Theorem shftfval 10763

Description: The value of the sequence shifter operation is a function on CC. A is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Hypothesis

Ref	Expression
shftfval.1	|- F e. _V

Assertion

Ref	Expression
shftfval	|- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )

Distinct variable groups:   x, y, A   x, F, y

Proof of Theorem shftfval

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 520	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x e. CC )
2		simpll 519	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> A e. CC )
3	1, 2	subcld 8209	. . . . . . . . . 10 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> ( x - A ) e. CC )
4		vex 2729	. . . . . . . . . . 11 |- y e. _V
5		breldmg 4810	. . . . . . . . . . 11 |- ( ( ( x - A ) e. CC /\ y e. _V /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
6	4, 5	mp3an2 1315	. . . . . . . . . 10 |- ( ( ( x - A ) e. CC /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
7	3, 6	sylancom 417	. . . . . . . . 9 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
8		npcan 8107	. . . . . . . . . . . 12 |- ( ( x e. CC /\ A e. CC ) -> ( ( x - A ) + A ) = x )
9	8	eqcomd 2171	. . . . . . . . . . 11 |- ( ( x e. CC /\ A e. CC ) -> x = ( ( x - A ) + A ) )
10	9	ancoms 266	. . . . . . . . . 10 |- ( ( A e. CC /\ x e. CC ) -> x = ( ( x - A ) + A ) )
11	10	adantr 274	. . . . . . . . 9 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x = ( ( x - A ) + A ) )
12		oveq1 5849	. . . . . . . . . . 11 |- ( w = ( x - A ) -> ( w + A ) = ( ( x - A ) + A ) )
13	12	eqeq2d 2177	. . . . . . . . . 10 |- ( w = ( x - A ) -> ( x = ( w + A ) <-> x = ( ( x - A ) + A ) ) )
14	13	rspcev 2830	. . . . . . . . 9 |- ( ( ( x - A ) e. dom F /\ x = ( ( x - A ) + A ) ) -> E. w e. dom F x = ( w + A ) )
15	7, 11, 14	syl2anc 409	. . . . . . . 8 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> E. w e. dom F x = ( w + A ) )
16		vex 2729	. . . . . . . . 9 |- x e. _V
17		eqeq1 2172	. . . . . . . . . 10 |- ( z = x -> ( z = ( w + A ) <-> x = ( w + A ) ) )
18	17	rexbidv 2467	. . . . . . . . 9 |- ( z = x -> ( E. w e. dom F z = ( w + A ) <-> E. w e. dom F x = ( w + A ) ) )
19	16, 18	elab 2870	. . . . . . . 8 |- ( x e. { z | E. w e. dom F z = ( w + A ) } <-> E. w e. dom F x = ( w + A ) )
20	15, 19	sylibr 133	. . . . . . 7 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x e. { z | E. w e. dom F z = ( w + A ) } )
21		brelrng 4835	. . . . . . . . 9 |- ( ( ( x - A ) e. CC /\ y e. _V /\ ( x - A ) F y ) -> y e. ran F )
22	4, 21	mp3an2 1315	. . . . . . . 8 |- ( ( ( x - A ) e. CC /\ ( x - A ) F y ) -> y e. ran F )
23	3, 22	sylancom 417	. . . . . . 7 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> y e. ran F )
24	20, 23	jca 304	. . . . . 6 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> ( x e. { z | E. w e. dom F z = ( w + A ) } /\ y e. ran F ) )
25	24	expl 376	. . . . 5 |- ( A e. CC -> ( ( x e. CC /\ ( x - A ) F y ) -> ( x e. { z | E. w e. dom F z = ( w + A ) } /\ y e. ran F ) ) )
26	25	ssopab2dv 4256	. . . 4 |- ( A e. CC -> { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } C_ { <. x , y >. | ( x e. { z | E. w e. dom F z = ( w + A ) } /\ y e. ran F ) } )
27		df-xp 4610	. . . 4 |- ( { z | E. w e. dom F z = ( w + A ) } X. ran F ) = { <. x , y >. | ( x e. { z | E. w e. dom F z = ( w + A ) } /\ y e. ran F ) }
28	26, 27	sseqtrrdi 3191	. . 3 |- ( A e. CC -> { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } C_ ( { z | E. w e. dom F z = ( w + A ) } X. ran F ) )
29		shftfval.1	. . . . . 6 |- F e. _V
30	29	dmex 4870	. . . . 5 |- dom F e. _V
31	30	abrexex 6085	. . . 4 |- { z | E. w e. dom F z = ( w + A ) } e. _V
32	29	rnex 4871	. . . 4 |- ran F e. _V
33	31, 32	xpex 4719	. . 3 |- ( { z | E. w e. dom F z = ( w + A ) } X. ran F ) e. _V
34		ssexg 4121	. . 3 |- ( ( { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } C_ ( { z | E. w e. dom F z = ( w + A ) } X. ran F ) /\ ( { z | E. w e. dom F z = ( w + A ) } X. ran F ) e. _V ) -> { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } e. _V )
35	28, 33, 34	sylancl 410	. 2 |- ( A e. CC -> { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } e. _V )
36		breq 3984	. . . . . 6 |- ( z = F -> ( ( x - w ) z y <-> ( x - w ) F y ) )
37	36	anbi2d 460	. . . . 5 |- ( z = F -> ( ( x e. CC /\ ( x - w ) z y ) <-> ( x e. CC /\ ( x - w ) F y ) ) )
38	37	opabbidv 4048	. . . 4 |- ( z = F -> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } = { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } )
39		oveq2 5850	. . . . . . 7 |- ( w = A -> ( x - w ) = ( x - A ) )
40	39	breq1d 3992	. . . . . 6 |- ( w = A -> ( ( x - w ) F y <-> ( x - A ) F y ) )
41	40	anbi2d 460	. . . . 5 |- ( w = A -> ( ( x e. CC /\ ( x - w ) F y ) <-> ( x e. CC /\ ( x - A ) F y ) ) )
42	41	opabbidv 4048	. . . 4 |- ( w = A -> { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
43		df-shft 10757	. . . 4 |- shift = ( z e. _V , w e. CC |-> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } )
44	38, 42, 43	ovmpog 5976	. . 3 |- ( ( F e. _V /\ A e. CC /\ { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } e. _V ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
45	29, 44	mp3an1 1314	. 2 |- ( ( A e. CC /\ { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } e. _V ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
46	35, 45	mpdan 418	1 |- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   {cab 2151   E.wrex 2445   _Vcvv 2726   C_ wss 3116   class class class wbr 3982   {copab 4042   X. cxp 4602   dom cdm 4604   ran crn 4605  (class class class)co 5842   CCcc 7751   + caddc 7756   - cmin 8069   shift cshi 10756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071  df-shft 10757

This theorem is referenced by:  shftdm  10764  shftfib  10765  shftfn  10766  2shfti  10773  shftidt2  10774