Theorem algrp1 12849

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 5699	. . 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 2331	. . . 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 12846	. . . 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 12845	. . . 4 |- ( ( ( ph /\ K e. Z ) /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
12	5, 8, 11	seq3p1 10923	. . 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 2283	. 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 12848	. . . . 5 |- ( ph -> R : Z --> S )
16	15	ffvelcdmda 5846	. . . 4 |- ( ( ph /\ K e. Z ) -> ( R ` K ) e. S )
17	4	peano2uzs 10000	. . . . . 6 |- ( K e. Z -> ( K + 1 ) e. Z )
18		fvconst2g 5932	. . . . . 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 2315	. . . 4 |- ( ( ph /\ K e. Z ) -> ( ( Z X. { A } ) ` ( K + 1 ) ) e. S )
21		algrflemg 6470	. . . 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 415	. . 3 |- ( ( ph /\ K e. Z ) -> ( ( R ` K ) ( F o. 1st ) ( ( Z X. { A } ) ` ( K + 1 ) ) ) = ( F ` ( R ` K ) ) )
23	1	fveq1i 5699	. . . 4 |- ( R ` K ) = ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` K )
24	23	oveq1i 6098	. . 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 2286	. 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 2274	1 |- ( ( ph /\ K e. Z ) -> ( R ` ( K + 1 ) ) = ( F ` ( R ` K ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {csn 3709   X. cxp 4774   o. ccom 4780   -->wf 5376   ` cfv 5380  (class class class)co 6088   1stc1st 6376   1c1 8186   + caddc 8188   ZZcz 9655   ZZ>=cuz 9937   seqcseq 10905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-seqfrec 10906

This theorem is used by:  alginv  12850  algcvg  12851  algcvga  12854  algfx  12855