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Theorem shftval2 10314
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 7735 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
213adant3 964 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  B )  e.  CC )
3 addcl 7521 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  +  C
)  e.  CC )
433adant2 963 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  C )  e.  CC )
5 shftfval.1 . . . 4  |-  F  e. 
_V
65shftval 10313 . . 3  |-  ( ( ( A  -  B
)  e.  CC  /\  ( A  +  C
)  e.  CC )  ->  ( ( F 
shift  ( A  -  B
) ) `  ( A  +  C )
)  =  ( F `
 ( ( A  +  C )  -  ( A  -  B
) ) ) )
72, 4, 6syl2anc 404 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( F  shift  ( A  -  B ) ) `
 ( A  +  C ) )  =  ( F `  (
( A  +  C
)  -  ( A  -  B ) ) ) )
8 pnncan 7777 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  B  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( C  +  B ) )
983com23 1150 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( C  +  B ) )
10 addcom 7673 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C
)  =  ( C  +  B ) )
11103adant1 962 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  =  ( C  +  B ) )
129, 11eqtr4d 2124 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( B  +  C ) )
1312fveq2d 5322 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( F `  ( ( A  +  C )  -  ( A  -  B ) ) )  =  ( F `  ( B  +  C
) ) )
147, 13eqtrd 2121 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 925    = wceq 1290    e. wcel 1439   _Vcvv 2620   ` cfv 5028  (class class class)co 5666   CCcc 7402    + caddc 7407    - cmin 7707    shift cshi 10302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-resscn 7491  ax-1cn 7492  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-sub 7709  df-shft 10303
This theorem is referenced by:  shftval3  10315
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