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Theorem recan 11820
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 8795 . . . . 5  |-  1  e.  CC
2 oveq1 5865 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  A )  =  ( 1  x.  A ) )
32fveq2d 5529 . . . . . . 7  |-  ( x  =  1  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  (
1  x.  A ) ) )
4 oveq1 5865 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  B )  =  ( 1  x.  B ) )
54fveq2d 5529 . . . . . . 7  |-  ( x  =  1  ->  (
Re `  ( x  x.  B ) )  =  ( Re `  (
1  x.  B ) ) )
63, 5eqeq12d 2297 . . . . . 6  |-  ( x  =  1  ->  (
( Re `  (
x  x.  A ) )  =  ( Re
`  ( x  x.  B ) )  <->  ( Re `  ( 1  x.  A
) )  =  ( Re `  ( 1  x.  B ) ) ) )
76rspcv 2880 . . . . 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 8 . . . 4  |-  ( A. x  e.  CC  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) )  ->  ( Re `  ( 1  x.  A
) )  =  ( Re `  ( 1  x.  B ) ) )
9 ax-icn 8796 . . . . . . 7  |-  _i  e.  CC
109negcli 9114 . . . . . 6  |-  -u _i  e.  CC
11 oveq1 5865 . . . . . . . . 9  |-  ( x  =  -u _i  ->  (
x  x.  A )  =  ( -u _i  x.  A ) )
1211fveq2d 5529 . . . . . . . 8  |-  ( x  =  -u _i  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  ( -u _i  x.  A ) ) )
13 oveq1 5865 . . . . . . . . 9  |-  ( x  =  -u _i  ->  (
x  x.  B )  =  ( -u _i  x.  B ) )
1413fveq2d 5529 . . . . . . . 8  |-  ( x  =  -u _i  ->  (
Re `  ( x  x.  B ) )  =  ( Re `  ( -u _i  x.  B ) ) )
1512, 14eqeq12d 2297 . . . . . . 7  |-  ( x  =  -u _i  ->  (
( Re `  (
x  x.  A ) )  =  ( Re
`  ( x  x.  B ) )  <->  ( Re `  ( -u _i  x.  A ) )  =  ( Re `  ( -u _i  x.  B ) ) ) )
1615rspcv 2880 . . . . . 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 8 . . . . 5  |-  ( A. x  e.  CC  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) )  ->  ( Re `  ( -u _i  x.  A ) )  =  ( Re `  ( -u _i  x.  B ) ) )
1817oveq2d 5874 . . . 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 5876 . . 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 11601 . . . . 5  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )
21 mulid2 8836 . . . . . . . 8  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2221eqcomd 2288 . . . . . . 7  |-  ( A  e.  CC  ->  A  =  ( 1  x.  A ) )
2322fveq2d 5529 . . . . . 6  |-  ( A  e.  CC  ->  (
Re `  A )  =  ( Re `  ( 1  x.  A
) ) )
24 imre 11593 . . . . . . 7  |-  ( A  e.  CC  ->  (
Im `  A )  =  ( Re `  ( -u _i  x.  A
) ) )
2524oveq2d 5874 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( Im `  A ) )  =  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) )
2623, 25oveq12d 5876 . . . . 5  |-  ( A  e.  CC  ->  (
( Re `  A
)  +  ( _i  x.  ( Im `  A ) ) )  =  ( ( Re
`  ( 1  x.  A ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) ) )
2720, 26eqtrd 2315 . . . 4  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  ( 1  x.  A ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) ) )
28 replim 11601 . . . . 5  |-  ( B  e.  CC  ->  B  =  ( ( Re
`  B )  +  ( _i  x.  (
Im `  B )
) ) )
29 mulid2 8836 . . . . . . . 8  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
3029eqcomd 2288 . . . . . . 7  |-  ( B  e.  CC  ->  B  =  ( 1  x.  B ) )
3130fveq2d 5529 . . . . . 6  |-  ( B  e.  CC  ->  (
Re `  B )  =  ( Re `  ( 1  x.  B
) ) )
32 imre 11593 . . . . . . 7  |-  ( B  e.  CC  ->  (
Im `  B )  =  ( Re `  ( -u _i  x.  B
) ) )
3332oveq2d 5874 . . . . . 6  |-  ( B  e.  CC  ->  (
_i  x.  ( Im `  B ) )  =  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) )
3431, 33oveq12d 5876 . . . . 5  |-  ( B  e.  CC  ->  (
( Re `  B
)  +  ( _i  x.  ( Im `  B ) ) )  =  ( ( Re
`  ( 1  x.  B ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) ) )
3528, 34eqtrd 2315 . . . 4  |-  ( B  e.  CC  ->  B  =  ( ( Re
`  ( 1  x.  B ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) ) )
3627, 35eqeqan12d 2298 . . 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 212 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) )  ->  A  =  B ) )
38 oveq2 5866 . . . 4  |-  ( A  =  B  ->  (
x  x.  A )  =  ( x  x.  B ) )
3938fveq2d 5529 . . 3  |-  ( A  =  B  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) ) )
4039ralrimivw 2627 . 2  |-  ( A  =  B  ->  A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) ) )
4137, 40impbid1 194 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742   -ucneg 9038   Recre 11582   Imcim 11583
This theorem is referenced by:  lnopunilem2  22591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586
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