Theorem shftval2 10837

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 8158	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2	1	3adant3 1017	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - B ) e. CC )
3		addcl 7938	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A + C ) e. CC )
4	3	3adant2 1016	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + C ) e. CC )
5		shftfval.1	. . . 4 |- F e. _V
6	5	shftval 10836	. . 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 411	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( F shift ( A - B ) ) ` ( A + C ) ) = ( F ` ( ( A + C ) - ( A - B ) ) ) )
8		pnncan 8200	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( C + B ) )
9	8	3com23 1209	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( C + B ) )
10		addcom 8096	. . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
11	10	3adant1 1015	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
12	9, 11	eqtr4d 2213	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( A - B ) ) = ( B + C ) )
13	12	fveq2d 5521	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( F ` ( ( A + C ) - ( A - B ) ) ) = ( F ` ( B + C ) ) )
14	7, 13	eqtrd 2210	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( F shift ( A - B ) ) ` ( A + C ) ) = ( F ` ( B + C ) ) )

Syntax hints:   -> wi 4   /\ w3a 978   = wceq 1353   e. wcel 2148   _Vcvv 2739   ` cfv 5218  (class class class)co 5877   CCcc 7811   + caddc 7816   - cmin 8130   shift cshi 10825

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-resscn 7905  ax-1cn 7906  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-sub 8132  df-shft 10826

This theorem is referenced by:  shftval3  10838