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Theorem crim 11477
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 8707 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 8676 . . . . 5  |-  _i  e.  CC
3 recn 8707 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 ax-mulcl 8679 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
52, 3, 4sylancr 647 . . . 4  |-  ( B  e.  RR  ->  (
_i  x.  B )  e.  CC )
6 ax-addcl 8677 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  +  ( _i  x.  B
) )  e.  CC )
71, 5, 6syl2an 465 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  ( _i  x.  B ) )  e.  CC )
8 imval 11469 . . 3  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
Im `  ( A  +  ( _i  x.  B ) ) )  =  ( Re `  ( ( A  +  ( _i  x.  B
) )  /  _i ) ) )
97, 8syl 17 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  ( Re `  ( ( A  +  ( _i  x.  B
) )  /  _i ) ) )
102, 4mpan 654 . . . . . 6  |-  ( B  e.  CC  ->  (
_i  x.  B )  e.  CC )
11 ine0 9095 . . . . . . 7  |-  _i  =/=  0
12 divdir 9327 . . . . . . . 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 1156 . . . . . . 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 669 . . . . . 6  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( ( A  +  ( _i  x.  B ) )  /  _i )  =  (
( A  /  _i )  +  ( (
_i  x.  B )  /  _i ) ) )
1510, 14sylan2 462 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( _i  x.  B
) )  /  _i )  =  ( ( A  /  _i )  +  ( ( _i  x.  B )  /  _i ) ) )
16 divrec2 9321 . . . . . . . 8  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  ( A  /  _i )  =  ( ( 1  /  _i )  x.  A
) )
172, 11, 16mp3an23 1274 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  /  _i )  =  ( ( 1  /  _i )  x.  A
) )
18 irec 11080 . . . . . . . . 9  |-  ( 1  /  _i )  = 
-u _i
1918oveq1i 5720 . . . . . . . 8  |-  ( ( 1  /  _i )  x.  A )  =  ( -u _i  x.  A )
2019a1i 12 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  /  _i )  x.  A )  =  ( -u _i  x.  A ) )
21 mulneg12 9098 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A
) )
222, 21mpan 654 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A ) )
2317, 20, 223eqtrd 2289 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  _i )  =  ( _i  x.  -u A
) )
24 divcan3 9328 . . . . . . 7  |-  ( ( B  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  B
)  /  _i )  =  B )
252, 11, 24mp3an23 1274 . . . . . 6  |-  ( B  e.  CC  ->  (
( _i  x.  B
)  /  _i )  =  B )
2623, 25oveqan12d 5729 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  /  _i )  +  (
( _i  x.  B
)  /  _i ) )  =  ( ( _i  x.  -u A
)  +  B ) )
27 negcl 8932 . . . . . . 7  |-  ( A  e.  CC  ->  -u A  e.  CC )
28 ax-mulcl 8679 . . . . . . 7  |-  ( ( _i  e.  CC  /\  -u A  e.  CC )  ->  ( _i  x.  -u A )  e.  CC )
292, 27, 28sylancr 647 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  e.  CC )
30 addcom 8878 . . . . . 6  |-  ( ( ( _i  x.  -u A
)  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  -u A )  +  B
)  =  ( B  +  ( _i  x.  -u A ) ) )
3129, 30sylan 459 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  -u A )  +  B
)  =  ( B  +  ( _i  x.  -u A ) ) )
3215, 26, 313eqtrrd 2290 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  ( _i  x.  -u A
) )  =  ( ( A  +  ( _i  x.  B ) )  /  _i ) )
331, 3, 32syl2an 465 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  ( _i  x.  -u A
) )  =  ( ( A  +  ( _i  x.  B ) )  /  _i ) )
3433fveq2d 5381 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( B  +  ( _i  x.  -u A ) ) )  =  ( Re
`  ( ( A  +  ( _i  x.  B ) )  /  _i ) ) )
35 id 21 . . 3  |-  ( B  e.  RR  ->  B  e.  RR )
36 renegcl 8990 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
37 crre 11476 . . 3  |-  ( ( B  e.  RR  /\  -u A  e.  RR )  ->  ( Re `  ( B  +  (
_i  x.  -u A ) ) )  =  B )
3835, 36, 37syl2anr 466 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( B  +  ( _i  x.  -u A ) ) )  =  B )
399, 34, 383eqtr2d 2291 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   ` cfv 4592  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618   _ici 8619    + caddc 8620    x. cmul 8622   -ucneg 8918    / cdiv 9303   Recre 11459   Imcim 11460
This theorem is referenced by:  replim  11478  reim0  11480  remullem  11490  imcj  11494  imneg  11495  imadd  11496  imi  11519  crimi  11555  crimd  11594  absreimsq  11654  4sqlem4  12873  logneg  19773  lognegb  19775  basellem3  20152  2sqlem2  20435
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-2 9684  df-cj 11461  df-re 11462  df-im 11463
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