Theorem ruclem25 7477

Description: Lemma for ruc 7492. 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 3709	. . . 4 |- (w = 1 -> (G` w) = (G` 1))
3		fveq2 3709	. . . 4 |- (w = 1 -> (H` w) = (H` 1))
4	2, 3	breq12d 2621	. . 3 |- (w = 1 -> ((G` w) < (H` w) <-> (G` 1) < (H` 1)))
5		fveq2 3709	. . . 4 |- (w = v -> (G` w) = (G` v))
6		fveq2 3709	. . . 4 |- (w = v -> (H` w) = (H` v))
7	5, 6	breq12d 2621	. . 3 |- (w = v -> ((G` w) < (H` w) <-> (G` v) < (H` v)))
8		fveq2 3709	. . . 4 |- (w = (v + 1) -> (G` w) = (G` (v + 1)))
9		fveq2 3709	. . . 4 |- (w = (v + 1) -> (H` w) = (H` (v + 1)))
10	8, 9	breq12d 2621	. . 3 |- (w = (v + 1) -> ((G` w) < (H` w) <-> (G` (v + 1)) < (H` (v + 1))))
11		fveq2 3709	. . . 4 |- (w = A -> (G` w) = (G` A))
12		fveq2 3709	. . . 4 |- (w = A -> (H` w) = (H` A))
13	11, 12	breq12d 2621	. . 3 |- (w = A -> ((G` w) < (H` w) <-> (G` A) < (H` A)))
14		1lt2 5975	. . . . 5 |- 1 < 2
15		1re 5407	. . . . . 6 |- 1 e. RR
16		2re 5926	. . . . . 6 |- 2 e. RR
17		ruclem.0	. . . . . . 7 |- F:NN-->RR
18		1nn 5882	. . . . . . 7 |- 1 e. NN
19		ffvelrn 3799	. . . . . . 7 |- ((F:NN-->RR /\ 1 e. NN) -> (F` 1) e. RR)
20	17, 18, 19	mp2an 695	. . . . . 6 |- (F` 1) e. RR
21	15, 16, 20	ltadd2 5564	. . . . 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 7468	. . . 4 |- (G` 1) = ((F` 1) + 1)
28	17, 23, 24, 25, 26	ruclem14 7466	. . . . . 6 |- ((D seq1 C)` 1) = <.((F` 1) + 1), ((F` 1) + 2)>.
29	28	fveq2i 3712	. . . . 5 |- (2nd` ((D seq1 C)` 1)) = (2nd` <.((F` 1) + 1), ((F` 1) + 2)>.)
30	24	ruclem9 7461	. . . . . 6 |- D e. V
31	17, 23	ruclem5 7457	. . . . . 6 |- C e. V
32	18, 30, 31, 26	ruclem11 7463	. . . . 5 |- (2nd` ((D seq1 C)` 1)) = (H` 1)
33		oprex 3968	. . . . . 6 |- ((F` 1) + 1) e. V
34		oprex 3968	. . . . . 6 |- ((F` 1) + 2) e. V
35	33, 34	op2nd 4070	. . . . 5 |- (2nd` <.((F` 1) + 1), ((F` 1) + 2)>.) = ((F` 1) + 2)
36	29, 32, 35	3eqtr3 1495	. . . 4 |- (H` 1) = ((F` 1) + 2)
37	22, 27, 36	3brtr4 2633	. . 3 |- (G` 1) < (H` 1)
38		fveq2 3709	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (G` v) = (G` if(v e. NN, v, 1)))
39		fveq2 3709	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (H` v) = (H` if(v e. NN, v, 1)))
40	38, 39	breq12d 2621	. . . . 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 3953	. . . . . . 7 |- (v = if(v e. NN, v, 1) -> (v + 1) = (if(v e. NN, v, 1) + 1))
42	41	fveq2d 3713	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (G` (v + 1)) = (G` (if(v e. NN, v, 1) + 1)))
43	41	fveq2d 3713	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (H` (v + 1)) = (H` (if(v e. NN, v, 1) + 1)))
44	42, 43	breq12d 2621	. . . . 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 624	. . . 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 2384	. . . . 5 |- if(v e. NN, v, 1) e. NN
47	17, 23, 24, 25, 26, 46	ruclem24 7476	. . . 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 2373	. . 3 |- (v e. NN -> ((G` v) < (H` v) -> (G` (v + 1)) < (H` (v + 1))))
49	4, 7, 10, 13, 37, 48	nnind 5885	. 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 953   e. wcel 955   \ cdif 2034   u. cun 2035  ifcif 2351  {csn 2399  <.cop 2401   class class class wbr 2609   X. cxp 3158   |` cres 3162   o. ccom 3164  -->wf 3168  ` cfv 3172  (class class class)co 3948  {copab2 3949  1stc1st 4061  2ndc2nd 4062  RRcr 5205  1c1 5207   + caddc 5209   x. cmul 5211   / cdiv 5266  NNcn 5268   < clt 5458  2c2 5908  3c3 5909   seq1 cseq1 6244

This theorem is referenced by:  ruclem26 7478  ruclem27 7479  ruclem32 7484

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-3 5918  df-n0 6047  df-z 6083  df-seq1 6245

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