Theorem algrp1 12000

Description: The value of the algorithm iterator R at ( K + 1 ). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Jim Kingdon, 12-Mar-2023.)

Hypotheses

Ref	Expression
algrf.1	|- Z = ( ZZ>= ` M )
algrf.2	|- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )
algrf.3	|- ( ph -> M e. ZZ )
algrf.4	|- ( ph -> A e. S )
algrf.5	|- ( ph -> F : S --> S )

Assertion

Ref	Expression
algrp1	|- ( ( ph /\ K e. Z ) -> ( R ` ( K + 1 ) ) = ( F ` ( R ` K ) ) )

Proof of Theorem algrp1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		algrf.2	. . . 4 |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )
2	1	fveq1i 5497	. . 3 |- ( R ` ( K + 1 ) ) = ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( K + 1 ) )
3		simpr 109	. . . . 5 |- ( ( ph /\ K e. Z ) -> K e. Z )
4		algrf.1	. . . . 5 |- Z = ( ZZ>= ` M )
5	3, 4	eleqtrdi 2263	. . . 4 |- ( ( ph /\ K e. Z ) -> K e. ( ZZ>= ` M ) )
6		algrf.4	. . . . . 6 |- ( ph -> A e. S )
7	6	adantr 274	. . . . 5 |- ( ( ph /\ K e. Z ) -> A e. S )
8	4, 7	ialgrlemconst 11997	. . . 4 |- ( ( ( ph /\ K e. Z ) /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) e. S )
9		algrf.5	. . . . . 6 |- ( ph -> F : S --> S )
10	9	adantr 274	. . . . 5 |- ( ( ph /\ K e. Z ) -> F : S --> S )
11	10	ialgrlem1st 11996	. . . 4 |- ( ( ( ph /\ K e. Z ) /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
12	5, 8, 11	seq3p1 10418	. . 3 |- ( ( ph /\ K e. Z ) -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( K + 1 ) ) = ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` K ) ( F o. 1st ) ( ( Z X. { A } ) ` ( K + 1 ) ) ) )
13	2, 12	eqtrid 2215	. 2 |- ( ( ph /\ K e. Z ) -> ( R ` ( K + 1 ) ) = ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` K ) ( F o. 1st ) ( ( Z X. { A } ) ` ( K + 1 ) ) ) )
14		algrf.3	. . . . . 6 |- ( ph -> M e. ZZ )
15	4, 1, 14, 6, 9	algrf 11999	. . . . 5 |- ( ph -> R : Z --> S )
16	15	ffvelrnda 5631	. . . 4 |- ( ( ph /\ K e. Z ) -> ( R ` K ) e. S )
17	4	peano2uzs 9543	. . . . . 6 |- ( K e. Z -> ( K + 1 ) e. Z )
18		fvconst2g 5710	. . . . . 6 |- ( ( A e. S /\ ( K + 1 ) e. Z ) -> ( ( Z X. { A } ) ` ( K + 1 ) ) = A )
19	6, 17, 18	syl2an 287	. . . . 5 |- ( ( ph /\ K e. Z ) -> ( ( Z X. { A } ) ` ( K + 1 ) ) = A )
20	19, 7	eqeltrd 2247	. . . 4 |- ( ( ph /\ K e. Z ) -> ( ( Z X. { A } ) ` ( K + 1 ) ) e. S )
21		algrflemg 6209	. . . 4 |- ( ( ( R ` K ) e. S /\ ( ( Z X. { A } ) ` ( K + 1 ) ) e. S ) -> ( ( R ` K ) ( F o. 1st ) ( ( Z X. { A } ) ` ( K + 1 ) ) ) = ( F ` ( R ` K ) ) )
22	16, 20, 21	syl2anc 409	. . 3 |- ( ( ph /\ K e. Z ) -> ( ( R ` K ) ( F o. 1st ) ( ( Z X. { A } ) ` ( K + 1 ) ) ) = ( F ` ( R ` K ) ) )
23	1	fveq1i 5497	. . . 4 |- ( R ` K ) = ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` K )
24	23	oveq1i 5863	. . 3 |- ( ( R ` K ) ( F o. 1st ) ( ( Z X. { A } ) ` ( K + 1 ) ) ) = ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` K ) ( F o. 1st ) ( ( Z X. { A } ) ` ( K + 1 ) ) )
25	22, 24	eqtr3di 2218	. 2 |- ( ( ph /\ K e. Z ) -> ( F ` ( R ` K ) ) = ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` K ) ( F o. 1st ) ( ( Z X. { A } ) ` ( K + 1 ) ) ) )
26	13, 25	eqtr4d 2206	1 |- ( ( ph /\ K e. Z ) -> ( R ` ( K + 1 ) ) = ( F ` ( R ` K ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   {csn 3583   X. cxp 4609   o. ccom 4615   -->wf 5194   ` cfv 5198  (class class class)co 5853   1stc1st 6117   1c1 7775   + caddc 7777   ZZcz 9212   ZZ>=cuz 9487   seqcseq 10401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402

This theorem is referenced by:  alginv  12001  algcvg  12002  algcvga  12005  algfx  12006