Theorem ruclem32 7753

Description: Lemma for ruc 7761. All values of function G are less than all values of function H.

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))
ruclem28.a	|- A e. NN
ruclem.b	|- B e. NN

Assertion

Ref	Expression
ruclem32	|- (G` A) < (H` B)

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

Proof of Theorem ruclem32

Step	Hyp	Ref	Expression
1		ruclem.b	. . . 4 |- B e. NN
2		opreq2 4027	. . . . . . 7 |- (w = 1 -> (A + w) = (A + 1))
3	2	fveq2d 3839	. . . . . 6 |- (w = 1 -> (G` (A + w)) = (G` (A + 1)))
4	3	breq2d 2703	. . . . 5 |- (w = 1 -> ((G` A) < (G` (A + w)) <-> (G` A) < (G` (A + 1))))
5		opreq2 4027	. . . . . . 7 |- (w = v -> (A + w) = (A + v))
6	5	fveq2d 3839	. . . . . 6 |- (w = v -> (G` (A + w)) = (G` (A + v)))
7	6	breq2d 2703	. . . . 5 |- (w = v -> ((G` A) < (G` (A + w)) <-> (G` A) < (G` (A + v))))
8		opreq2 4027	. . . . . . 7 |- (w = (v + 1) -> (A + w) = (A + (v + 1)))
9	8	fveq2d 3839	. . . . . 6 |- (w = (v + 1) -> (G` (A + w)) = (G` (A + (v + 1))))
10	9	breq2d 2703	. . . . 5 |- (w = (v + 1) -> ((G` A) < (G` (A + w)) <-> (G` A) < (G` (A + (v + 1)))))
11		opreq2 4027	. . . . . . 7 |- (w = B -> (A + w) = (A + B))
12	11	fveq2d 3839	. . . . . 6 |- (w = B -> (G` (A + w)) = (G` (A + B)))
13	12	breq2d 2703	. . . . 5 |- (w = B -> ((G` A) < (G` (A + w)) <-> (G` A) < (G` (A + B))))
14		ruclem.0	. . . . . 6 |- F:NN-->RR
15		ruclem.1	. . . . . 6 |- C = ({<.1, <.((F` 1) + 1), ((F` 1) + 2)>.>.} u. (F |` (NN \ {1})))
16		ruclem.2	. . . . . 6 |- 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)>.))}
17		ruclem.3	. . . . . 6 |- G = (1st o. (D seq1 C))
18		ruclem.4	. . . . . 6 |- H = (2nd o. (D seq1 C))
19		ruclem28.a	. . . . . 6 |- A e. NN
20	14, 15, 16, 17, 18, 19	ruclem26 7747	. . . . 5 |- (G` A) < (G` (A + 1))
21		opreq2 4027	. . . . . . . . 9 |- (v = if(v e. NN, v, 1) -> (A + v) = (A + if(v e. NN, v, 1)))
22	21	fveq2d 3839	. . . . . . . 8 |- (v = if(v e. NN, v, 1) -> (G` (A + v)) = (G` (A + if(v e. NN, v, 1))))
23	22	breq2d 2703	. . . . . . 7 |- (v = if(v e. NN, v, 1) -> ((G` A) < (G` (A + v)) <-> (G` A) < (G` (A + if(v e. NN, v, 1)))))
24		opreq1 4026	. . . . . . . . . 10 |- (v = if(v e. NN, v, 1) -> (v + 1) = (if(v e. NN, v, 1) + 1))
25	24	opreq2d 4034	. . . . . . . . 9 |- (v = if(v e. NN, v, 1) -> (A + (v + 1)) = (A + (if(v e. NN, v, 1) + 1)))
26	25	fveq2d 3839	. . . . . . . 8 |- (v = if(v e. NN, v, 1) -> (G` (A + (v + 1))) = (G` (A + (if(v e. NN, v, 1) + 1))))
27	26	breq2d 2703	. . . . . . 7 |- (v = if(v e. NN, v, 1) -> ((G` A) < (G` (A + (v + 1))) <-> (G` A) < (G` (A + (if(v e. NN, v, 1) + 1)))))
28	23, 27	imbi12d 629	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (((G` A) < (G` (A + v)) -> (G` A) < (G` (A + (v + 1)))) <-> ((G` A) < (G` (A + if(v e. NN, v, 1))) -> (G` A) < (G` (A + (if(v e. NN, v, 1) + 1))))))
29		1nn 6079	. . . . . . . 8 |- 1 e. NN
30	29	elimel 2451	. . . . . . 7 |- if(v e. NN, v, 1) e. NN
31	14, 15, 16, 17, 18, 19, 30	ruclem30 7751	. . . . . 6 |- ((G` A) < (G` (A + if(v e. NN, v, 1))) -> (G` A) < (G` (A + (if(v e. NN, v, 1) + 1))))
32	28, 31	dedth 2437	. . . . 5 |- (v e. NN -> ((G` A) < (G` (A + v)) -> (G` A) < (G` (A + (v + 1)))))
33	4, 7, 10, 13, 20, 32	nnind 6082	. . . 4 |- (B e. NN -> (G` A) < (G` (A + B)))
34	1, 33	ax-mp 7	. . 3 |- (G` A) < (G` (A + B))
35		nnaddcl 6085	. . . . 5 |- ((A e. NN /\ B e. NN) -> (A + B) e. NN)
36	19, 1, 35	mp2an 701	. . . 4 |- (A + B) e. NN
37	14, 15, 16, 17, 18, 36	ruclem25 7746	. . 3 |- (G` (A + B)) < (H` (A + B))
38	14, 15, 16, 17, 18, 19	ruclem22 7743	. . . 4 |- (G` A) e. RR
39	14, 15, 16, 17, 18, 36	ruclem22 7743	. . . 4 |- (G` (A + B)) e. RR
40	14, 15, 16, 17, 18, 36	ruclem23 7744	. . . 4 |- (H` (A + B)) e. RR
41	38, 39, 40	lttri 5739	. . 3 |- (((G` A) < (G` (A + B)) /\ (G` (A + B)) < (H` (A + B))) -> (G` A) < (H` (A + B)))
42	34, 37, 41	mp2an 701	. 2 |- (G` A) < (H` (A + B))
43		opreq1 4026	. . . . . 6 |- (w = 1 -> (w + B) = (1 + B))
44	43	fveq2d 3839	. . . . 5 |- (w = 1 -> (H` (w + B)) = (H` (1 + B)))
45	44	breq1d 2702	. . . 4 |- (w = 1 -> ((H` (w + B)) < (H` B) <-> (H` (1 + B)) < (H` B)))
46		opreq1 4026	. . . . . 6 |- (w = v -> (w + B) = (v + B))
47	46	fveq2d 3839	. . . . 5 |- (w = v -> (H` (w + B)) = (H` (v + B)))
48	47	breq1d 2702	. . . 4 |- (w = v -> ((H` (w + B)) < (H` B) <-> (H` (v + B)) < (H` B)))
49		opreq1 4026	. . . . . 6 |- (w = (v + 1) -> (w + B) = ((v + 1) + B))
50	49	fveq2d 3839	. . . . 5 |- (w = (v + 1) -> (H` (w + B)) = (H` ((v + 1) + B)))
51	50	breq1d 2702	. . . 4 |- (w = (v + 1) -> ((H` (w + B)) < (H` B) <-> (H` ((v + 1) + B)) < (H` B)))
52		opreq1 4026	. . . . . 6 |- (w = A -> (w + B) = (A + B))
53	52	fveq2d 3839	. . . . 5 |- (w = A -> (H` (w + B)) = (H` (A + B)))
54	53	breq1d 2702	. . . 4 |- (w = A -> ((H` (w + B)) < (H` B) <-> (H` (A + B)) < (H` B)))
55		ax1cn 5423	. . . . . . 7 |- 1 e. CC
56	1	nncni 6077	. . . . . . 7 |- B e. CC
57	55, 56	addcomi 5476	. . . . . 6 |- (1 + B) = (B + 1)
58	57	fveq2i 3838	. . . . 5 |- (H` (1 + B)) = (H` (B + 1))
59	14, 15, 16, 17, 18, 1	ruclem27 7748	. . . . 5 |- (H` (B + 1)) < (H` B)
60	58, 59	eqbrtri 2707	. . . 4 |- (H` (1 + B)) < (H` B)
61		opreq1 4026	. . . . . . . 8 |- (v = if(v e. NN, v, 1) -> (v + B) = (if(v e. NN, v, 1) + B))
62	61	fveq2d 3839	. . . . . . 7 |- (v = if(v e. NN, v, 1) -> (H` (v + B)) = (H` (if(v e. NN, v, 1) + B)))
63	62	breq1d 2702	. . . . . 6 |- (v = if(v e. NN, v, 1) -> ((H` (v + B)) < (H` B) <-> (H` (if(v e. NN, v, 1) + B)) < (H` B)))
64	24	opreq1d 4033	. . . . . . . 8 |- (v = if(v e. NN, v, 1) -> ((v + 1) + B) = ((if(v e. NN, v, 1) + 1) + B))
65	64	fveq2d 3839	. . . . . . 7 |- (v = if(v e. NN, v, 1) -> (H` ((v + 1) + B)) = (H` ((if(v e. NN, v, 1) + 1) + B)))
66	65	breq1d 2702	. . . . . 6 |- (v = if(v e. NN, v, 1) -> ((H` ((v + 1) + B)) < (H` B) <-> (H` ((if(v e. NN, v, 1) + 1) + B)) < (H` B)))
67	63, 66	imbi12d 629	. . . . 5 |- (v = if(v e. NN, v, 1) -> (((H` (v + B)) < (H` B) -> (H` ((v + 1) + B)) < (H` B)) <-> ((H` (if(v e. NN, v, 1) + B)) < (H` B) -> (H` ((if(v e. NN, v, 1) + 1) + B)) < (H` B))))
68	14, 15, 16, 17, 18, 30, 1	ruclem31 7752	. . . . 5 |- ((H` (if(v e. NN, v, 1) + B)) < (H` B) -> (H` ((if(v e. NN, v, 1) + 1) + B)) < (H` B))
69	67, 68	dedth 2437	. . . 4 |- (v e. NN -> ((H` (v + B)) < (H` B) -> (H` ((v + 1) + B)) < (H` B)))
70	45, 48, 51, 54, 60, 69	nnind 6082	. . 3 |- (A e. NN -> (H` (A + B)) < (H` B))
71	19, 70	ax-mp 7	. 2 |- (H` (A + B)) < (H` B)
72	14, 15, 16, 17, 18, 1	ruclem23 7744	. . 3 |- (H` B) e. RR
73	38, 40, 72	lttri 5739	. 2 |- (((G` A) < (H` (A + B)) /\ (H` (A + B)) < (H` B)) -> (G` A) < (H` B))
74	42, 71, 73	mp2an 701	1 |- (G` A) < (H` B)

Syntax hints:   -> wi 3   /\ wa 221   = wceq 992   e. wcel 994   \ cdif 2096   u. cun 2097  ifcif 2415  {csn 2467  <.cop 2469   class class class wbr 2692   X. cxp 3249   |` cres 3253   o. ccom 3255  -->wf 3259  ` cfv 3263  (class class class)co 4021  {copab2 4022  1stc1st 4138  2ndc2nd 4139  RRcr 5387  1c1 5389   + caddc 5391   x. cmul 5393   / cdiv 5448  NNcn 5450   < clt 5640  2c2 6107  3c3 6108   seq1 cseq1 6672

This theorem is referenced by:  ruclem33 7754  ruclem35 7756

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-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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