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Theorem crim 12596
Description: The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
Assertion
Ref Expression
crim  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B )

Proof of Theorem crim
StepHypRef Expression
1 recn 9364 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 9333 . . . . 5  |-  _i  e.  CC
3 recn 9364 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 9358 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
52, 3, 4sylancr 663 . . . 4  |-  ( B  e.  RR  ->  (
_i  x.  B )  e.  CC )
6 addcl 9356 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  +  ( _i  x.  B
) )  e.  CC )
71, 5, 6syl2an 477 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  ( _i  x.  B ) )  e.  CC )
8 imval 12588 . . 3  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
Im `  ( A  +  ( _i  x.  B ) ) )  =  ( Re `  ( ( A  +  ( _i  x.  B
) )  /  _i ) ) )
97, 8syl 16 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  ( Re `  ( ( A  +  ( _i  x.  B
) )  /  _i ) ) )
102, 4mpan 670 . . . . . 6  |-  ( B  e.  CC  ->  (
_i  x.  B )  e.  CC )
11 ine0 9772 . . . . . . 7  |-  _i  =/=  0
12 divdir 10009 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC  /\  ( _i  e.  CC  /\  _i  =/=  0 ) )  ->  ( ( A  +  ( _i  x.  B ) )  /  _i )  =  (
( A  /  _i )  +  ( (
_i  x.  B )  /  _i ) ) )
13123expa 1187 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  /\  ( _i  e.  CC  /\  _i  =/=  0
) )  ->  (
( A  +  ( _i  x.  B ) )  /  _i )  =  ( ( A  /  _i )  +  ( ( _i  x.  B )  /  _i ) ) )
142, 11, 13mpanr12 685 . . . . . 6  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( ( A  +  ( _i  x.  B ) )  /  _i )  =  (
( A  /  _i )  +  ( (
_i  x.  B )  /  _i ) ) )
1510, 14sylan2 474 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( _i  x.  B
) )  /  _i )  =  ( ( A  /  _i )  +  ( ( _i  x.  B )  /  _i ) ) )
16 divrec2 10003 . . . . . . . 8  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  ( A  /  _i )  =  ( ( 1  /  _i )  x.  A
) )
172, 11, 16mp3an23 1306 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  /  _i )  =  ( ( 1  /  _i )  x.  A
) )
18 irec 11957 . . . . . . . . 9  |-  ( 1  /  _i )  = 
-u _i
1918oveq1i 6096 . . . . . . . 8  |-  ( ( 1  /  _i )  x.  A )  =  ( -u _i  x.  A )
2019a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  /  _i )  x.  A )  =  ( -u _i  x.  A ) )
21 mulneg12 9775 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A
) )
222, 21mpan 670 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A ) )
2317, 20, 223eqtrd 2474 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  _i )  =  ( _i  x.  -u A
) )
24 divcan3 10010 . . . . . . 7  |-  ( ( B  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  B
)  /  _i )  =  B )
252, 11, 24mp3an23 1306 . . . . . 6  |-  ( B  e.  CC  ->  (
( _i  x.  B
)  /  _i )  =  B )
2623, 25oveqan12d 6105 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  /  _i )  +  (
( _i  x.  B
)  /  _i ) )  =  ( ( _i  x.  -u A
)  +  B ) )
27 negcl 9602 . . . . . . 7  |-  ( A  e.  CC  ->  -u A  e.  CC )
28 mulcl 9358 . . . . . . 7  |-  ( ( _i  e.  CC  /\  -u A  e.  CC )  ->  ( _i  x.  -u A )  e.  CC )
292, 27, 28sylancr 663 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  e.  CC )
30 addcom 9547 . . . . . 6  |-  ( ( ( _i  x.  -u A
)  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  -u A )  +  B
)  =  ( B  +  ( _i  x.  -u A ) ) )
3129, 30sylan 471 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  -u A )  +  B
)  =  ( B  +  ( _i  x.  -u A ) ) )
3215, 26, 313eqtrrd 2475 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  ( _i  x.  -u A
) )  =  ( ( A  +  ( _i  x.  B ) )  /  _i ) )
331, 3, 32syl2an 477 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  ( _i  x.  -u A
) )  =  ( ( A  +  ( _i  x.  B ) )  /  _i ) )
3433fveq2d 5690 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( B  +  ( _i  x.  -u A ) ) )  =  ( Re
`  ( ( A  +  ( _i  x.  B ) )  /  _i ) ) )
35 id 22 . . 3  |-  ( B  e.  RR  ->  B  e.  RR )
36 renegcl 9664 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
37 crre 12595 . . 3  |-  ( ( B  e.  RR  /\  -u A  e.  RR )  ->  ( Re `  ( B  +  (
_i  x.  -u A ) ) )  =  B )
3835, 36, 37syl2anr 478 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( B  +  ( _i  x.  -u A ) ) )  =  B )
399, 34, 383eqtr2d 2476 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275   _ici 9276    + caddc 9277    x. cmul 9279   -ucneg 9588    / cdiv 9985   Recre 12578   Imcim 12579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-2 10372  df-cj 12580  df-re 12581  df-im 12582
This theorem is referenced by:  replim  12597  reim0  12599  remullem  12609  imcj  12613  imneg  12614  imadd  12615  imi  12638  crimi  12674  crimd  12713  absreimsq  12773  4sqlem4  14005  logneg  22011  lognegb  22013  basellem3  22395  2sqlem2  22678  cnre2csqima  26293
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