Theorem shftfvalg 10983

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.)

Assertion

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

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

Allowed substitution hints:   V( x, y)

Proof of Theorem shftfvalg

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

Step	Hyp	Ref	Expression
1		simpr 110	. 2 |- ( ( A e. CC /\ F e. V ) -> F e. V )
2		simpl 109	. 2 |- ( ( A e. CC /\ F e. V ) -> A e. CC )
3		simplr 528	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x e. CC )
4		simpll 527	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> A e. CC )
5	3, 4	subcld 8337	. . . . . . . . . 10 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> ( x - A ) e. CC )
6		vex 2766	. . . . . . . . . . 11 |- y e. _V
7		breldmg 4872	. . . . . . . . . . 11 |- ( ( ( x - A ) e. CC /\ y e. _V /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
8	6, 7	mp3an2 1336	. . . . . . . . . 10 |- ( ( ( x - A ) e. CC /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
9	5, 8	sylancom 420	. . . . . . . . 9 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
10		npcan 8235	. . . . . . . . . . . 12 |- ( ( x e. CC /\ A e. CC ) -> ( ( x - A ) + A ) = x )
11	10	eqcomd 2202	. . . . . . . . . . 11 |- ( ( x e. CC /\ A e. CC ) -> x = ( ( x - A ) + A ) )
12	11	ancoms 268	. . . . . . . . . 10 |- ( ( A e. CC /\ x e. CC ) -> x = ( ( x - A ) + A ) )
13	12	adantr 276	. . . . . . . . 9 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x = ( ( x - A ) + A ) )
14		oveq1 5929	. . . . . . . . . . 11 |- ( w = ( x - A ) -> ( w + A ) = ( ( x - A ) + A ) )
15	14	eqeq2d 2208	. . . . . . . . . 10 |- ( w = ( x - A ) -> ( x = ( w + A ) <-> x = ( ( x - A ) + A ) ) )
16	15	rspcev 2868	. . . . . . . . 9 |- ( ( ( x - A ) e. dom F /\ x = ( ( x - A ) + A ) ) -> E. w e. dom F x = ( w + A ) )
17	9, 13, 16	syl2anc 411	. . . . . . . 8 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> E. w e. dom F x = ( w + A ) )
18		vex 2766	. . . . . . . . 9 |- x e. _V
19		eqeq1 2203	. . . . . . . . . 10 |- ( z = x -> ( z = ( w + A ) <-> x = ( w + A ) ) )
20	19	rexbidv 2498	. . . . . . . . 9 |- ( z = x -> ( E. w e. dom F z = ( w + A ) <-> E. w e. dom F x = ( w + A ) ) )
21	18, 20	elab 2908	. . . . . . . 8 |- ( x e. { z | E. w e. dom F z = ( w + A ) } <-> E. w e. dom F x = ( w + A ) )
22	17, 21	sylibr 134	. . . . . . 7 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x e. { z | E. w e. dom F z = ( w + A ) } )
23		brelrng 4897	. . . . . . . . 9 |- ( ( ( x - A ) e. CC /\ y e. _V /\ ( x - A ) F y ) -> y e. ran F )
24	6, 23	mp3an2 1336	. . . . . . . 8 |- ( ( ( x - A ) e. CC /\ ( x - A ) F y ) -> y e. ran F )
25	5, 24	sylancom 420	. . . . . . 7 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> y e. ran F )
26	22, 25	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 ) )
27	26	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 ) ) )
28	27	ssopab2dv 4313	. . . 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 ) } )
29		df-xp 4669	. . . 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 ) }
30	28, 29	sseqtrrdi 3232	. . 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 ) )
31		dmexg 4930	. . . . 5 |- ( F e. V -> dom F e. _V )
32		abrexexg 6175	. . . . 5 |- ( dom F e. _V -> { z | E. w e. dom F z = ( w + A ) } e. _V )
33	31, 32	syl 14	. . . 4 |- ( F e. V -> { z | E. w e. dom F z = ( w + A ) } e. _V )
34		rnexg 4931	. . . 4 |- ( F e. V -> ran F e. _V )
35		xpexg 4777	. . . 4 |- ( ( { z | E. w e. dom F z = ( w + A ) } e. _V /\ ran F e. _V ) -> ( { z | E. w e. dom F z = ( w + A ) } X. ran F ) e. _V )
36	33, 34, 35	syl2anc 411	. . 3 |- ( F e. V -> ( { z | E. w e. dom F z = ( w + A ) } X. ran F ) e. _V )
37		ssexg 4172	. . 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 )
38	30, 36, 37	syl2an 289	. 2 |- ( ( A e. CC /\ F e. V ) -> { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } e. _V )
39		elex 2774	. . 3 |- ( F e. V -> F e. _V )
40		breq 4035	. . . . . 6 |- ( z = F -> ( ( x - w ) z y <-> ( x - w ) F y ) )
41	40	anbi2d 464	. . . . 5 |- ( z = F -> ( ( x e. CC /\ ( x - w ) z y ) <-> ( x e. CC /\ ( x - w ) F y ) ) )
42	41	opabbidv 4099	. . . 4 |- ( z = F -> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } = { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } )
43		oveq2 5930	. . . . . . 7 |- ( w = A -> ( x - w ) = ( x - A ) )
44	43	breq1d 4043	. . . . . 6 |- ( w = A -> ( ( x - w ) F y <-> ( x - A ) F y ) )
45	44	anbi2d 464	. . . . 5 |- ( w = A -> ( ( x e. CC /\ ( x - w ) F y ) <-> ( x e. CC /\ ( x - A ) F y ) ) )
46	45	opabbidv 4099	. . . 4 |- ( w = A -> { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
47		df-shft 10980	. . . 4 |- shift = ( z e. _V , w e. CC |-> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } )
48	42, 46, 47	ovmpog 6057	. . 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 ) } )
49	39, 48	syl3an1 1282	. 2 |- ( ( 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 ) } )
50	1, 2, 38, 49	syl3anc 1249	1 |- ( ( A e. CC /\ F e. V ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   {cab 2182   E.wrex 2476   _Vcvv 2763   C_ wss 3157   class class class wbr 4033   {copab 4093   X. cxp 4661   dom cdm 4663   ran crn 4664  (class class class)co 5922   CCcc 7877   + caddc 7882   - cmin 8197   shift cshi 10979

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-resscn 7971  ax-1cn 7972  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-sub 8199  df-shft 10980

This theorem is referenced by:  ovshftex  10984  shftfibg  10985  2shfti  10996