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Theorem algrp1 12952
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 447 . . . 4  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  Z )
2 algrf.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
31, 2syl6eleq 2456 . . 3  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  ( ZZ>= `  M )
)
4 seqp1 11225 . . 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 15 . 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 5633 . 2  |-  ( R `
 ( K  + 
1 ) )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  ( K  +  1 ) )
86fveq1i 5633 . . . 4  |-  ( R `
 K )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K )
98fveq2i 5635 . . 3  |-  ( F `
 ( R `  K ) )  =  ( F `  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K ) )
10 fvex 5646 . . . 4  |-  (  seq 
M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  K )  e.  _V
11 fvex 5646 . . . 4  |-  ( ( Z  X.  { A } ) `  ( K  +  1 ) )  e.  _V
1210, 11algrflem 6352 . . 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 2389 . 2  |-  ( F `
 ( R `  K ) )  =  ( (  seq  M
( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  K
) ( F  o.  1st ) ( ( Z  X.  { A }
) `  ( K  +  1 ) ) )
145, 7, 133eqtr4g 2423 1  |-  ( (
ph  /\  K  e.  Z )  ->  ( R `  ( K  +  1 ) )  =  ( F `  ( R `  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   {csn 3729    X. cxp 4790    o. ccom 4796   -->wf 5354   ` cfv 5358  (class class class)co 5981   1stc1st 6247   1c1 8885    + caddc 8887   ZZcz 10175   ZZ>=cuz 10381    seq cseq 11210
This theorem is referenced by:  alginv  12953  algcvg  12954  algcvga  12957  algfx  12958  ovolicc2lem3  19093  ovolicc2lem4  19094  bfplem1  26052  bfplem2  26053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-seq 11211
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