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Theorem recan 11113
Description: Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)
Assertion
Ref Expression
recan  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) )  <-> 
A  =  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem recan
StepHypRef Expression
1 ax-1cn 7903 . . . . 5  |-  1  e.  CC
2 oveq1 5881 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  A )  =  ( 1  x.  A ) )
32fveq2d 5519 . . . . . . 7  |-  ( x  =  1  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  (
1  x.  A ) ) )
4 oveq1 5881 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  B )  =  ( 1  x.  B ) )
54fveq2d 5519 . . . . . . 7  |-  ( x  =  1  ->  (
Re `  ( x  x.  B ) )  =  ( Re `  (
1  x.  B ) ) )
63, 5eqeq12d 2192 . . . . . 6  |-  ( x  =  1  ->  (
( Re `  (
x  x.  A ) )  =  ( Re
`  ( x  x.  B ) )  <->  ( Re `  ( 1  x.  A
) )  =  ( Re `  ( 1  x.  B ) ) ) )
76rspcv 2837 . . . . 5  |-  ( 1  e.  CC  ->  ( A. x  e.  CC  ( Re `  ( x  x.  A ) )  =  ( Re `  ( x  x.  B
) )  ->  (
Re `  ( 1  x.  A ) )  =  ( Re `  (
1  x.  B ) ) ) )
81, 7ax-mp 5 . . . 4  |-  ( A. x  e.  CC  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) )  ->  ( Re `  ( 1  x.  A
) )  =  ( Re `  ( 1  x.  B ) ) )
9 negicn 8156 . . . . . 6  |-  -u _i  e.  CC
10 oveq1 5881 . . . . . . . . 9  |-  ( x  =  -u _i  ->  (
x  x.  A )  =  ( -u _i  x.  A ) )
1110fveq2d 5519 . . . . . . . 8  |-  ( x  =  -u _i  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  ( -u _i  x.  A ) ) )
12 oveq1 5881 . . . . . . . . 9  |-  ( x  =  -u _i  ->  (
x  x.  B )  =  ( -u _i  x.  B ) )
1312fveq2d 5519 . . . . . . . 8  |-  ( x  =  -u _i  ->  (
Re `  ( x  x.  B ) )  =  ( Re `  ( -u _i  x.  B ) ) )
1411, 13eqeq12d 2192 . . . . . . 7  |-  ( x  =  -u _i  ->  (
( Re `  (
x  x.  A ) )  =  ( Re
`  ( x  x.  B ) )  <->  ( Re `  ( -u _i  x.  A ) )  =  ( Re `  ( -u _i  x.  B ) ) ) )
1514rspcv 2837 . . . . . 6  |-  ( -u _i  e.  CC  ->  ( A. x  e.  CC  ( Re `  ( x  x.  A ) )  =  ( Re `  ( x  x.  B
) )  ->  (
Re `  ( -u _i  x.  A ) )  =  ( Re `  ( -u _i  x.  B ) ) ) )
169, 15ax-mp 5 . . . . 5  |-  ( A. x  e.  CC  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) )  ->  ( Re `  ( -u _i  x.  A ) )  =  ( Re `  ( -u _i  x.  B ) ) )
1716oveq2d 5890 . . . 4  |-  ( A. x  e.  CC  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) )  ->  ( _i  x.  ( Re `  ( -u _i  x.  A ) ) )  =  ( _i  x.  ( Re
`  ( -u _i  x.  B ) ) ) )
188, 17oveq12d 5892 . . 3  |-  ( A. x  e.  CC  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) )  ->  ( (
Re `  ( 1  x.  A ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) )  =  ( ( Re `  ( 1  x.  B ) )  +  ( _i  x.  ( Re `  ( -u _i  x.  B ) ) ) ) )
19 replim 10863 . . . . 5  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )
20 mullid 7954 . . . . . . . 8  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2120eqcomd 2183 . . . . . . 7  |-  ( A  e.  CC  ->  A  =  ( 1  x.  A ) )
2221fveq2d 5519 . . . . . 6  |-  ( A  e.  CC  ->  (
Re `  A )  =  ( Re `  ( 1  x.  A
) ) )
23 imre 10855 . . . . . . 7  |-  ( A  e.  CC  ->  (
Im `  A )  =  ( Re `  ( -u _i  x.  A
) ) )
2423oveq2d 5890 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( Im `  A ) )  =  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) )
2522, 24oveq12d 5892 . . . . 5  |-  ( A  e.  CC  ->  (
( Re `  A
)  +  ( _i  x.  ( Im `  A ) ) )  =  ( ( Re
`  ( 1  x.  A ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) ) )
2619, 25eqtrd 2210 . . . 4  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  ( 1  x.  A ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) ) )
27 replim 10863 . . . . 5  |-  ( B  e.  CC  ->  B  =  ( ( Re
`  B )  +  ( _i  x.  (
Im `  B )
) ) )
28 mullid 7954 . . . . . . . 8  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
2928eqcomd 2183 . . . . . . 7  |-  ( B  e.  CC  ->  B  =  ( 1  x.  B ) )
3029fveq2d 5519 . . . . . 6  |-  ( B  e.  CC  ->  (
Re `  B )  =  ( Re `  ( 1  x.  B
) ) )
31 imre 10855 . . . . . . 7  |-  ( B  e.  CC  ->  (
Im `  B )  =  ( Re `  ( -u _i  x.  B
) ) )
3231oveq2d 5890 . . . . . 6  |-  ( B  e.  CC  ->  (
_i  x.  ( Im `  B ) )  =  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) )
3330, 32oveq12d 5892 . . . . 5  |-  ( B  e.  CC  ->  (
( Re `  B
)  +  ( _i  x.  ( Im `  B ) ) )  =  ( ( Re
`  ( 1  x.  B ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) ) )
3427, 33eqtrd 2210 . . . 4  |-  ( B  e.  CC  ->  B  =  ( ( Re
`  ( 1  x.  B ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) ) )
3526, 34eqeqan12d 2193 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  B  <-> 
( ( Re `  ( 1  x.  A
) )  +  ( _i  x.  ( Re
`  ( -u _i  x.  A ) ) ) )  =  ( ( Re `  ( 1  x.  B ) )  +  ( _i  x.  ( Re `  ( -u _i  x.  B ) ) ) ) ) )
3618, 35imbitrrid 156 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) )  ->  A  =  B ) )
37 oveq2 5882 . . . 4  |-  ( A  =  B  ->  (
x  x.  A )  =  ( x  x.  B ) )
3837fveq2d 5519 . . 3  |-  ( A  =  B  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) ) )
3938ralrimivw 2551 . 2  |-  ( A  =  B  ->  A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) ) )
4036, 39impbid1 142 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   ` cfv 5216  (class class class)co 5874   CCcc 7808   1c1 7811   _ici 7812    + caddc 7813    x. cmul 7815   -ucneg 8127   Recre 10844   Imcim 10845
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-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928
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-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-po 4296  df-iso 4297  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-2 8976  df-cj 10846  df-re 10847  df-im 10848
This theorem is referenced by: (None)
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