ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recan Unicode version

Theorem recan 10369
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 7341 . . . . 5  |-  1  e.  CC
2 oveq1 5598 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  A )  =  ( 1  x.  A ) )
32fveq2d 5257 . . . . . . 7  |-  ( x  =  1  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  (
1  x.  A ) ) )
4 oveq1 5598 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  B )  =  ( 1  x.  B ) )
54fveq2d 5257 . . . . . . 7  |-  ( x  =  1  ->  (
Re `  ( x  x.  B ) )  =  ( Re `  (
1  x.  B ) ) )
63, 5eqeq12d 2097 . . . . . 6  |-  ( x  =  1  ->  (
( Re `  (
x  x.  A ) )  =  ( Re
`  ( x  x.  B ) )  <->  ( Re `  ( 1  x.  A
) )  =  ( Re `  ( 1  x.  B ) ) ) )
76rspcv 2708 . . . . 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 7 . . . 4  |-  ( A. x  e.  CC  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) )  ->  ( Re `  ( 1  x.  A
) )  =  ( Re `  ( 1  x.  B ) ) )
9 negicn 7586 . . . . . 6  |-  -u _i  e.  CC
10 oveq1 5598 . . . . . . . . 9  |-  ( x  =  -u _i  ->  (
x  x.  A )  =  ( -u _i  x.  A ) )
1110fveq2d 5257 . . . . . . . 8  |-  ( x  =  -u _i  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  ( -u _i  x.  A ) ) )
12 oveq1 5598 . . . . . . . . 9  |-  ( x  =  -u _i  ->  (
x  x.  B )  =  ( -u _i  x.  B ) )
1312fveq2d 5257 . . . . . . . 8  |-  ( x  =  -u _i  ->  (
Re `  ( x  x.  B ) )  =  ( Re `  ( -u _i  x.  B ) ) )
1411, 13eqeq12d 2097 . . . . . . 7  |-  ( x  =  -u _i  ->  (
( Re `  (
x  x.  A ) )  =  ( Re
`  ( x  x.  B ) )  <->  ( Re `  ( -u _i  x.  A ) )  =  ( Re `  ( -u _i  x.  B ) ) ) )
1514rspcv 2708 . . . . . 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 7 . . . . 5  |-  ( A. x  e.  CC  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) )  ->  ( Re `  ( -u _i  x.  A ) )  =  ( Re `  ( -u _i  x.  B ) ) )
1716oveq2d 5607 . . . 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 5609 . . 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 10120 . . . . 5  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )
20 mulid2 7389 . . . . . . . 8  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2120eqcomd 2088 . . . . . . 7  |-  ( A  e.  CC  ->  A  =  ( 1  x.  A ) )
2221fveq2d 5257 . . . . . 6  |-  ( A  e.  CC  ->  (
Re `  A )  =  ( Re `  ( 1  x.  A
) ) )
23 imre 10112 . . . . . . 7  |-  ( A  e.  CC  ->  (
Im `  A )  =  ( Re `  ( -u _i  x.  A
) ) )
2423oveq2d 5607 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( Im `  A ) )  =  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) )
2522, 24oveq12d 5609 . . . . 5  |-  ( A  e.  CC  ->  (
( Re `  A
)  +  ( _i  x.  ( Im `  A ) ) )  =  ( ( Re
`  ( 1  x.  A ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) ) )
2619, 25eqtrd 2115 . . . 4  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  ( 1  x.  A ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) ) )
27 replim 10120 . . . . 5  |-  ( B  e.  CC  ->  B  =  ( ( Re
`  B )  +  ( _i  x.  (
Im `  B )
) ) )
28 mulid2 7389 . . . . . . . 8  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
2928eqcomd 2088 . . . . . . 7  |-  ( B  e.  CC  ->  B  =  ( 1  x.  B ) )
3029fveq2d 5257 . . . . . 6  |-  ( B  e.  CC  ->  (
Re `  B )  =  ( Re `  ( 1  x.  B
) ) )
31 imre 10112 . . . . . . 7  |-  ( B  e.  CC  ->  (
Im `  B )  =  ( Re `  ( -u _i  x.  B
) ) )
3231oveq2d 5607 . . . . . 6  |-  ( B  e.  CC  ->  (
_i  x.  ( Im `  B ) )  =  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) )
3330, 32oveq12d 5609 . . . . 5  |-  ( B  e.  CC  ->  (
( Re `  B
)  +  ( _i  x.  ( Im `  B ) ) )  =  ( ( Re
`  ( 1  x.  B ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) ) )
3427, 33eqtrd 2115 . . . 4  |-  ( B  e.  CC  ->  B  =  ( ( Re
`  ( 1  x.  B ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) ) )
3526, 34eqeqan12d 2098 . . 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, 35syl5ibr 154 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) )  ->  A  =  B ) )
37 oveq2 5599 . . . 4  |-  ( A  =  B  ->  (
x  x.  A )  =  ( x  x.  B ) )
3837fveq2d 5257 . . 3  |-  ( A  =  B  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) ) )
3938ralrimivw 2441 . 2  |-  ( A  =  B  ->  A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) ) )
4036, 39impbid1 140 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2353   ` cfv 4969  (class class class)co 5591   CCcc 7251   1c1 7254   _ici 7255    + caddc 7256    x. cmul 7258   -ucneg 7557   Recre 10101   Imcim 10102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-mulrcl 7347  ax-addcom 7348  ax-mulcom 7349  ax-addass 7350  ax-mulass 7351  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-1rid 7355  ax-0id 7356  ax-rnegex 7357  ax-precex 7358  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-apti 7363  ax-pre-ltadd 7364  ax-pre-mulgt0 7365  ax-pre-mulext 7366
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-po 4087  df-iso 4088  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-reap 7952  df-ap 7959  df-div 8038  df-2 8375  df-cj 10103  df-re 10104  df-im 10105
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator