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Theorem shftval2 9427
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
StepHypRef Expression
1 subcl 7210 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
213adant3 924 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  B )  e.  CC )
3 addcl 7006 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  +  C
)  e.  CC )
433adant2 923 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  C )  e.  CC )
5 shftfval.1 . . . 4  |-  F  e. 
_V
65shftval 9426 . . 3  |-  ( ( ( A  -  B
)  e.  CC  /\  ( A  +  C
)  e.  CC )  ->  ( ( F 
shift  ( A  -  B
) ) `  ( A  +  C )
)  =  ( F `
 ( ( A  +  C )  -  ( A  -  B
) ) ) )
72, 4, 6syl2anc 391 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( F  shift  ( A  -  B ) ) `
 ( A  +  C ) )  =  ( F `  (
( A  +  C
)  -  ( A  -  B ) ) ) )
8 pnncan 7252 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  B  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( C  +  B ) )
983com23 1110 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( C  +  B ) )
10 addcom 7150 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C
)  =  ( C  +  B ) )
11103adant1 922 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  =  ( C  +  B ) )
129, 11eqtr4d 2075 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( B  +  C ) )
1312fveq2d 5182 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( F `  ( ( A  +  C )  -  ( A  -  B ) ) )  =  ( F `  ( B  +  C
) ) )
147, 13eqtrd 2072 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( F  shift  ( A  -  B ) ) `
 ( A  +  C ) )  =  ( F `  ( B  +  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 885    = wceq 1243    e. wcel 1393   _Vcvv 2557   ` cfv 4902  (class class class)co 5512   CCcc 6887    + caddc 6892    - cmin 7182    shift cshi 9415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184  df-shft 9416
This theorem is referenced by:  shftval3  9428
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