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Theorem shftval2 10610
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 7973 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
213adant3 1001 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  B )  e.  CC )
3 addcl 7757 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  +  C
)  e.  CC )
433adant2 1000 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  C )  e.  CC )
5 shftfval.1 . . . 4  |-  F  e. 
_V
65shftval 10609 . . 3  |-  ( ( ( A  -  B
)  e.  CC  /\  ( A  +  C
)  e.  CC )  ->  ( ( F 
shift  ( A  -  B
) ) `  ( A  +  C )
)  =  ( F `
 ( ( A  +  C )  -  ( A  -  B
) ) ) )
72, 4, 6syl2anc 408 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( F  shift  ( A  -  B ) ) `
 ( A  +  C ) )  =  ( F `  (
( A  +  C
)  -  ( A  -  B ) ) ) )
8 pnncan 8015 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  B  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( C  +  B ) )
983com23 1187 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( C  +  B ) )
10 addcom 7911 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C
)  =  ( C  +  B ) )
11103adant1 999 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  =  ( C  +  B ) )
129, 11eqtr4d 2175 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( B  +  C ) )
1312fveq2d 5425 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( F `  ( ( A  +  C )  -  ( A  -  B ) ) )  =  ( F `  ( B  +  C
) ) )
147, 13eqtrd 2172 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 962    = wceq 1331    e. wcel 1480   _Vcvv 2686   ` cfv 5123  (class class class)co 5774   CCcc 7630    + caddc 7635    - cmin 7945    shift cshi 10598
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-resscn 7724  ax-1cn 7725  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7947  df-shft 10599
This theorem is referenced by:  shftval3  10611
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