + 1)) < S)
51	49, 50	mpbir 188	. . 3 |- S <_ (H` (A + 1))
52	1, 2, 3, 4, 5, 6	ruclem27 7748	. . 3 |- (H` (A + 1)) < (H` A)
53	1, 2, 3, 4, 5, 6	ruclem23 7744	. . . 4 |- (H` A) e. RR
54	22, 25, 53	lelttri 5740	. . 3 |- ((S <_ (H` (A + 1)) /\ (H` (A + 1)) < (H` A)) -> S < (H` A))
55	51, 52, 54	mp2an 701	. 2 |- S < (H` A)
56	24, 55	pm3.2i 283	1 |- ((G` A) < S /\ S < (H` A))

Syntax hints:  -. wn 2   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  A.wral 1691  E.wrex 1692   \ cdif 2096   u. cun 2097  ifcif 2415  {csn 2467  <.cop 2469   class class class wbr 2692   X. cxp 3249  ran crn 3252   |` cres 3253   o. ccom 3255   Fn wfn 3258  -->wf 3259  ` cfv 3263  (class class class)co 4021  {copab2 4022  1stc1st 4138  2ndc2nd 4139  supcsup 4716  RRcr 5387  1c1 5389   + caddc 5391   x. cmul 5393   / cdiv 5448   <_ cle 5449  NNcn 5450   < clt 5640  2c2 6107  3c3 6108   seq1 cseq1 6672

This theorem is referenced by:  ruclem36 7757

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-sup 4717  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-div 5855  df-n 6070  df-2 6116  df-3 6117  df-n0 6268  df-z 6304  df-seq1 6673

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