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Theorem shftval2 10783
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 8111 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
213adant3 1012 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  B )  e.  CC )
3 addcl 7892 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  +  C
)  e.  CC )
433adant2 1011 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  C )  e.  CC )
5 shftfval.1 . . . 4  |-  F  e. 
_V
65shftval 10782 . . 3  |-  ( ( ( A  -  B
)  e.  CC  /\  ( A  +  C
)  e.  CC )  ->  ( ( F 
shift  ( A  -  B
) ) `  ( A  +  C )
)  =  ( F `
 ( ( A  +  C )  -  ( A  -  B
) ) ) )
72, 4, 6syl2anc 409 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( F  shift  ( A  -  B ) ) `
 ( A  +  C ) )  =  ( F `  (
( A  +  C
)  -  ( A  -  B ) ) ) )
8 pnncan 8153 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  B  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( C  +  B ) )
983com23 1204 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( C  +  B ) )
10 addcom 8049 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C
)  =  ( C  +  B ) )
11103adant1 1010 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  =  ( C  +  B ) )
129, 11eqtr4d 2206 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  -  ( A  -  B ) )  =  ( B  +  C ) )
1312fveq2d 5498 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( F `  ( ( A  +  C )  -  ( A  -  B ) ) )  =  ( F `  ( B  +  C
) ) )
147, 13eqtrd 2203 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 973    = wceq 1348    e. wcel 2141   _Vcvv 2730   ` cfv 5196  (class class class)co 5851   CCcc 7765    + caddc 7770    - cmin 8083    shift cshi 10771
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-resscn 7859  ax-1cn 7860  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-addcom 7867  ax-addass 7869  ax-distr 7871  ax-i2m1 7872  ax-0id 7875  ax-rnegex 7876  ax-cnre 7878
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-sub 8085  df-shft 10772
This theorem is referenced by:  shftval3  10784
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