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Theorem recan 12178
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 9086 . . . . 5  |-  1  e.  CC
2 oveq1 6124 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  A )  =  ( 1  x.  A ) )
32fveq2d 5767 . . . . . . 7  |-  ( x  =  1  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  (
1  x.  A ) ) )
4 oveq1 6124 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  B )  =  ( 1  x.  B ) )
54fveq2d 5767 . . . . . . 7  |-  ( x  =  1  ->  (
Re `  ( x  x.  B ) )  =  ( Re `  (
1  x.  B ) ) )
63, 5eqeq12d 2457 . . . . . 6  |-  ( x  =  1  ->  (
( Re `  (
x  x.  A ) )  =  ( Re
`  ( x  x.  B ) )  <->  ( Re `  ( 1  x.  A
) )  =  ( Re `  ( 1  x.  B ) ) ) )
76rspcv 3057 . . . . 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 ax-icn 9087 . . . . . . 7  |-  _i  e.  CC
109negcli 9406 . . . . . 6  |-  -u _i  e.  CC
11 oveq1 6124 . . . . . . . . 9  |-  ( x  =  -u _i  ->  (
x  x.  A )  =  ( -u _i  x.  A ) )
1211fveq2d 5767 . . . . . . . 8  |-  ( x  =  -u _i  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  ( -u _i  x.  A ) ) )
13 oveq1 6124 . . . . . . . . 9  |-  ( x  =  -u _i  ->  (
x  x.  B )  =  ( -u _i  x.  B ) )
1413fveq2d 5767 . . . . . . . 8  |-  ( x  =  -u _i  ->  (
Re `  ( x  x.  B ) )  =  ( Re `  ( -u _i  x.  B ) ) )
1512, 14eqeq12d 2457 . . . . . . 7  |-  ( x  =  -u _i  ->  (
( Re `  (
x  x.  A ) )  =  ( Re
`  ( x  x.  B ) )  <->  ( Re `  ( -u _i  x.  A ) )  =  ( Re `  ( -u _i  x.  B ) ) ) )
1615rspcv 3057 . . . . . 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 ) ) ) )
1710, 16ax-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 ) ) )
1817oveq2d 6133 . . . 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 ) ) ) )
198, 18oveq12d 6135 . . 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 ) ) ) ) )
20 replim 11959 . . . . 5  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )
21 mulid2 9127 . . . . . . . 8  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2221eqcomd 2448 . . . . . . 7  |-  ( A  e.  CC  ->  A  =  ( 1  x.  A ) )
2322fveq2d 5767 . . . . . 6  |-  ( A  e.  CC  ->  (
Re `  A )  =  ( Re `  ( 1  x.  A
) ) )
24 imre 11951 . . . . . . 7  |-  ( A  e.  CC  ->  (
Im `  A )  =  ( Re `  ( -u _i  x.  A
) ) )
2524oveq2d 6133 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( Im `  A ) )  =  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) )
2623, 25oveq12d 6135 . . . . 5  |-  ( A  e.  CC  ->  (
( Re `  A
)  +  ( _i  x.  ( Im `  A ) ) )  =  ( ( Re
`  ( 1  x.  A ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) ) )
2720, 26eqtrd 2475 . . . 4  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  ( 1  x.  A ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) ) )
28 replim 11959 . . . . 5  |-  ( B  e.  CC  ->  B  =  ( ( Re
`  B )  +  ( _i  x.  (
Im `  B )
) ) )
29 mulid2 9127 . . . . . . . 8  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
3029eqcomd 2448 . . . . . . 7  |-  ( B  e.  CC  ->  B  =  ( 1  x.  B ) )
3130fveq2d 5767 . . . . . 6  |-  ( B  e.  CC  ->  (
Re `  B )  =  ( Re `  ( 1  x.  B
) ) )
32 imre 11951 . . . . . . 7  |-  ( B  e.  CC  ->  (
Im `  B )  =  ( Re `  ( -u _i  x.  B
) ) )
3332oveq2d 6133 . . . . . 6  |-  ( B  e.  CC  ->  (
_i  x.  ( Im `  B ) )  =  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) )
3431, 33oveq12d 6135 . . . . 5  |-  ( B  e.  CC  ->  (
( Re `  B
)  +  ( _i  x.  ( Im `  B ) ) )  =  ( ( Re
`  ( 1  x.  B ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) ) )
3528, 34eqtrd 2475 . . . 4  |-  ( B  e.  CC  ->  B  =  ( ( Re
`  ( 1  x.  B ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) ) )
3627, 35eqeqan12d 2458 . . 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 ) ) ) ) ) )
3719, 36syl5ibr 214 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) )  ->  A  =  B ) )
38 oveq2 6125 . . . 4  |-  ( A  =  B  ->  (
x  x.  A )  =  ( x  x.  B ) )
3938fveq2d 5767 . . 3  |-  ( A  =  B  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) ) )
4039ralrimivw 2797 . 2  |-  ( A  =  B  ->  A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) ) )
4137, 40impbid1 196 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    <-> wb 178    /\ wa 360    = wceq 1654    e. wcel 1728   A.wral 2712   ` cfv 5489  (class class class)co 6117   CCcc 9026   1c1 9029   _ici 9030    + caddc 9031    x. cmul 9033   -ucneg 9330   Recre 11940   Imcim 11941
This theorem is referenced by:  lnopunilem2  23552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-po 4538  df-so 4539  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-riota 6585  df-er 6941  df-en 7146  df-dom 7147  df-sdom 7148  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-div 9716  df-2 10096  df-cj 11942  df-re 11943  df-im 11944
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