Theorem shftval2 11591

Description: Value of a sequence shifted by A - B. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Hypothesis

Ref	Expression
shftfval.1	|- F e. _V

Assertion

Ref	Expression
shftval2	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( F shift ( A - B ) ) ` ( A + C ) ) = ( F ` ( B + C ) ) )

Proof of Theorem shftval2

Step	Hyp	Ref	Expression
1		subcl 8525	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2	1	3adant3 1048	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - B ) e. CC )
3		addcl 8304	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A + C ) e. CC )
4	3	3adant2 1047	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + C ) e. CC )
5		shftfval.1	. . . 4 |- F e. _V
6	5	shftval 11590	. . 3 |- ( ( ( A - B ) e. CC /\ ( A + C ) e. CC ) -> ( ( F shift ( A - B ) ) ` ( A + C ) ) = ( F ` ( ( A + C ) - ( A - B ) ) ) )
7	2, 4, 6	syl2anc 415	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( F shift ( A - B ) ) ` ( A + C ) ) = ( F ` ( ( A + C ) - ( A - B ) ) ) )
8		pnncan 8567	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( C + B ) )
9	8	3com23 1240	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( C + B ) )
10		addcom 8463	. . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
11	10	3adant1 1046	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
12	9, 11	eqtr4d 2274	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( B + C ) )
13	12	fveq2d 5699	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( F ` ( ( A + C ) - ( A - B ) ) ) = ( F ` ( B + C ) ) )
14	7, 13	eqtrd 2271	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( F shift ( A - B ) ) ` ( A + C ) ) = ( F ` ( B + C ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   ` cfv 5377  (class class class)co 6085   CCcc 8177   + caddc 8182   - cmin 8497   shift cshi 11579

This proof depends on 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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-resscn 8271  ax-1cn 8272  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-sub 8499  df-shft 11580

This theorem is used by:  shftval3  11592