Theorem ruclem21 7545

Description: Lemma for ruc 7564. The value of our constructed function H when the value of the input function F does not lie between the previous values of G and H. This assignment to H just shrinks the interval between G and H by some arbitrary amount.

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
ruclem21	|- (-. ((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (H` (A + 1)) = (((G` A) + (2 x. (H` A))) / 3))

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

Proof of Theorem ruclem21

Step	Hyp	Ref	Expression
1		iffalse 2377	. . . 4 |- (-. ((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> if(((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)), <.(((2 x. (F` (A + 1))) + (H` A)) / 3), (((F` (A + 1)) + (2 x. (H` A))) / 3)>., <.(((2 x. (G` A)) + (H` A)) / 3), (((G` A) + (2 x. (H` A))) / 3)>.) = <.(((2 x. (G` A)) + (H` A)) / 3), (((G` A) + (2 x. (H` A))) / 3)>.)
2		ruclem.0	. . . . 5 |- F:NN-->RR
3		ruclem.1	. . . . 5 |- C = ({<.1, <.((F` 1) + 1), ((F` 1) + 2)>.>.} u. (F |` (NN \ {1})))
4		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)>.))}
5		ruclem.3	. . . . 5 |- G = (1st o. (D seq1 C))
6		ruclem.4	. . . . 5 |- H = (2nd o. (D seq1 C))
7		ruclem18.a	. . . . 5 |- A e. NN
8	2, 3, 4, 5, 6, 7	ruclem15 7539	. . . 4 |- ((D seq1 C)` (A + 1)) = if(((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)), <.(((2 x. (F` (A + 1))) + (H` A)) / 3), (((F` (A + 1)) + (2 x. (H` A))) / 3)>., <.(((2 x. (G` A)) + (H` A)) / 3), (((G` A) + (2 x. (H` A))) / 3)>.)
9	1, 8	syl5eq 1526	. . 3 |- (-. ((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> ((D seq1 C)` (A + 1)) = <.(((2 x. (G` A)) + (H` A)) / 3), (((G` A) + (2 x. (H` A))) / 3)>.)
10	9	fveq2d 3742	. 2 |- (-. ((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (2nd` ((D seq1 C)` (A + 1))) = (2nd` <.(((2 x. (G` A)) + (H` A)) / 3), (((G` A) + (2 x. (H` A))) / 3)>.))
11		peano2nn 5941	. . . 4 |- (A e. NN -> (A + 1) e. NN)
12	7, 11	ax-mp 7	. . 3 |- (A + 1) e. NN
13	4	ruclem9 7533	. . 3 |- D e. V
14	2, 3	ruclem5 7529	. . 3 |- C e. V
15	12, 13, 14, 6	ruclem11 7535	. 2 |- (2nd` ((D seq1 C)` (A + 1))) = (H` (A + 1))
16		oprex 3997	. . 3 |- (((2 x. (G` A)) + (H` A)) / 3) e. V
17		oprex 3997	. . 3 |- (((G` A) + (2 x. (H` A))) / 3) e. V
18	16, 17	op2nd 4100	. 2 |- (2nd` <.(((2 x. (G` A)) + (H` A)) / 3), (((G` A) + (2 x. (H` A))) / 3)>.) = (((G` A) + (2 x. (H` A))) / 3)
19	10, 15, 18	3eqtr3g 1537	1 |- (-. ((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (H` (A + 1)) = (((G` A) + (2 x. (H` A))) / 3))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 960   e. wcel 962   \ cdif 2053   u. cun 2054  ifcif 2371  {csn 2419  <.cop 2421   class class class wbr 2632   X. cxp 3182   |` cres 3186   o. ccom 3188  -->wf 3192  ` cfv 3196  (class class class)co 3977  {copab2 3978  1stc1st 4091  2ndc2nd 4092  RRcr 5246  1c1 5248   + caddc 5250   x. cmul 5252   / cdiv 5307  NNcn 5309   < clt 5499  2c2 5967  3c3 5968   seq1 cseq1 6490

This theorem is referenced by:  ruclem24 7548  ruclem27 7551

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-rep 2706  ax-sep 2716  ax-nul 2723  ax-pow 2756  ax-pr 2793  ax-un 2880  ax-inf2 4637

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 780  df-3an 781  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-nel 1595  df-ral 1656  df-rex 1657  df-reu 1658  df-rab 1659  df-v 1819  df-sbc 1949  df-csb 2010  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-pss 2064  df-nul 2290  df-if 2372  df-pw 2412  df-sn 2422  df-pr 2423  df-tp 2425  df-op 2426  df-uni 2516  df-int 2546  df-iun 2580  df-br 2633  df-opab 2680  df-tr 2694  df-eprel 2846  df-id 2849  df-po 2854  df-so 2864  df-fr 2931  df-we 2948  df-ord 2965  df-on 2966  df-lim 2967  df-suc 2968  df-om 3146  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fn 3207  df-f 3208  df-f1 3209  df-fo 3210  df-f1o 3211  df-fv 3212  df-rdg 3946  df-opr 3979  df-oprab 3980  df-1st 4093  df-2nd 4094  df-1o 4147  df-oadd 4149  df-omul 4150  df-er 4275  df-ec 4277  df-qs 4280  df-en 4382  df-dom 4383  df-sdom 4384  df-ni 5013  df-pli 5014  df-mi 5015  df-lti 5016  df-plpq 5048  df-mpq 5049  df-enq 5050  df-nq 5051  df-plq 5052  df-mq 5053  df-rq 5054  df-ltq 5055  df-1q 5056  df-np 5099  df-1p 5100  df-plp 5101  df-mp 5102  df-ltp 5103  df-plpr 5177  df-mpr 5178  df-enr 5179  df-nr 5180  df-plr 5181  df-mr 5182  df-ltr 5183  df-0r 5184  df-1r 5185  df-m1r 5186  df-c 5253  df-0 5254  df-1 5255  df-i 5256  df-r 5257  df-plus 5258  df-mul 5259  df-lt 5260  df-sub 5369  df-neg 5371  df-pnf 5500  df-mnf 5501  df-xr 5502  df-ltxr 5503  df-le 5504  df-div 5716  df-n 5931  df-2 5976  df-3 5977  df-n0 6106  df-z 6142  df-seq1 6491

Copyright terms: Public domain