Theorem shftf 11573

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 5528	. . 3 |- ( F : B --> C -> F Fn B )
2		shftfval.1	. . . 4 |- F e. _V
3	2	shftfn 11567	. . 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 6082	. . . . . 6 |- ( x = y -> ( x - A ) = ( y - A ) )
6	5	eleq1d 2307	. . . . 5 |- ( x = y -> ( ( x - A ) e. B <-> ( y - A ) e. B ) )
7	6	elrab 2982	. . . 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 11568	. . . . . 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 5832	. . . . . 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 2315	. . . 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 2623	. 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 5857	. 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 421	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 1402   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   Fn wfn 5367   -->wf 5368   ` cfv 5372  (class class class)co 6075   CCcc 8167   - cmin 8487   shift cshi 11557

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-resscn 8261  ax-1cn 8262  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sub 8489  df-shft 11558

This theorem is referenced by: (None)