MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  recan Structured version   Unicode version

Theorem recan 13135
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 9551 . . . . 5  |-  1  e.  CC
2 oveq1 6292 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  A )  =  ( 1  x.  A ) )
32fveq2d 5870 . . . . . . 7  |-  ( x  =  1  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  (
1  x.  A ) ) )
4 oveq1 6292 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  B )  =  ( 1  x.  B ) )
54fveq2d 5870 . . . . . . 7  |-  ( x  =  1  ->  (
Re `  ( x  x.  B ) )  =  ( Re `  (
1  x.  B ) ) )
63, 5eqeq12d 2489 . . . . . 6  |-  ( x  =  1  ->  (
( Re `  (
x  x.  A ) )  =  ( Re
`  ( x  x.  B ) )  <->  ( Re `  ( 1  x.  A
) )  =  ( Re `  ( 1  x.  B ) ) ) )
76rspcv 3210 . . . . 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 9822 . . . . . 6  |-  -u _i  e.  CC
10 oveq1 6292 . . . . . . . . 9  |-  ( x  =  -u _i  ->  (
x  x.  A )  =  ( -u _i  x.  A ) )
1110fveq2d 5870 . . . . . . . 8  |-  ( x  =  -u _i  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  ( -u _i  x.  A ) ) )
12 oveq1 6292 . . . . . . . . 9  |-  ( x  =  -u _i  ->  (
x  x.  B )  =  ( -u _i  x.  B ) )
1312fveq2d 5870 . . . . . . . 8  |-  ( x  =  -u _i  ->  (
Re `  ( x  x.  B ) )  =  ( Re `  ( -u _i  x.  B ) ) )
1411, 13eqeq12d 2489 . . . . . . 7  |-  ( x  =  -u _i  ->  (
( Re `  (
x  x.  A ) )  =  ( Re
`  ( x  x.  B ) )  <->  ( Re `  ( -u _i  x.  A ) )  =  ( Re `  ( -u _i  x.  B ) ) ) )
1514rspcv 3210 . . . . . 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 6301 . . . 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 6303 . . 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 12915 . . . . 5  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )
20 mulid2 9595 . . . . . . . 8  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2120eqcomd 2475 . . . . . . 7  |-  ( A  e.  CC  ->  A  =  ( 1  x.  A ) )
2221fveq2d 5870 . . . . . 6  |-  ( A  e.  CC  ->  (
Re `  A )  =  ( Re `  ( 1  x.  A
) ) )
23 imre 12907 . . . . . . 7  |-  ( A  e.  CC  ->  (
Im `  A )  =  ( Re `  ( -u _i  x.  A
) ) )
2423oveq2d 6301 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( Im `  A ) )  =  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) )
2522, 24oveq12d 6303 . . . . 5  |-  ( A  e.  CC  ->  (
( Re `  A
)  +  ( _i  x.  ( Im `  A ) ) )  =  ( ( Re
`  ( 1  x.  A ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) ) )
2619, 25eqtrd 2508 . . . 4  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  ( 1  x.  A ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  A ) ) ) ) )
27 replim 12915 . . . . 5  |-  ( B  e.  CC  ->  B  =  ( ( Re
`  B )  +  ( _i  x.  (
Im `  B )
) ) )
28 mulid2 9595 . . . . . . . 8  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
2928eqcomd 2475 . . . . . . 7  |-  ( B  e.  CC  ->  B  =  ( 1  x.  B ) )
3029fveq2d 5870 . . . . . 6  |-  ( B  e.  CC  ->  (
Re `  B )  =  ( Re `  ( 1  x.  B
) ) )
31 imre 12907 . . . . . . 7  |-  ( B  e.  CC  ->  (
Im `  B )  =  ( Re `  ( -u _i  x.  B
) ) )
3231oveq2d 6301 . . . . . 6  |-  ( B  e.  CC  ->  (
_i  x.  ( Im `  B ) )  =  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) )
3330, 32oveq12d 6303 . . . . 5  |-  ( B  e.  CC  ->  (
( Re `  B
)  +  ( _i  x.  ( Im `  B ) ) )  =  ( ( Re
`  ( 1  x.  B ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) ) )
3427, 33eqtrd 2508 . . . 4  |-  ( B  e.  CC  ->  B  =  ( ( Re
`  ( 1  x.  B ) )  +  ( _i  x.  (
Re `  ( -u _i  x.  B ) ) ) ) )
3526, 34eqeqan12d 2490 . . 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 221 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) )  ->  A  =  B ) )
37 oveq2 6293 . . . 4  |-  ( A  =  B  ->  (
x  x.  A )  =  ( x  x.  B ) )
3837fveq2d 5870 . . 3  |-  ( A  =  B  ->  (
Re `  ( x  x.  A ) )  =  ( Re `  (
x  x.  B ) ) )
3938ralrimivw 2879 . 2  |-  ( A  =  B  ->  A. x  e.  CC  ( Re `  ( x  x.  A
) )  =  ( Re `  ( x  x.  B ) ) )
4036, 39impbid1 203 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   ` cfv 5588  (class class class)co 6285   CCcc 9491   1c1 9494   _ici 9495    + caddc 9496    x. cmul 9498   -ucneg 9807   Recre 12896   Imcim 12897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-2 10595  df-cj 12898  df-re 12899  df-im 12900
This theorem is referenced by:  lnopunilem2  26703
  Copyright terms: Public domain W3C validator