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Theorem algrp1 11987
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.
StepHypRef Expression
1 algrf.2 . . . 4  |-  R  =  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )
21fveq1i 5495 . . 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 )
53, 4eleqtrdi 2263 . . . 4  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  ( ZZ>= `  M )
)
6 algrf.4 . . . . . 6  |-  ( ph  ->  A  e.  S )
76adantr 274 . . . . 5  |-  ( (
ph  /\  K  e.  Z )  ->  A  e.  S )
84, 7ialgrlemconst 11984 . . . 4  |-  ( ( ( ph  /\  K  e.  Z )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( ( Z  X.  { A }
) `  x )  e.  S )
9 algrf.5 . . . . . 6  |-  ( ph  ->  F : S --> S )
109adantr 274 . . . . 5  |-  ( (
ph  /\  K  e.  Z )  ->  F : S --> S )
1110ialgrlem1st 11983 . . . 4  |-  ( ( ( ph  /\  K  e.  Z )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x
( F  o.  1st ) y )  e.  S )
125, 8, 11seq3p1 10405 . . 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 ) ) ) )
132, 12eqtrid 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 )
154, 1, 14, 6, 9algrf 11986 . . . . 5  |-  ( ph  ->  R : Z --> S )
1615ffvelrnda 5628 . . . 4  |-  ( (
ph  /\  K  e.  Z )  ->  ( R `  K )  e.  S )
174peano2uzs 9530 . . . . . 6  |-  ( K  e.  Z  ->  ( K  +  1 )  e.  Z )
18 fvconst2g 5707 . . . . . 6  |-  ( ( A  e.  S  /\  ( K  +  1
)  e.  Z )  ->  ( ( Z  X.  { A }
) `  ( K  +  1 ) )  =  A )
196, 17, 18syl2an 287 . . . . 5  |-  ( (
ph  /\  K  e.  Z )  ->  (
( Z  X.  { A } ) `  ( K  +  1 ) )  =  A )
2019, 7eqeltrd 2247 . . . 4  |-  ( (
ph  /\  K  e.  Z )  ->  (
( Z  X.  { A } ) `  ( K  +  1 ) )  e.  S )
21 algrflemg 6206 . . . 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 ) ) )
2216, 20, 21syl2anc 409 . . 3  |-  ( (
ph  /\  K  e.  Z )  ->  (
( R `  K
) ( F  o.  1st ) ( ( Z  X.  { A }
) `  ( K  +  1 ) ) )  =  ( F `
 ( R `  K ) ) )
231fveq1i 5495 . . . 4  |-  ( R `
 K )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K )
2423oveq1i 5860 . . 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
) ) )
2522, 24eqtr3di 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 ) ) ) )
2613, 25eqtr4d 2206 1  |-  ( (
ph  /\  K  e.  Z )  ->  ( R `  ( K  +  1 ) )  =  ( F `  ( R `  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   {csn 3581    X. cxp 4607    o. ccom 4613   -->wf 5192   ` cfv 5196  (class class class)co 5850   1stc1st 6114   1c1 7762    + caddc 7764   ZZcz 9199   ZZ>=cuz 9474    seqcseq 10388
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 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-ltadd 7877
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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-seqfrec 10389
This theorem is referenced by:  alginv  11988  algcvg  11989  algcvga  11992  algfx  11993
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