Theorem shftfval 13810

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		ovex 6678	. . . . . . . . . 10 |- ( x - A ) e. _V
2		vex 3203	. . . . . . . . . 10 |- y e. _V
3	1, 2	breldm 5329	. . . . . . . . 9 |- ( ( x - A ) F y -> ( x - A ) e. dom F )
4		npcan 10290	. . . . . . . . . . 11 |- ( ( x e. CC /\ A e. CC ) -> ( ( x - A ) + A ) = x )
5	4	eqcomd 2628	. . . . . . . . . 10 |- ( ( x e. CC /\ A e. CC ) -> x = ( ( x - A ) + A ) )
6	5	ancoms 469	. . . . . . . . 9 |- ( ( A e. CC /\ x e. CC ) -> x = ( ( x - A ) + A ) )
7		oveq1 6657	. . . . . . . . . . 11 |- ( w = ( x - A ) -> ( w + A ) = ( ( x - A ) + A ) )
8	7	eqeq2d 2632	. . . . . . . . . 10 |- ( w = ( x - A ) -> ( x = ( w + A ) <-> x = ( ( x - A ) + A ) ) )
9	8	rspcev 3309	. . . . . . . . 9 |- ( ( ( x - A ) e. dom F /\ x = ( ( x - A ) + A ) ) -> E. w e. dom F x = ( w + A ) )
10	3, 6, 9	syl2anr 495	. . . . . . . 8 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> E. w e. dom F x = ( w + A ) )
11		vex 3203	. . . . . . . . 9 |- x e. _V
12		eqeq1 2626	. . . . . . . . . 10 |- ( z = x -> ( z = ( w + A ) <-> x = ( w + A ) ) )
13	12	rexbidv 3052	. . . . . . . . 9 |- ( z = x -> ( E. w e. dom F z = ( w + A ) <-> E. w e. dom F x = ( w + A ) ) )
14	11, 13	elab 3350	. . . . . . . 8 |- ( x e. { z | E. w e. dom F z = ( w + A ) } <-> E. w e. dom F x = ( w + A ) )
15	10, 14	sylibr 224	. . . . . . 7 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x e. { z | E. w e. dom F z = ( w + A ) } )
16	1, 2	brelrn 5356	. . . . . . . 8 |- ( ( x - A ) F y -> y e. ran F )
17	16	adantl 482	. . . . . . 7 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> y e. ran F )
18	15, 17	jca 554	. . . . . 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 ) )
19	18	expl 648	. . . . 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 ) ) )
20	19	ssopab2dv 5004	. . . 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 ) } )
21		df-xp 5120	. . . 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 ) }
22	20, 21	syl6sseqr 3652	. . 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 ) )
23		shftfval.1	. . . . . 6 |- F e. _V
24	23	dmex 7099	. . . . 5 |- dom F e. _V
25	24	abrexex 7141	. . . 4 |- { z | E. w e. dom F z = ( w + A ) } e. _V
26	23	rnex 7100	. . . 4 |- ran F e. _V
27	25, 26	xpex 6962	. . 3 |- ( { z | E. w e. dom F z = ( w + A ) } X. ran F ) e. _V
28		ssexg 4804	. . 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 )
29	22, 27, 28	sylancl 694	. 2 |- ( A e. CC -> { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } e. _V )
30		breq 4655	. . . . . 6 |- ( z = F -> ( ( x - w ) z y <-> ( x - w ) F y ) )
31	30	anbi2d 740	. . . . 5 |- ( z = F -> ( ( x e. CC /\ ( x - w ) z y ) <-> ( x e. CC /\ ( x - w ) F y ) ) )
32	31	opabbidv 4716	. . . 4 |- ( z = F -> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } = { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } )
33		oveq2 6658	. . . . . . 7 |- ( w = A -> ( x - w ) = ( x - A ) )
34	33	breq1d 4663	. . . . . 6 |- ( w = A -> ( ( x - w ) F y <-> ( x - A ) F y ) )
35	34	anbi2d 740	. . . . 5 |- ( w = A -> ( ( x e. CC /\ ( x - w ) F y ) <-> ( x e. CC /\ ( x - A ) F y ) ) )
36	35	opabbidv 4716	. . . 4 |- ( w = A -> { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
37		df-shft 13807	. . . 4 |- shift = ( z e. _V , w e. CC |-> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } )
38	32, 36, 37	ovmpt2g 6795	. . 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 ) } )
39	23, 38	mp3an1 1411	. 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 ) } )
40	29, 39	mpdan 702	1 |- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   {copab 4712   X. cxp 5112   dom cdm 5114   ran crn 5115  (class class class)co 6650   CCcc 9934   + caddc 9939   - cmin 10266   shift cshi 13806

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-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

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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-shft 13807

This theorem is referenced by:  shftdm  13811  shftfib  13812  shftfn  13813  2shfti  13820  shftidt2  13821