Theorem shftval2 10768

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 8097	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2	1	3adant3 1007	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - B ) e. CC )
3		addcl 7878	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A + C ) e. CC )
4	3	3adant2 1006	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + C ) e. CC )
5		shftfval.1	. . . 4 |- F e. _V
6	5	shftval 10767	. . 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 8139	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( C + B ) )
9	8	3com23 1199	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( C + B ) )
10		addcom 8035	. . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
11	10	3adant1 1005	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
12	9, 11	eqtr4d 2201	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( B + C ) )
13	12	fveq2d 5490	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( F ` ( ( A + C ) - ( A - B ) ) ) = ( F ` ( B + C ) ) )
14	7, 13	eqtrd 2198	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( F shift ( A - B ) ) ` ( A + C ) ) = ( F ` ( B + C ) ) )

Syntax hints:   -> wi 4   /\ w3a 968   = wceq 1343   e. wcel 2136   _Vcvv 2726   ` cfv 5188  (class class class)co 5842   CCcc 7751   + caddc 7756   - cmin 8069   shift cshi 10756

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071  df-shft 10757

This theorem is referenced by:  shftval3  10769