Theorem shftf 11540

Description: Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Hypothesis

Ref	Expression
shftfval.1	|- F e. _V

Assertion

Ref	Expression
shftf	|- ( ( F : B --> C /\ A e. CC ) -> ( F shift A ) : { x e. CC | ( x - A ) e. B } --> C )

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

Allowed substitution hint:   C( x)

Proof of Theorem shftf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5513	. . 3 |- ( F : B --> C -> F Fn B )
2		shftfval.1	. . . 4 |- F e. _V
3	2	shftfn 11534	. . 3 |- ( ( F Fn B /\ A e. CC ) -> ( F shift A ) Fn { x e. CC | ( x - A ) e. B } )
4	1, 3	sylan 283	. 2 |- ( ( F : B --> C /\ A e. CC ) -> ( F shift A ) Fn { x e. CC | ( x - A ) e. B } )
5		oveq1 6065	. . . . . 6 |- ( x = y -> ( x - A ) = ( y - A ) )
6	5	eleq1d 2303	. . . . 5 |- ( x = y -> ( ( x - A ) e. B <-> ( y - A ) e. B ) )
7	6	elrab 2976	. . . 4 |- ( y e. { x e. CC | ( x - A ) e. B } <-> ( y e. CC /\ ( y - A ) e. B ) )
8		simpr 110	. . . . . 6 |- ( ( F : B --> C /\ A e. CC ) -> A e. CC )
9		simpl 109	. . . . . 6 |- ( ( y e. CC /\ ( y - A ) e. B ) -> y e. CC )
10	2	shftval 11535	. . . . . 6 |- ( ( A e. CC /\ y e. CC ) -> ( ( F shift A ) ` y ) = ( F ` ( y - A ) ) )
11	8, 9, 10	syl2an 289	. . . . 5 |- ( ( ( F : B --> C /\ A e. CC ) /\ ( y e. CC /\ ( y - A ) e. B ) ) -> ( ( F shift A ) ` y ) = ( F ` ( y - A ) ) )
12		simpl 109	. . . . . 6 |- ( ( F : B --> C /\ A e. CC ) -> F : B --> C )
13		simpr 110	. . . . . 6 |- ( ( y e. CC /\ ( y - A ) e. B ) -> ( y - A ) e. B )
14		ffvelcdm 5815	. . . . . 6 |- ( ( F : B --> C /\ ( y - A ) e. B ) -> ( F ` ( y - A ) ) e. C )
15	12, 13, 14	syl2an 289	. . . . 5 |- ( ( ( F : B --> C /\ A e. CC ) /\ ( y e. CC /\ ( y - A ) e. B ) ) -> ( F ` ( y - A ) ) e. C )
16	11, 15	eqeltrd 2311	. . . 4 |- ( ( ( F : B --> C /\ A e. CC ) /\ ( y e. CC /\ ( y - A ) e. B ) ) -> ( ( F shift A ) ` y ) e. C )
17	7, 16	sylan2b 287	. . 3 |- ( ( ( F : B --> C /\ A e. CC ) /\ y e. { x e. CC | ( x - A ) e. B } ) -> ( ( F shift A ) ` y ) e. C )
18	17	ralrimiva 2617	. 2 |- ( ( F : B --> C /\ A e. CC ) -> A. y e. { x e. CC | ( x - A ) e. B } ( ( F shift A ) ` y ) e. C )
19		ffnfv 5840	. 2 |- ( ( F shift A ) : { x e. CC | ( x - A ) e. B } --> C <-> ( ( F shift A ) Fn { x e. CC | ( x - A ) e. B } /\ A. y e. { x e. CC | ( x - A ) e. B } ( ( F shift A ) ` y ) e. C ) )
20	4, 18, 19	sylanbrc 417	1 |- ( ( F : B --> C /\ A e. CC ) -> ( F shift A ) : { x e. CC | ( x - A ) e. B } --> C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   _Vcvv 2815   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   CCcc 8141   - cmin 8460   shift cshi 11524

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-resscn 8235  ax-1cn 8236  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-sub 8462  df-shft 11525

This theorem is referenced by: (None)