ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ialgrp1 Unicode version

Theorem ialgrp1 10561
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 } ) ,  S )
algrf.3  |-  ( ph  ->  M  e.  ZZ )
algrf.4  |-  ( ph  ->  A  e.  S )
algrf.5  |-  ( ph  ->  F : S --> S )
algrf.s  |-  ( ph  ->  S  e.  V )
Assertion
Ref Expression
ialgrp1  |-  ( (
ph  /\  K  e.  Z )  ->  ( R `  ( K  +  1 ) )  =  ( F `  ( R `  K ) ) )

Proof of Theorem ialgrp1
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 } ) ,  S )
21fveq1i 5204 . . 3  |-  ( R `
 ( K  + 
1 ) )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S ) `  ( K  +  1
) )
3 simpr 108 . . . . 5  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  Z )
4 algrf.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
53, 4syl6eleq 2172 . . . 4  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  ( ZZ>= `  M )
)
6 algrf.4 . . . . . 6  |-  ( ph  ->  A  e.  S )
76adantr 270 . . . . 5  |-  ( (
ph  /\  K  e.  Z )  ->  A  e.  S )
84, 7ialgrlemconst 10558 . . . 4  |-  ( ( ( ph  /\  K  e.  Z )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( ( Z  X.  { A }
) `  x )  e.  S )
9 algrf.5 . . . . . 6  |-  ( ph  ->  F : S --> S )
109adantr 270 . . . . 5  |-  ( (
ph  /\  K  e.  Z )  ->  F : S --> S )
1110ialgrlem1st 10557 . . . 4  |-  ( ( ( ph  /\  K  e.  Z )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x
( F  o.  1st ) y )  e.  S )
125, 8, 11iseqp1 9527 . . 3  |-  ( (
ph  /\  K  e.  Z )  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S
) `  ( K  +  1 ) )  =  ( (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S
) `  K )
( F  o.  1st ) ( ( Z  X.  { A }
) `  ( K  +  1 ) ) ) )
132, 12syl5eq 2126 . 2  |-  ( (
ph  /\  K  e.  Z )  ->  ( R `  ( K  +  1 ) )  =  ( (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S
) `  K )
( F  o.  1st ) ( ( Z  X.  { A }
) `  ( K  +  1 ) ) ) )
145, 8, 11iseqcl 9526 . . . 4  |-  ( (
ph  /\  K  e.  Z )  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S
) `  K )  e.  S )
154peano2uzs 8742 . . . . . 6  |-  ( K  e.  Z  ->  ( K  +  1 )  e.  Z )
16 fvconst2g 5401 . . . . . 6  |-  ( ( A  e.  S  /\  ( K  +  1
)  e.  Z )  ->  ( ( Z  X.  { A }
) `  ( K  +  1 ) )  =  A )
176, 15, 16syl2an 283 . . . . 5  |-  ( (
ph  /\  K  e.  Z )  ->  (
( Z  X.  { A } ) `  ( K  +  1 ) )  =  A )
1817, 7eqeltrd 2156 . . . 4  |-  ( (
ph  /\  K  e.  Z )  ->  (
( Z  X.  { A } ) `  ( K  +  1 ) )  e.  S )
19 algrflemg 5876 . . . 4  |-  ( ( (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S ) `  K )  e.  S  /\  ( ( Z  X.  { A } ) `  ( K  +  1
) )  e.  S
)  ->  ( (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S
) `  K )
( F  o.  1st ) ( ( Z  X.  { A }
) `  ( K  +  1 ) ) )  =  ( F `
 (  seq M
( ( F  o.  1st ) ,  ( Z  X.  { A }
) ,  S ) `
 K ) ) )
2014, 18, 19syl2anc 403 . . 3  |-  ( (
ph  /\  K  e.  Z )  ->  (
(  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S ) `  K
) ( F  o.  1st ) ( ( Z  X.  { A }
) `  ( K  +  1 ) ) )  =  ( F `
 (  seq M
( ( F  o.  1st ) ,  ( Z  X.  { A }
) ,  S ) `
 K ) ) )
211fveq1i 5204 . . . 4  |-  ( R `
 K )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S ) `  K )
2221fveq2i 5206 . . 3  |-  ( F `
 ( R `  K ) )  =  ( F `  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S
) `  K )
)
2320, 22syl6reqr 2133 . 2  |-  ( (
ph  /\  K  e.  Z )  ->  ( F `  ( R `  K ) )  =  ( (  seq M
( ( F  o.  1st ) ,  ( Z  X.  { A }
) ,  S ) `
 K ) ( F  o.  1st )
( ( Z  X.  { A } ) `  ( K  +  1
) ) ) )
2413, 23eqtr4d 2117 1  |-  ( (
ph  /\  K  e.  Z )  ->  ( R `  ( K  +  1 ) )  =  ( F `  ( R `  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   {csn 3400    X. cxp 4363    o. ccom 4369   -->wf 4922   ` cfv 4926  (class class class)co 5537   1stc1st 5790   1c1 7033    + caddc 7035   ZZcz 8421   ZZ>=cuz 8689    seqcseq 9510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-addcom 7127  ax-addass 7129  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-0id 7135  ax-rnegex 7136  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-ltadd 7143
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-id 4050  df-iord 4123  df-on 4125  df-ilim 4126  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-frec 6034  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-inn 8096  df-n0 8345  df-z 8422  df-uz 8690  df-iseq 9511
This theorem is referenced by:  ialginv  10562  ialgcvg  10563  ialgcvga  10566  ialgfx  10567
  Copyright terms: Public domain W3C validator