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Theorem algrp1 13754
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 461 . . . 4  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  Z )
2 algrf.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
31, 2syl6eleq 2533 . . 3  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  ( ZZ>= `  M )
)
4 seqp1 11826 . . 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 5697 . 2  |-  ( R `
 ( K  + 
1 ) )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  ( K  +  1 ) )
86fveq1i 5697 . . . 4  |-  ( R `
 K )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K )
98fveq2i 5699 . . 3  |-  ( F `
 ( R `  K ) )  =  ( F `  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K ) )
10 fvex 5706 . . . 4  |-  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K )  e.  _V
11 fvex 5706 . . . 4  |-  ( ( Z  X.  { A } ) `  ( K  +  1 ) )  e.  _V
1210, 11algrflem 6686 . . 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 2466 . 2  |-  ( F `
 ( R `  K ) )  =  ( (  seq M
( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  K
) ( F  o.  1st ) ( ( Z  X.  { A }
) `  ( K  +  1 ) ) )
145, 7, 133eqtr4g 2500 1  |-  ( (
ph  /\  K  e.  Z )  ->  ( R `  ( K  +  1 ) )  =  ( F `  ( R `  K ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3882    X. cxp 4843    o. ccom 4849   -->wf 5419   ` cfv 5423  (class class class)co 6096   1stc1st 6580   1c1 9288    + caddc 9290   ZZcz 10651   ZZ>=cuz 10866    seqcseq 11811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-seq 11812
This theorem is referenced by:  alginv  13755  algcvg  13756  algcvga  13759  algfx  13760  ovolicc2lem3  21007  ovolicc2lem4  21008  bfplem1  28726  bfplem2  28727
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