Theorem algrp1 11763

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 5430	. . 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 2233	. . . 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 11760	. . . 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 11759	. . . 4 |- ( ( ( ph /\ K e. Z ) /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
12	5, 8, 11	seq3p1 10266	. . 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	syl5eq 2185	. 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	1	fveq1i 5430	. . . 4 |- ( R ` K ) = ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` K )
15	14	oveq1i 5792	. . 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 ) ) )
16		algrf.3	. . . . . 6 |- ( ph -> M e. ZZ )
17	4, 1, 16, 6, 9	algrf 11762	. . . . 5 |- ( ph -> R : Z --> S )
18	17	ffvelrnda 5563	. . . 4 |- ( ( ph /\ K e. Z ) -> ( R ` K ) e. S )
19	4	peano2uzs 9406	. . . . . 6 |- ( K e. Z -> ( K + 1 ) e. Z )
20		fvconst2g 5642	. . . . . 6 |- ( ( A e. S /\ ( K + 1 ) e. Z ) -> ( ( Z X. { A } ) ` ( K + 1 ) ) = A )
21	6, 19, 20	syl2an 287	. . . . 5 |- ( ( ph /\ K e. Z ) -> ( ( Z X. { A } ) ` ( K + 1 ) ) = A )
22	21, 7	eqeltrd 2217	. . . 4 |- ( ( ph /\ K e. Z ) -> ( ( Z X. { A } ) ` ( K + 1 ) ) e. S )
23		algrflemg 6135	. . . 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 ) ) )
24	18, 22, 23	syl2anc 409	. . 3 |- ( ( ph /\ K e. Z ) -> ( ( R ` K ) ( F o. 1st ) ( ( Z X. { A } ) ` ( K + 1 ) ) ) = ( F ` ( R ` K ) ) )
25	15, 24	syl5reqr 2188	. 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 2176	1 |- ( ( ph /\ K e. Z ) -> ( R ` ( K + 1 ) ) = ( F ` ( R ` K ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   {csn 3532   X. cxp 4545   o. ccom 4551   -->wf 5127   ` cfv 5131  (class class class)co 5782   1stc1st 6044   1c1 7645   + caddc 7647   ZZcz 9078   ZZ>=cuz 9350   seqcseq 10249

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-seqfrec 10250

This theorem is referenced by:  alginv  11764  algcvg  11765  algcvga  11768  algfx  11769