Theorem ruclem25 7746

Description: Lemma for ruc 7761. At any index A, the value of G is less than the value of 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))
ruclem18.a	|- A e. NN

Assertion

Ref	Expression
ruclem25	|- (G` A) < (H` A)

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

Proof of Theorem ruclem25

Step	Hyp	Ref	Expression
1		ruclem18.a	. 2 |- A e. NN
2		fveq2 3835	. . . 4 |- (w = 1 -> (G` w) = (G` 1))
3		fveq2 3835	. . . 4 |- (w = 1 -> (H` w) = (H` 1))
4	2, 3	breq12d 2704	. . 3 |- (w = 1 -> ((G` w) < (H` w) <-> (G` 1) < (H` 1)))
5		fveq2 3835	. . . 4 |- (w = v -> (G` w) = (G` v))
6		fveq2 3835	. . . 4 |- (w = v -> (H` w) = (H` v))
7	5, 6	breq12d 2704	. . 3 |- (w = v -> ((G` w) < (H` w) <-> (G` v) < (H` v)))
8		fveq2 3835	. . . 4 |- (w = (v + 1) -> (G` w) = (G` (v + 1)))
9		fveq2 3835	. . . 4 |- (w = (v + 1) -> (H` w) = (H` (v + 1)))
10	8, 9	breq12d 2704	. . 3 |- (w = (v + 1) -> ((G` w) < (H` w) <-> (G` (v + 1)) < (H` (v + 1))))
11		fveq2 3835	. . . 4 |- (w = A -> (G` w) = (G` A))
12		fveq2 3835	. . . 4 |- (w = A -> (H` w) = (H` A))
13	11, 12	breq12d 2704	. . 3 |- (w = A -> ((G` w) < (H` w) <-> (G` A) < (H` A)))
14		1lt2 6174	. . . . 5 |- 1 < 2
15		1re 5589	. . . . . 6 |- 1 e. RR
16		2re 6125	. . . . . 6 |- 2 e. RR
17		ruclem.0	. . . . . . 7 |- F:NN-->RR
18		1nn 6079	. . . . . . 7 |- 1 e. NN
19		ffvelrn 3928	. . . . . . 7 |- ((F:NN-->RR /\ 1 e. NN) -> (F` 1) e. RR)
20	17, 18, 19	mp2an 701	. . . . . 6 |- (F` 1) e. RR
21	15, 16, 20	ltadd2i 5744	. . . . 5 |- (1 < 2 <-> ((F` 1) + 1) < ((F` 1) + 2))
22	14, 21	mpbi 187	. . . 4 |- ((F` 1) + 1) < ((F` 1) + 2)
23		ruclem.1	. . . . 5 |- C = ({<.1, <.((F` 1) + 1), ((F` 1) + 2)>.>.} u. (F |` (NN \ {1})))
24		ruclem.2	. . . . 5 |- 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)>.))}
25		ruclem.3	. . . . 5 |- G = (1st o. (D seq1 C))
26		ruclem.4	. . . . 5 |- H = (2nd o. (D seq1 C))
27	17, 23, 24, 25, 26	ruclem16 7737	. . . 4 |- (G` 1) = ((F` 1) + 1)
28	17, 23, 24, 25, 26	ruclem14 7735	. . . . . 6 |- ((D seq1 C)` 1) = <.((F` 1) + 1), ((F` 1) + 2)>.
29	28	fveq2i 3838	. . . . 5 |- (2nd` ((D seq1 C)` 1)) = (2nd` <.((F` 1) + 1), ((F` 1) + 2)>.)
30	24	ruclem9 7730	. . . . . 6 |- D e. V
31	17, 23	ruclem5 7726	. . . . . 6 |- C e. V
32	18, 30, 31, 26	ruclem11 7732	. . . . 5 |- (2nd` ((D seq1 C)` 1)) = (H` 1)
33		oprex 4041	. . . . . 6 |- ((F` 1) + 1) e. V
34		oprex 4041	. . . . . 6 |- ((F` 1) + 2) e. V
35	33, 34	op2nd 4147	. . . . 5 |- (2nd` <.((F` 1) + 1), ((F` 1) + 2)>.) = ((F` 1) + 2)
36	29, 32, 35	3eqtr3i 1546	. . . 4 |- (H` 1) = ((F` 1) + 2)
37	22, 27, 36	3brtr4i 2716	. . 3 |- (G` 1) < (H` 1)
38		fveq2 3835	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (G` v) = (G` if(v e. NN, v, 1)))
39		fveq2 3835	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (H` v) = (H` if(v e. NN, v, 1)))
40	38, 39	breq12d 2704	. . . . 5 |- (v = if(v e. NN, v, 1) -> ((G` v) < (H` v) <-> (G` if(v e. NN, v, 1)) < (H` if(v e. NN, v, 1))))
41		opreq1 4026	. . . . . . 7 |- (v = if(v e. NN, v, 1) -> (v + 1) = (if(v e. NN, v, 1) + 1))
42	41	fveq2d 3839	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (G` (v + 1)) = (G` (if(v e. NN, v, 1) + 1)))
43	41	fveq2d 3839	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (H` (v + 1)) = (H` (if(v e. NN, v, 1) + 1)))
44	42, 43	breq12d 2704	. . . . 5 |- (v = if(v e. NN, v, 1) -> ((G` (v + 1)) < (H` (v + 1)) <-> (G` (if(v e. NN, v, 1) + 1)) < (H` (if(v e. NN, v, 1) + 1))))
45	40, 44	imbi12d 629	. . . 4 |- (v = if(v e. NN, v, 1) -> (((G` v) < (H` v) -> (G` (v + 1)) < (H` (v + 1))) <-> ((G` if(v e. NN, v, 1)) < (H` if(v e. NN, v, 1)) -> (G` (if(v e. NN, v, 1) + 1)) < (H` (if(v e. NN, v, 1) + 1)))))
46	18	elimel 2451	. . . . 5 |- if(v e. NN, v, 1) e. NN
47	17, 23, 24, 25, 26, 46	ruclem24 7745	. . . 4 |- ((G` if(v e. NN, v, 1)) < (H` if(v e. NN, v, 1)) -> (G` (if(v e. NN, v, 1) + 1)) < (H` (if(v e. NN, v, 1) + 1)))
48	45, 47	dedth 2437	. . 3 |- (v e. NN -> ((G` v) < (H` v) -> (G` (v + 1)) < (H` (v + 1))))
49	4, 7, 10, 13, 37, 48	nnind 6082	. 2 |- (A e. NN -> (G` A) < (H` A))
50	1, 49	ax-mp 7	1 |- (G` A) < (H` A)

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:  ruclem26 7747  ruclem27 7748  ruclem32 7753

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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