Theorem shftval2 10783

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 8111	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2	1	3adant3 1012	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - B ) e. CC )
3		addcl 7892	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A + C ) e. CC )
4	3	3adant2 1011	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + C ) e. CC )
5		shftfval.1	. . . 4 |- F e. _V
6	5	shftval 10782	. . 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 409	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( F shift ( A - B ) ) ` ( A + C ) ) = ( F ` ( ( A + C ) - ( A - B ) ) ) )
8		pnncan 8153	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( C + B ) )
9	8	3com23 1204	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( C + B ) )
10		addcom 8049	. . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
11	10	3adant1 1010	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
12	9, 11	eqtr4d 2206	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( B + C ) )
13	12	fveq2d 5498	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( F ` ( ( A + C ) - ( A - B ) ) ) = ( F ` ( B + C ) ) )
14	7, 13	eqtrd 2203	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( F shift ( A - B ) ) ` ( A + C ) ) = ( F ` ( B + C ) ) )

Syntax hints:   -> wi 4   /\ w3a 973   = wceq 1348   e. wcel 2141   _Vcvv 2730   ` cfv 5196  (class class class)co 5851   CCcc 7765   + caddc 7770   - cmin 8083   shift cshi 10771

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-resscn 7859  ax-1cn 7860  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-addcom 7867  ax-addass 7869  ax-distr 7871  ax-i2m1 7872  ax-0id 7875  ax-rnegex 7876  ax-cnre 7878

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-sub 8085  df-shft 10772

This theorem is referenced by:  shftval3  10784