Theorem algrp1 12401

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 5579	. . 3 |- ( R ` ( K + 1 ) ) = ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( K + 1 ) )
3		simpr 110	. . . . 5 |- ( ( ph /\ K e. Z ) -> K e. Z )
4		algrf.1	. . . . 5 |- Z = ( ZZ>= ` M )
5	3, 4	eleqtrdi 2298	. . . 4 |- ( ( ph /\ K e. Z ) -> K e. ( ZZ>= ` M ) )
6		algrf.4	. . . . . 6 |- ( ph -> A e. S )
7	6	adantr 276	. . . . 5 |- ( ( ph /\ K e. Z ) -> A e. S )
8	4, 7	ialgrlemconst 12398	. . . 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 276	. . . . 5 |- ( ( ph /\ K e. Z ) -> F : S --> S )
11	10	ialgrlem1st 12397	. . . 4 |- ( ( ( ph /\ K e. Z ) /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
12	5, 8, 11	seq3p1 10612	. . 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 2250	. 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 12400	. . . . 5 |- ( ph -> R : Z --> S )
16	15	ffvelcdmda 5717	. . . 4 |- ( ( ph /\ K e. Z ) -> ( R ` K ) e. S )
17	4	peano2uzs 9707	. . . . . 6 |- ( K e. Z -> ( K + 1 ) e. Z )
18		fvconst2g 5800	. . . . . 6 |- ( ( A e. S /\ ( K + 1 ) e. Z ) -> ( ( Z X. { A } ) ` ( K + 1 ) ) = A )
19	6, 17, 18	syl2an 289	. . . . 5 |- ( ( ph /\ K e. Z ) -> ( ( Z X. { A } ) ` ( K + 1 ) ) = A )
20	19, 7	eqeltrd 2282	. . . 4 |- ( ( ph /\ K e. Z ) -> ( ( Z X. { A } ) ` ( K + 1 ) ) e. S )
21		algrflemg 6318	. . . 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 411	. . 3 |- ( ( ph /\ K e. Z ) -> ( ( R ` K ) ( F o. 1st ) ( ( Z X. { A } ) ` ( K + 1 ) ) ) = ( F ` ( R ` K ) ) )
23	1	fveq1i 5579	. . . 4 |- ( R ` K ) = ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` K )
24	23	oveq1i 5956	. . 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 2253	. 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 2241	1 |- ( ( ph /\ K e. Z ) -> ( R ` ( K + 1 ) ) = ( F ` ( R ` K ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   {csn 3633   X. cxp 4674   o. ccom 4680   -->wf 5268   ` cfv 5272  (class class class)co 5946   1stc1st 6226   1c1 7928   + caddc 7930   ZZcz 9374   ZZ>=cuz 9650   seqcseq 10594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-inn 9039  df-n0 9298  df-z 9375  df-uz 9651  df-seqfrec 10595

This theorem is referenced by:  alginv  12402  algcvg  12403  algcvga  12406  algfx  12407