Theorem algrp1 11727

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 5422	. . 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 2232	. . . 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 11724	. . . 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 11723	. . . 4 |- ( ( ( ph /\ K e. Z ) /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
12	5, 8, 11	seq3p1 10235	. . 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 2184	. 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 5422	. . . 4 |- ( R ` K ) = ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` K )
15	14	oveq1i 5784	. . 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 11726	. . . . 5 |- ( ph -> R : Z --> S )
18	17	ffvelrnda 5555	. . . 4 |- ( ( ph /\ K e. Z ) -> ( R ` K ) e. S )
19	4	peano2uzs 9379	. . . . . 6 |- ( K e. Z -> ( K + 1 ) e. Z )
20		fvconst2g 5634	. . . . . 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 2216	. . . 4 |- ( ( ph /\ K e. Z ) -> ( ( Z X. { A } ) ` ( K + 1 ) ) e. S )
23		algrflemg 6127	. . . 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 408	. . 3 |- ( ( ph /\ K e. Z ) -> ( ( R ` K ) ( F o. 1st ) ( ( Z X. { A } ) ` ( K + 1 ) ) ) = ( F ` ( R ` K ) ) )
25	15, 24	syl5reqr 2187	. 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 2175	1 |- ( ( ph /\ K e. Z ) -> ( R ` ( K + 1 ) ) = ( F ` ( R ` K ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   {csn 3527   X. cxp 4537   o. ccom 4543   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036   1c1 7621   + caddc 7623   ZZcz 9054   ZZ>=cuz 9326   seqcseq 10218

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-seqfrec 10219

This theorem is referenced by:  alginv  11728  algcvg  11729  algcvga  11732  algfx  11733