Theorem fargshiftfva 41379

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, k, 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

Step	Hyp	Ref	Expression
1		fz0add1fz1 12537	. . . . . . 7 |- ( ( N e. NN0 /\ l e. ( 0..^ N ) ) -> ( l + 1 ) e. ( 1 ... N ) )
2		simpl 473	. . . . . . . . . . 11 |- ( ( ( l + 1 ) e. ( 1 ... N ) /\ ( N e. NN0 /\ l e. ( 0..^ N ) ) ) -> ( l + 1 ) e. ( 1 ... N ) )
3	2	adantr 481	. . . . . . . . . 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 6191	. . . . . . . . . . . . . 14 |- ( k = ( l + 1 ) -> ( F ` k ) = ( F ` ( l + 1 ) ) )
5	4	fveq2d 6195	. . . . . . . . . . . . 13 |- ( k = ( l + 1 ) -> ( E ` ( F ` k ) ) = ( E ` ( F ` ( l + 1 ) ) ) )
6		csbeq1 3536	. . . . . . . . . . . . 13 |- ( k = ( l + 1 ) -> [_ k / x ]_ P = [_ ( l + 1 ) / x ]_ P )
7	5, 6	eqeq12d 2637	. . . . . . . . . . . 12 |- ( k = ( l + 1 ) -> ( ( E ` ( F ` k ) ) = [_ k / x ]_ P <-> ( E ` ( F ` ( l + 1 ) ) ) = [_ ( l + 1 ) / x ]_ P ) )
8	7	adantl 482	. . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN0 /\ l e. ( 0..^ N ) ) -> N e. NN0 )
10	9	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( l + 1 ) e. ( 1 ... N ) /\ ( N e. NN0 /\ l e. ( 0..^ N ) ) ) -> N e. NN0 )
11	10	anim1i 592	. . . . . . . . . . . . . . 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 ) )
12	11	adantr 481	. . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ l e. ( 0..^ N ) ) -> l e. ( 0..^ N ) )
14	13	ad3antlr 767	. . . . . . . . . . . . . 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 ) ) )
16	15	fargshiftfv 41375	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ F : ( 1 ... N ) --> dom E ) -> ( l e. ( 0..^ N ) -> ( G ` l ) = ( F ` ( l + 1 ) ) ) )
17	16	imp 445	. . . . . . . . . . . . . . 15 |- ( ( ( N e. NN0 /\ F : ( 1 ... N ) --> dom E ) /\ l e. ( 0..^ N ) ) -> ( G ` l ) = ( F ` ( l + 1 ) ) )
18	17	eqcomd 2628	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN0 /\ F : ( 1 ... N ) --> dom E ) /\ l e. ( 0..^ N ) ) -> ( F ` ( l + 1 ) ) = ( G ` l ) )
19	12, 14, 18	syl2anc 693	. . . . . . . . . . . . 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 ) )
20	19	fveq2d 6195	. . . . . . . . . . . 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 ) ) )
21	20	eqeq1d 2624	. . . . . . . . . . 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 ) )
22	8, 21	bitrd 268	. . . . . . . . . 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 ) )
23	3, 22	rspcdv 3312	. . . . . . . . 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 ) )
24	23	ex 450	. . . . . . . 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 ) ) )
25	24	com23 86	. . . . . . 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 ) ) )
26	1, 25	mpancom 703	. . . . . 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 ) ) )
27	26	ex 450	. . . . 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 ) ) ) )
28	27	com24 95	. . . 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 ) ) ) )
29	28	imp31 448	. . 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 ) )
30	29	ralrimiv 2965	. 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 )
31	30	ex 450	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 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   [_csb 3533   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118

This theorem is referenced by: (None)