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Theorem fargshiftfva 23678
Description: The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
Hypothesis
Ref Expression
fargshift.g  |-  G  =  ( x  e.  ( 0..^ ( # `  F
) )  |->  ( F `
 ( x  + 
1 ) ) )
Assertion
Ref Expression
fargshiftfva  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  ->  ( A. k  e.  ( 1 ... N ) ( E `  ( F `
 k ) )  =  [_ k  /  x ]_ P  ->  A. l  e.  ( 0..^ N ) ( E `  ( G `  l )
)  =  [_ (
l  +  1 )  /  x ]_ P
) )
Distinct variable groups:    x, F    x, E    k, F, l, x    x, N    k, E    k, G    k, N    P, k    E, l    N, l    P, l
Allowed substitution hints:    P( x)    G( x, l)

Proof of Theorem fargshiftfva
StepHypRef Expression
1 fargshiftlem 23673 . . . . . . 7  |-  ( ( N  e.  NN0  /\  l  e.  ( 0..^ N ) )  -> 
( l  +  1 )  e.  ( 1 ... N ) )
2 simpl 457 . . . . . . . . . . 11  |-  ( ( ( l  +  1 )  e.  ( 1 ... N )  /\  ( N  e.  NN0  /\  l  e.  ( 0..^ N ) ) )  ->  ( l  +  1 )  e.  ( 1 ... N ) )
32adantr 465 . . . . . . . . . 10  |-  ( ( ( ( l  +  1 )  e.  ( 1 ... N )  /\  ( N  e. 
NN0  /\  l  e.  ( 0..^ N ) ) )  /\  F :
( 1 ... N
) --> dom  E )  ->  ( l  +  1 )  e.  ( 1 ... N ) )
4 fveq2 5800 . . . . . . . . . . . . . 14  |-  ( k  =  ( l  +  1 )  ->  ( F `  k )  =  ( F `  ( l  +  1 ) ) )
54fveq2d 5804 . . . . . . . . . . . . 13  |-  ( k  =  ( l  +  1 )  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  ( l  +  1 ) ) ) )
6 csbeq1 3399 . . . . . . . . . . . . 13  |-  ( k  =  ( l  +  1 )  ->  [_ k  /  x ]_ P  = 
[_ ( l  +  1 )  /  x ]_ P )
75, 6eqeq12d 2476 . . . . . . . . . . . 12  |-  ( k  =  ( l  +  1 )  ->  (
( E `  ( F `  k )
)  =  [_ k  /  x ]_ P  <->  ( E `  ( F `  (
l  +  1 ) ) )  =  [_ ( l  +  1 )  /  x ]_ P ) )
87adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( ( l  +  1 )  e.  ( 1 ... N
)  /\  ( N  e.  NN0  /\  l  e.  ( 0..^ N ) ) )  /\  F : ( 1 ... N ) --> dom  E
)  /\  k  =  ( l  +  1 ) )  ->  (
( E `  ( F `  k )
)  =  [_ k  /  x ]_ P  <->  ( E `  ( F `  (
l  +  1 ) ) )  =  [_ ( l  +  1 )  /  x ]_ P ) )
9 simpl 457 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  l  e.  ( 0..^ N ) )  ->  N  e.  NN0 )
109adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( l  +  1 )  e.  ( 1 ... N )  /\  ( N  e.  NN0  /\  l  e.  ( 0..^ N ) ) )  ->  N  e.  NN0 )
1110anim1i 568 . . . . . . . . . . . . . . 15  |-  ( ( ( ( l  +  1 )  e.  ( 1 ... N )  /\  ( N  e. 
NN0  /\  l  e.  ( 0..^ N ) ) )  /\  F :
( 1 ... N
) --> dom  E )  ->  ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom 
E ) )
1211adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( l  +  1 )  e.  ( 1 ... N
)  /\  ( N  e.  NN0  /\  l  e.  ( 0..^ N ) ) )  /\  F : ( 1 ... N ) --> dom  E
)  /\  k  =  ( l  +  1 ) )  ->  ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
) )
13 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  l  e.  ( 0..^ N ) )  -> 
l  e.  ( 0..^ N ) )
1413ad3antlr 730 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( l  +  1 )  e.  ( 1 ... N
)  /\  ( N  e.  NN0  /\  l  e.  ( 0..^ N ) ) )  /\  F : ( 1 ... N ) --> dom  E
)  /\  k  =  ( l  +  1 ) )  ->  l  e.  ( 0..^ N ) )
15 fargshift.g . . . . . . . . . . . . . . . . 17  |-  G  =  ( x  e.  ( 0..^ ( # `  F
) )  |->  ( F `
 ( x  + 
1 ) ) )
1615fargshiftfv 23674 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  ->  ( l  e.  ( 0..^ N )  ->  ( G `  l )  =  ( F `  ( l  +  1 ) ) ) )
1716imp 429 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom 
E )  /\  l  e.  ( 0..^ N ) )  ->  ( G `  l )  =  ( F `  ( l  +  1 ) ) )
1817eqcomd 2462 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom 
E )  /\  l  e.  ( 0..^ N ) )  ->  ( F `  ( l  +  1 ) )  =  ( G `  l ) )
1912, 14, 18syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ( ( l  +  1 )  e.  ( 1 ... N
)  /\  ( N  e.  NN0  /\  l  e.  ( 0..^ N ) ) )  /\  F : ( 1 ... N ) --> dom  E
)  /\  k  =  ( l  +  1 ) )  ->  ( F `  ( l  +  1 ) )  =  ( G `  l ) )
2019fveq2d 5804 . . . . . . . . . . . 12  |-  ( ( ( ( ( l  +  1 )  e.  ( 1 ... N
)  /\  ( N  e.  NN0  /\  l  e.  ( 0..^ N ) ) )  /\  F : ( 1 ... N ) --> dom  E
)  /\  k  =  ( l  +  1 ) )  ->  ( E `  ( F `  ( l  +  1 ) ) )  =  ( E `  ( G `  l )
) )
2120eqeq1d 2456 . . . . . . . . . . 11  |-  ( ( ( ( ( l  +  1 )  e.  ( 1 ... N
)  /\  ( N  e.  NN0  /\  l  e.  ( 0..^ N ) ) )  /\  F : ( 1 ... N ) --> dom  E
)  /\  k  =  ( l  +  1 ) )  ->  (
( E `  ( F `  ( l  +  1 ) ) )  =  [_ (
l  +  1 )  /  x ]_ P  <->  ( E `  ( G `
 l ) )  =  [_ ( l  +  1 )  /  x ]_ P ) )
228, 21bitrd 253 . . . . . . . . . 10  |-  ( ( ( ( ( l  +  1 )  e.  ( 1 ... N
)  /\  ( N  e.  NN0  /\  l  e.  ( 0..^ N ) ) )  /\  F : ( 1 ... N ) --> dom  E
)  /\  k  =  ( l  +  1 ) )  ->  (
( E `  ( F `  k )
)  =  [_ k  /  x ]_ P  <->  ( E `  ( G `  l
) )  =  [_ ( l  +  1 )  /  x ]_ P ) )
233, 22rspcdv 3182 . . . . . . . . 9  |-  ( ( ( ( l  +  1 )  e.  ( 1 ... N )  /\  ( N  e. 
NN0  /\  l  e.  ( 0..^ N ) ) )  /\  F :
( 1 ... N
) --> dom  E )  ->  ( A. k  e.  ( 1 ... N
) ( E `  ( F `  k ) )  =  [_ k  /  x ]_ P  -> 
( E `  ( G `  l )
)  =  [_ (
l  +  1 )  /  x ]_ P
) )
2423ex 434 . . . . . . . 8  |-  ( ( ( l  +  1 )  e.  ( 1 ... N )  /\  ( N  e.  NN0  /\  l  e.  ( 0..^ N ) ) )  ->  ( F :
( 1 ... N
) --> dom  E  ->  ( A. k  e.  ( 1 ... N ) ( E `  ( F `  k )
)  =  [_ k  /  x ]_ P  -> 
( E `  ( G `  l )
)  =  [_ (
l  +  1 )  /  x ]_ P
) ) )
2524com23 78 . . . . . . 7  |-  ( ( ( l  +  1 )  e.  ( 1 ... N )  /\  ( N  e.  NN0  /\  l  e.  ( 0..^ N ) ) )  ->  ( A. k  e.  ( 1 ... N
) ( E `  ( F `  k ) )  =  [_ k  /  x ]_ P  -> 
( F : ( 1 ... N ) --> dom  E  ->  ( E `  ( G `  l ) )  = 
[_ ( l  +  1 )  /  x ]_ P ) ) )
261, 25mpancom 669 . . . . . 6  |-  ( ( N  e.  NN0  /\  l  e.  ( 0..^ N ) )  -> 
( A. k  e.  ( 1 ... N
) ( E `  ( F `  k ) )  =  [_ k  /  x ]_ P  -> 
( F : ( 1 ... N ) --> dom  E  ->  ( E `  ( G `  l ) )  = 
[_ ( l  +  1 )  /  x ]_ P ) ) )
2726ex 434 . . . . 5  |-  ( N  e.  NN0  ->  ( l  e.  ( 0..^ N )  ->  ( A. k  e.  ( 1 ... N ) ( E `  ( F `
 k ) )  =  [_ k  /  x ]_ P  ->  ( F : ( 1 ... N ) --> dom  E  ->  ( E `  ( G `  l )
)  =  [_ (
l  +  1 )  /  x ]_ P
) ) ) )
2827com24 87 . . . 4  |-  ( N  e.  NN0  ->  ( F : ( 1 ... N ) --> dom  E  ->  ( A. k  e.  ( 1 ... N
) ( E `  ( F `  k ) )  =  [_ k  /  x ]_ P  -> 
( l  e.  ( 0..^ N )  -> 
( E `  ( G `  l )
)  =  [_ (
l  +  1 )  /  x ]_ P
) ) ) )
2928imp31 432 . . 3  |-  ( ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom 
E )  /\  A. k  e.  ( 1 ... N ) ( E `  ( F `
 k ) )  =  [_ k  /  x ]_ P )  -> 
( l  e.  ( 0..^ N )  -> 
( E `  ( G `  l )
)  =  [_ (
l  +  1 )  /  x ]_ P
) )
3029ralrimiv 2828 . 2  |-  ( ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom 
E )  /\  A. k  e.  ( 1 ... N ) ( E `  ( F `
 k ) )  =  [_ k  /  x ]_ P )  ->  A. l  e.  (
0..^ N ) ( E `  ( G `
 l ) )  =  [_ ( l  +  1 )  /  x ]_ P )
3130ex 434 1  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  ->  ( A. k  e.  ( 1 ... N ) ( E `  ( F `
 k ) )  =  [_ k  /  x ]_ P  ->  A. l  e.  ( 0..^ N ) ( E `  ( G `  l )
)  =  [_ (
l  +  1 )  /  x ]_ P
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   [_csb 3396    |-> cmpt 4459   dom cdm 4949   -->wf 5523   ` cfv 5527  (class class class)co 6201   0cc0 9394   1c1 9395    + caddc 9397   NN0cn0 10691   ...cfz 11555  ..^cfzo 11666   #chash 12221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222
This theorem is referenced by: (None)
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