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Theorem algrp1 11738
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 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 )
53, 4eleqtrdi 2232 . . . 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 11735 . . . 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 11734 . . . 4  |-  ( ( ( ph  /\  K  e.  Z )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x
( F  o.  1st ) y )  e.  S )
125, 8, 11seq3p1 10247 . . 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, 12syl5eq 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 ) ) ) )
141fveq1i 5422 . . . 4  |-  ( R `
 K )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K )
1514oveq1i 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 )
174, 1, 16, 6, 9algrf 11737 . . . . 5  |-  ( ph  ->  R : Z --> S )
1817ffvelrnda 5555 . . . 4  |-  ( (
ph  /\  K  e.  Z )  ->  ( R `  K )  e.  S )
194peano2uzs 9391 . . . . . 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 )
216, 19, 20syl2an 287 . . . . 5  |-  ( (
ph  /\  K  e.  Z )  ->  (
( Z  X.  { A } ) `  ( K  +  1 ) )  =  A )
2221, 7eqeltrd 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 ) ) )
2418, 22, 23syl2anc 408 . . 3  |-  ( (
ph  /\  K  e.  Z )  ->  (
( R `  K
) ( F  o.  1st ) ( ( Z  X.  { A }
) `  ( K  +  1 ) ) )  =  ( F `
 ( R `  K ) ) )
2515, 24syl5reqr 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 ) ) ) )
2613, 25eqtr4d 2175 1  |-  ( (
ph  /\  K  e.  Z )  ->  ( R `  ( K  +  1 ) )  =  ( F `  ( R `  K ) ) )
Colors of variables: wff set class
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 7633    + caddc 7635   ZZcz 9066   ZZ>=cuz 9338    seqcseq 10230
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 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-ltadd 7748
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 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-seqfrec 10231
This theorem is referenced by:  alginv  11739  algcvg  11740  algcvga  11743  algfx  11744
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