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Theorem cjreim2 11853
Description: The conjugate of the representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
Assertion
Ref Expression
cjreim2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  -  ( _i  x.  B ) ) )  =  ( A  +  ( _i  x.  B
) ) )

Proof of Theorem cjreim2
StepHypRef Expression
1 cjreim 11852 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  +  ( _i  x.  B ) ) )  =  ( A  -  ( _i  x.  B
) ) )
21fveq2d 5636 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  (
* `  ( A  +  ( _i  x.  B ) ) ) )  =  ( * `
 ( A  -  ( _i  x.  B
) ) ) )
3 simpl 443 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
43recnd 9008 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
5 ax-icn 8943 . . . . . 6  |-  _i  e.  CC
65a1i 10 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  _i  e.  CC )
7 simpr 447 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
87recnd 9008 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
96, 8mulcld 9002 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  B
)  e.  CC )
104, 9addcld 9001 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  ( _i  x.  B ) )  e.  CC )
11 cjcj 11832 . . 3  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
* `  ( * `  ( A  +  ( _i  x.  B ) ) ) )  =  ( A  +  ( _i  x.  B ) ) )
1210, 11syl 15 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  (
* `  ( A  +  ( _i  x.  B ) ) ) )  =  ( A  +  ( _i  x.  B ) ) )
132, 12eqtr3d 2400 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  -  ( _i  x.  B ) ) )  =  ( A  +  ( _i  x.  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   ` cfv 5358  (class class class)co 5981   CCcc 8882   RRcr 8883   _ici 8886    + caddc 8887    x. cmul 8889    - cmin 9184   *ccj 11788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-2 9951  df-cj 11791  df-re 11792  df-im 11793
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