Theorem ruclem25 7535

Description: Lemma for ruc 7550. 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 3730	. . . 4 |- (w = 1 -> (G` w) = (G` 1))
3		fveq2 3730	. . . 4 |- (w = 1 -> (H` w) = (H` 1))
4	2, 3	breq12d 2636	. . 3 |- (w = 1 -> ((G` w) < (H` w) <-> (G` 1) < (H` 1)))
5		fveq2 3730	. . . 4 |- (w = v -> (G` w) = (G` v))
6		fveq2 3730	. . . 4 |- (w = v -> (H` w) = (H` v))
7	5, 6	breq12d 2636	. . 3 |- (w = v -> ((G` w) < (H` w) <-> (G` v) < (H` v)))
8		fveq2 3730	. . . 4 |- (w = (v + 1) -> (G` w) = (G` (v + 1)))
9		fveq2 3730	. . . 4 |- (w = (v + 1) -> (H` w) = (H` (v + 1)))
10	8, 9	breq12d 2636	. . 3 |- (w = (v + 1) -> ((G` w) < (H` w) <-> (G` (v + 1)) < (H` (v + 1))))
11		fveq2 3730	. . . 4 |- (w = A -> (G` w) = (G` A))
12		fveq2 3730	. . . 4 |- (w = A -> (H` w) = (H` A))
13	11, 12	breq12d 2636	. . 3 |- (w = A -> ((G` w) < (H` w) <-> (G` A) < (H` A)))
14		1lt2 6030	. . . . 5 |- 1 < 2
15		1re 5447	. . . . . 6 |- 1 e. RR
16		2re 5981	. . . . . 6 |- 2 e. RR
17		ruclem.0	. . . . . . 7 |- F:NN-->RR
18		1nn 5936	. . . . . . 7 |- 1 e. NN
19		ffvelrn 3820	. . . . . . 7 |- ((F:NN-->RR /\ 1 e. NN) -> (F` 1) e. RR)
20	17, 18, 19	mp2an 699	. . . . . 6 |- (F` 1) e. RR
21	15, 16, 20	ltadd2 5602	. . . . 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 7526	. . . 4 |- (G` 1) = ((F` 1) + 1)
28	17, 23, 24, 25, 26	ruclem14 7524	. . . . . 6 |- ((D seq1 C)` 1) = <.((F` 1) + 1), ((F` 1) + 2)>.
29	28	fveq2i 3733	. . . . 5 |- (2nd` ((D seq1 C)` 1)) = (2nd` <.((F` 1) + 1), ((F` 1) + 2)>.)
30	24	ruclem9 7519	. . . . . 6 |- D e. V
31	17, 23	ruclem5 7515	. . . . . 6 |- C e. V
32	18, 30, 31, 26	ruclem11 7521	. . . . 5 |- (2nd` ((D seq1 C)` 1)) = (H` 1)
33		oprex 3989	. . . . . 6 |- ((F` 1) + 1) e. V
34		oprex 3989	. . . . . 6 |- ((F` 1) + 2) e. V
35	33, 34	op2nd 4092	. . . . 5 |- (2nd` <.((F` 1) + 1), ((F` 1) + 2)>.) = ((F` 1) + 2)
36	29, 32, 35	3eqtr3 1506	. . . 4 |- (H` 1) = ((F` 1) + 2)
37	22, 27, 36	3brtr4 2648	. . 3 |- (G` 1) < (H` 1)
38		fveq2 3730	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (G` v) = (G` if(v e. NN, v, 1)))
39		fveq2 3730	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (H` v) = (H` if(v e. NN, v, 1)))
40	38, 39	breq12d 2636	. . . . 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 3974	. . . . . . 7 |- (v = if(v e. NN, v, 1) -> (v + 1) = (if(v e. NN, v, 1) + 1))
42	41	fveq2d 3734	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (G` (v + 1)) = (G` (if(v e. NN, v, 1) + 1)))
43	41	fveq2d 3734	. . . . . 6 |- (v = if(v e. NN, v, 1) -> (H` (v + 1)) = (H` (if(v e. NN, v, 1) + 1)))
44	42, 43	breq12d 2636	. . . . 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 628	. . . 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 2398	. . . . 5 |- if(v e. NN, v, 1) e. NN
47	17, 23, 24, 25, 26, 46	ruclem24 7534	. . . 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 2387	. . 3 |- (v e. NN -> ((G` v) < (H` v) -> (G` (v + 1)) < (H` (v + 1))))
49	4, 7, 10, 13, 37, 48	nnind 5939	. 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 958   e. wcel 960   \ cdif 2047   u. cun 2048  ifcif 2365  {csn 2413  <.cop 2415   class class class wbr 2624   X. cxp 3174   |` cres 3178   o. ccom 3180  -->wf 3184  ` cfv 3188  (class class class)co 3969  {copab2 3970  1stc1st 4083  2ndc2nd 4084  RRcr 5245  1c1 5247   + caddc 5249   x. cmul 5251   / cdiv 5306  NNcn 5308   < clt 5498  2c2 5963  3c3 5964   seq1 cseq1 6308

This theorem is referenced by:  ruclem26 7536  ruclem27 7537  ruclem32 7542

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-3 5973  df-n0 6102  df-z 6138  df-seq1 6309

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