Theorem ruclem25 7534

Description: Lemma for ruc 7549. 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 3724	. . . 4 |- (w = 1 -> (G` w) = (G` 1))
3		fveq2 3724	. . . 4 |- (w = 1 -> (H` w) = (H` 1))
4	2, 3	breq12d 2631	. . 3 |- (w = 1 -> ((G` w) < (H` w) <-> (G` 1) < (H` 1)))
5		fveq2 3724	. . . 4 |- (w = v -> (G` w) = (G` v))
6		fveq2 3724	. . . 4 |- (w = v -> (H` w) = (H` v))
7	5, 6	breq12d 2631	. . 3 |- (w = v -> ((G` w) < (H` w) <-> (G` v) < (H` v)))
8		fveq2 3724	. . . 4 |- (w = (v + 1) -> (G` w) = (G` (v + 1)))
9		fveq2 3724	. . . 4 |- (w = (v + 1) -> (H` w) = (H` (v + 1)))
10	8, 9	breq12d 2631	. . 3 |- (w = (v + 1) -> ((G` w) < (H` w) <-> (G` (v + 1)) < (H` (v + 1))))
11		fveq2 3724	. . . 4 |- (w = A -> (G` w) = (G` A))
12		fveq2 3724	. . . 4 |- (w = A -> (H` w) = (H` A))
13	11, 12	breq12d 2631	. . 3 |- (w = A -> ((G` w) < (H` w) <-> (G` A) < (H` A)))
14		1lt2 6028	. . . . 5 |- 1 < 2
15		1re 5435	. . . . . 6 |- 1 e. RR
16		2re 5979	. . . . . 6 |- 2 e. RR
17		ruclem.0	. . . . . . 7 |- F:NN-->RR
18		1nn 5934	. . . . . . 7 |- 1 e. NN
19		ffvelrn 3814	. . . . . . 7 |- ((F:NN-->RR /\ 1 e. NN) -> (F` 1) e. RR)
20	17, 18, 19	mp2an 697	. . . . . 6 |- (F` 1) e. RR
21	15, 16, 20	ltadd2 5590	. . . . 5 |- (1 < 2 <-> ((F` 1) + 1) < ((F` 1) + 2))
22	14, 21	mpbi 189	. . . 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 7525	. . . 4 |- (G` 1) = ((F` 1) + 1)
28	17, 23, 24, 25, 26	ruclem14 7523	. . . . . 6 |- ((D seq1 C)` 1) = <.((F` 1) + 1), ((F` 1) + 2)>.
29	28	fveq2i 3727	. . . . 5 |- (2nd` ((D seq1 C)` 1)) = (2nd` <.((F` 1) + 1), ((F` 1) + 2)>.)
30	24	ruclem9 7518	. . . . . 6 |- D e. V
31	17, 23	ruclem5 7514	. . . . . 6 |- C e. V
32	18, 30, 31, 26	ruclem11 7520	. . . . 5 |- (2nd` ((D seq1 C)` 1)) = (H` 1)
33		oprex 3983	. . . . . 6 |- ((F` 1) + 1) e. V
34		oprex 3983	. . . . . 6 |- ((F` 1) + 2) e. V
35	33, 34	op2nd 4086	. . . . 5 |- (2nd` <.((F` 1) + 1), ((F` 1) + 2)>.) = ((F` 1) + 2)
36	29, 32, 35	3eqtr3 1503	. . . 4 |- (H` 1) = ((F` 1) + 2)
37	22, 27, 36	3brtr4 2643	. . 3 |- (G` 1) < (H` 1)
38		fveq2 3724	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (G` v) = (G` if(v e. NN, v, 1)))
39		fveq2 3724	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (H` v) = (H` if(v e. NN, v, 1)))
40	38, 39	breq12d 2631	. . . . 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 3968	. . . . . . 7 |- (v = if(v e. NN, v, 1) -> (v + 1) = (if(v e. NN, v, 1) + 1))
42	41	fveq2d 3728	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (G` (v + 1)) = (G` (if(v e. NN, v, 1) + 1)))
43	41	fveq2d 3728	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (H` (v + 1)) = (H` (if(v e. NN, v, 1) + 1)))
44	42, 43	breq12d 2631	. . . . 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 626	. . . 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 2394	. . . . 5 |- if(v e. NN, v, 1) e. NN
47	17, 23, 24, 25, 26, 46	ruclem24 7533	. . . 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 2383	. . 3 |- (v e. NN -> ((G` v) < (H` v) -> (G` (v + 1)) < (H` (v + 1))))
49	4, 7, 10, 13, 37, 48	nnind 5937	. 2 |- (A e. NN -> (G` A) < (H` A))
50	1, 49	ax-mp 7	1 |- (G` A) < (H` A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958   \ cdif 2044   u. cun 2045  ifcif 2361  {csn 2409  <.cop 2411   class class class wbr 2619   X. cxp 3168   |` cres 3172   o. ccom 3174  -->wf 3178  ` cfv 3182  (class class class)co 3963  {copab2 3964  1stc1st 4077  2ndc2nd 4078  RRcr 5233  1c1 5235   + caddc 5237   x. cmul 5239   / cdiv 5294  NNcn 5296   < clt 5486  2c2 5961  3c3 5962   seq1 cseq1 6307

This theorem is referenced by:  ruclem26 7535  ruclem27 7536  ruclem32 7541

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-2 5970  df-3 5971  df-n0 6100  df-z 6136  df-seq1 6308

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