Theorem shftfval 11381

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 529	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x e. CC )
2		simpll 527	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> A e. CC )
3	1, 2	subcld 8489	. . . . . . . . . 10 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> ( x - A ) e. CC )
4		vex 2805	. . . . . . . . . . 11 |- y e. _V
5		breldmg 4937	. . . . . . . . . . 11 |- ( ( ( x - A ) e. CC /\ y e. _V /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
6	4, 5	mp3an2 1361	. . . . . . . . . 10 |- ( ( ( x - A ) e. CC /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
7	3, 6	sylancom 420	. . . . . . . . 9 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
8		npcan 8387	. . . . . . . . . . . 12 |- ( ( x e. CC /\ A e. CC ) -> ( ( x - A ) + A ) = x )
9	8	eqcomd 2237	. . . . . . . . . . 11 |- ( ( x e. CC /\ A e. CC ) -> x = ( ( x - A ) + A ) )
10	9	ancoms 268	. . . . . . . . . 10 |- ( ( A e. CC /\ x e. CC ) -> x = ( ( x - A ) + A ) )
11	10	adantr 276	. . . . . . . . 9 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x = ( ( x - A ) + A ) )
12		oveq1 6024	. . . . . . . . . . 11 |- ( w = ( x - A ) -> ( w + A ) = ( ( x - A ) + A ) )
13	12	eqeq2d 2243	. . . . . . . . . 10 |- ( w = ( x - A ) -> ( x = ( w + A ) <-> x = ( ( x - A ) + A ) ) )
14	13	rspcev 2910	. . . . . . . . 9 |- ( ( ( x - A ) e. dom F /\ x = ( ( x - A ) + A ) ) -> E. w e. dom F x = ( w + A ) )
15	7, 11, 14	syl2anc 411	. . . . . . . 8 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> E. w e. dom F x = ( w + A ) )
16		vex 2805	. . . . . . . . 9 |- x e. _V
17		eqeq1 2238	. . . . . . . . . 10 |- ( z = x -> ( z = ( w + A ) <-> x = ( w + A ) ) )
18	17	rexbidv 2533	. . . . . . . . 9 |- ( z = x -> ( E. w e. dom F z = ( w + A ) <-> E. w e. dom F x = ( w + A ) ) )
19	16, 18	elab 2950	. . . . . . . 8 |- ( x e. { z | E. w e. dom F z = ( w + A ) } <-> E. w e. dom F x = ( w + A ) )
20	15, 19	sylibr 134	. . . . . . 7 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x e. { z | E. w e. dom F z = ( w + A ) } )
21		brelrng 4963	. . . . . . . . 9 |- ( ( ( x - A ) e. CC /\ y e. _V /\ ( x - A ) F y ) -> y e. ran F )
22	4, 21	mp3an2 1361	. . . . . . . 8 |- ( ( ( x - A ) e. CC /\ ( x - A ) F y ) -> y e. ran F )
23	3, 22	sylancom 420	. . . . . . 7 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> y e. ran F )
24	20, 23	jca 306	. . . . . 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 378	. . . . 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 4373	. . . 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 4731	. . . 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 3276	. . 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 4999	. . . . 5 |- dom F e. _V
31	30	abrexex 6278	. . . 4 |- { z | E. w e. dom F z = ( w + A ) } e. _V
32	29	rnex 5000	. . . 4 |- ran F e. _V
33	31, 32	xpex 4842	. . 3 |- ( { z | E. w e. dom F z = ( w + A ) } X. ran F ) e. _V
34		ssexg 4228	. . 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 413	. 2 |- ( A e. CC -> { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } e. _V )
36		breq 4090	. . . . . 6 |- ( z = F -> ( ( x - w ) z y <-> ( x - w ) F y ) )
37	36	anbi2d 464	. . . . 5 |- ( z = F -> ( ( x e. CC /\ ( x - w ) z y ) <-> ( x e. CC /\ ( x - w ) F y ) ) )
38	37	opabbidv 4155	. . . 4 |- ( z = F -> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } = { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } )
39		oveq2 6025	. . . . . . 7 |- ( w = A -> ( x - w ) = ( x - A ) )
40	39	breq1d 4098	. . . . . 6 |- ( w = A -> ( ( x - w ) F y <-> ( x - A ) F y ) )
41	40	anbi2d 464	. . . . 5 |- ( w = A -> ( ( x e. CC /\ ( x - w ) F y ) <-> ( x e. CC /\ ( x - A ) F y ) ) )
42	41	opabbidv 4155	. . . 4 |- ( w = A -> { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
43		df-shft 11375	. . . 4 |- shift = ( z e. _V , w e. CC |-> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } )
44	38, 42, 43	ovmpog 6155	. . 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 1360	. 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 421	1 |- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   {cab 2217   E.wrex 2511   _Vcvv 2802   C_ wss 3200   class class class wbr 4088   {copab 4149   X. cxp 4723   dom cdm 4725   ran crn 4726  (class class class)co 6017   CCcc 8029   + caddc 8034   - cmin 8349   shift cshi 11374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-resscn 8123  ax-1cn 8124  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-sub 8351  df-shft 11375

This theorem is referenced by:  shftdm  11382  shftfib  11383  shftfn  11384  2shfti  11391  shftidt2  11392