Theorem shftfvalg 11244

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 8418	. . . . . . . . . 10 |- ( ( ( A e. CC /\ x e. CC ) /\ ( x - A ) F y ) -> ( x - A ) e. CC )
6		vex 2779	. . . . . . . . . . 11 |- y e. _V
7		breldmg 4903	. . . . . . . . . . 11 |- ( ( ( x - A ) e. CC /\ y e. _V /\ ( x - A ) F y ) -> ( x - A ) e. dom F )
8	6, 7	mp3an2 1338	. . . . . . . . . 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 8316	. . . . . . . . . . . 12 |- ( ( x e. CC /\ A e. CC ) -> ( ( x - A ) + A ) = x )
11	10	eqcomd 2213	. . . . . . . . . . 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 5974	. . . . . . . . . . 11 |- ( w = ( x - A ) -> ( w + A ) = ( ( x - A ) + A ) )
15	14	eqeq2d 2219	. . . . . . . . . 10 |- ( w = ( x - A ) -> ( x = ( w + A ) <-> x = ( ( x - A ) + A ) ) )
16	15	rspcev 2884	. . . . . . . . 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 2779	. . . . . . . . 9 |- x e. _V
19		eqeq1 2214	. . . . . . . . . 10 |- ( z = x -> ( z = ( w + A ) <-> x = ( w + A ) ) )
20	19	rexbidv 2509	. . . . . . . . 9 |- ( z = x -> ( E. w e. dom F z = ( w + A ) <-> E. w e. dom F x = ( w + A ) ) )
21	18, 20	elab 2924	. . . . . . . 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 4928	. . . . . . . . 9 |- ( ( ( x - A ) e. CC /\ y e. _V /\ ( x - A ) F y ) -> y e. ran F )
24	6, 23	mp3an2 1338	. . . . . . . 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 4343	. . . 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 4699	. . . 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 3250	. . 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 4961	. . . . 5 |- ( F e. V -> dom F e. _V )
32		abrexexg 6226	. . . . 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 4962	. . . 4 |- ( F e. V -> ran F e. _V )
35		xpexg 4807	. . . 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 4199	. . 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 2788	. . 3 |- ( F e. V -> F e. _V )
40		breq 4061	. . . . . 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 4126	. . . 4 |- ( z = F -> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } = { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } )
43		oveq2 5975	. . . . . . 7 |- ( w = A -> ( x - w ) = ( x - A ) )
44	43	breq1d 4069	. . . . . 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 4126	. . . 4 |- ( w = A -> { <. x , y >. | ( x e. CC /\ ( x - w ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
47		df-shft 11241	. . . 4 |- shift = ( z e. _V , w e. CC |-> { <. x , y >. | ( x e. CC /\ ( x - w ) z y ) } )
48	42, 46, 47	ovmpog 6103	. . 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 1283	. 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 1250	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 1373   e. wcel 2178   {cab 2193   E.wrex 2487   _Vcvv 2776   C_ wss 3174   class class class wbr 4059   {copab 4120   X. cxp 4691   dom cdm 4693   ran crn 4694  (class class class)co 5967   CCcc 7958   + caddc 7963   - cmin 8278   shift cshi 11240

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-resscn 8052  ax-1cn 8053  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-sub 8280  df-shft 11241

This theorem is referenced by:  ovshftex  11245  shftfibg  11246  2shfti  11257