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Theorem algrp1 13065
Description: The value of the algorithm iterator  R at  ( K  +  1 ). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
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
StepHypRef Expression
1 simpr 448 . . . 4  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  Z )
2 algrf.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
31, 2syl6eleq 2526 . . 3  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  ( ZZ>= `  M )
)
4 seqp1 11338 . . 3  |-  ( K  e.  ( ZZ>= `  M
)  ->  (  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
) ) ) )
53, 4syl 16 . 2  |-  ( (
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 ) ) ) )
6 algrf.2 . . 3  |-  R  =  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )
76fveq1i 5729 . 2  |-  ( R `
 ( K  + 
1 ) )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  ( K  +  1 ) )
86fveq1i 5729 . . . 4  |-  ( R `
 K )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K )
98fveq2i 5731 . . 3  |-  ( F `
 ( R `  K ) )  =  ( F `  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K ) )
10 fvex 5742 . . . 4  |-  (  seq 
M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K )  e.  _V
11 fvex 5742 . . . 4  |-  ( ( Z  X.  { A } ) `  ( K  +  1 ) )  e.  _V
1210, 11algrflem 6455 . . 3  |-  ( (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 K ) ( F  o.  1st )
( ( Z  X.  { A } ) `  ( K  +  1
) ) )  =  ( F `  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K ) )
139, 12eqtr4i 2459 . 2  |-  ( F `
 ( R `  K ) )  =  ( (  seq  M
( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  K
) ( F  o.  1st ) ( ( Z  X.  { A }
) `  ( K  +  1 ) ) )
145, 7, 133eqtr4g 2493 1  |-  ( (
ph  /\  K  e.  Z )  ->  ( R `  ( K  +  1 ) )  =  ( F `  ( R `  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3814    X. cxp 4876    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   1c1 8991    + caddc 8993   ZZcz 10282   ZZ>=cuz 10488    seq cseq 11323
This theorem is referenced by:  alginv  13066  algcvg  13067  algcvga  13070  algfx  13071  ovolicc2lem3  19415  ovolicc2lem4  19416  bfplem1  26531  bfplem2  26532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-seq 11324
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