Theorem shftfvalg 10590

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 109	. 2 |- ( ( A e. CC /\ F e. V ) -> F e. V )
2		simpl 108	. 2 |- ( ( A e. CC /\ F e. V ) -> A e. CC )
3		simplr 519	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x e. CC )
4		simpll 518	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> A e. CC )
5	3, 4	subcld 8073	. . . . . . . . . 10 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> ( x - A ) e. CC )
6		vex 2689	. . . . . . . . . . 11 |- y e. _V
7		breldmg 4745	. . . . . . . . . . 11 |- ( ( ( x - A ) e. CC /\ y e. _V /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
8	6, 7	mp3an2 1303	. . . . . . . . . 10 |- ( ( ( x - A ) e. CC /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
9	5, 8	sylancom 416	. . . . . . . . 9 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
10		npcan 7971	. . . . . . . . . . . 12 |- ( ( x e. CC /\ A e. CC ) -> ( ( x - A ) + A ) = x )
11	10	eqcomd 2145	. . . . . . . . . . 11 |- ( ( x e. CC /\ A e. CC ) -> x = ( ( x - A ) + A ) )
12	11	ancoms 266	. . . . . . . . . 10 |- ( ( A e. CC /\ x e. CC ) -> x = ( ( x - A ) + A ) )
13	12	adantr 274	. . . . . . . . 9 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x = ( ( x - A ) + A ) )
14		oveq1 5781	. . . . . . . . . . 11 |- ( w = ( x - A ) -> ( w + A ) = ( ( x - A ) + A ) )
15	14	eqeq2d 2151	. . . . . . . . . 10 |- ( w = ( x - A ) -> ( x = ( w + A ) <-> x = ( ( x - A ) + A ) ) )
16	15	rspcev 2789	. . . . . . . . 9 |- ( ( ( x - A ) e. dom F /\ x = ( ( x - A ) + A ) ) -> E. w e. dom F x = ( w + A ) )
17	9, 13, 16	syl2anc 408	. . . . . . . 8 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> E. w e. dom F x = ( w + A ) )
18		vex 2689	. . . . . . . . 9 |- x e. _V
19		eqeq1 2146	. . . . . . . . . 10 |- ( z = x -> ( z = ( w + A ) <-> x = ( w + A ) ) )
20	19	rexbidv 2438	. . . . . . . . 9 |- ( z = x -> ( E. w e. dom F z = ( w + A ) <-> E. w e. dom F x = ( w + A ) ) )
21	18, 20	elab 2828	. . . . . . . 8 |- ( x e. { z | E. w e. dom F z = ( w + A ) } <-> E. w e. dom F x = ( w + A ) )
22	17, 21	sylibr 133	. . . . . . 7 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> x e. { z | E. w e. dom F z = ( w + A ) } )
23		brelrng 4770	. . . . . . . . 9 |- ( ( ( x - A ) e. CC /\ y e. _V /\ ( x - A ) F y ) -> y e. ran F )
24	6, 23	mp3an2 1303	. . . . . . . 8 |- ( ( ( x - A ) e. CC /\ ( x - A ) F y ) -> y e. ran F )
25	5, 24	sylancom 416	. . . . . . 7 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> y e. ran F )
26	22, 25	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 ) )
27	26	expl 375	. . . . 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 4200	. . . 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 4545	. . . 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 3146	. . 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 4803	. . . . 5 |- ( F e. V -> dom F e. _V )
32		abrexexg 6016	. . . . 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 4804	. . . 4 |- ( F e. V -> ran F e. _V )
35		xpexg 4653	. . . 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 408	. . 3 |- ( F e. V -> ( { z | E. w e. dom F z = ( w + A ) } X. ran F ) e. _V )
37		ssexg 4067	. . 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 287	. 2 |- ( ( A e. CC /\ F e. V ) -> { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } e. _V )
39		elex 2697	. . 3 |- ( F e. V -> F e. _V )
40		breq 3931	. . . . . 6 |- ( z = F -> ( ( x - w ) z y <-> ( x - w ) F y ) )
41	40	anbi2d 459	. . . . 5 |- ( z = F -> ( ( x e. CC /\ ( x - w ) z y ) <-> ( x e. CC /\ ( x - w ) F y ) ) )
42	41	opabbidv 3994	. . . 4 |- ( z = F -> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } = { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } )
43		oveq2 5782	. . . . . . 7 |- ( w = A -> ( x - w ) = ( x - A ) )
44	43	breq1d 3939	. . . . . 6 |- ( w = A -> ( ( x - w ) F y <-> ( x - A ) F y ) )
45	44	anbi2d 459	. . . . 5 |- ( w = A -> ( ( x e. CC /\ ( x - w ) F y ) <-> ( x e. CC /\ ( x - A ) F y ) ) )
46	45	opabbidv 3994	. . . 4 |- ( w = A -> { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
47		df-shft 10587	. . . 4 |- shift = ( z e. _V , w e. CC |-> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } )
48	42, 46, 47	ovmpog 5905	. . 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 1249	. 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 1216	1 |- ( ( A e. CC /\ F e. V ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   {cab 2125   E.wrex 2417   _Vcvv 2686   C_ wss 3071   class class class wbr 3929   {copab 3988   X. cxp 4537   dom cdm 4539   ran crn 4540  (class class class)co 5774   CCcc 7618   + caddc 7623   - cmin 7933   shift cshi 10586

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-resscn 7712  ax-1cn 7713  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7935  df-shft 10587

This theorem is referenced by:  ovshftex  10591  shftfibg  10592  2shfti  10603