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

Theorem crim 11875
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 9036 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 9005 . . . . 5  |-  _i  e.  CC
3 recn 9036 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 9030 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
52, 3, 4sylancr 645 . . . 4  |-  ( B  e.  RR  ->  (
_i  x.  B )  e.  CC )
6 addcl 9028 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  +  ( _i  x.  B
) )  e.  CC )
71, 5, 6syl2an 464 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  ( _i  x.  B ) )  e.  CC )
8 imval 11867 . . 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 652 . . . . . 6  |-  ( B  e.  CC  ->  (
_i  x.  B )  e.  CC )
11 ine0 9425 . . . . . . 7  |-  _i  =/=  0
12 divdir 9657 . . . . . . . 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 1153 . . . . . . 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 667 . . . . . 6  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( ( A  +  ( _i  x.  B ) )  /  _i )  =  (
( A  /  _i )  +  ( (
_i  x.  B )  /  _i ) ) )
1510, 14sylan2 461 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( _i  x.  B
) )  /  _i )  =  ( ( A  /  _i )  +  ( ( _i  x.  B )  /  _i ) ) )
16 divrec2 9651 . . . . . . . 8  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  ( A  /  _i )  =  ( ( 1  /  _i )  x.  A
) )
172, 11, 16mp3an23 1271 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  /  _i )  =  ( ( 1  /  _i )  x.  A
) )
18 irec 11435 . . . . . . . . 9  |-  ( 1  /  _i )  = 
-u _i
1918oveq1i 6050 . . . . . . . 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 9428 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A
) )
222, 21mpan 652 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A ) )
2317, 20, 223eqtrd 2440 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  _i )  =  ( _i  x.  -u A
) )
24 divcan3 9658 . . . . . . 7  |-  ( ( B  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  B
)  /  _i )  =  B )
252, 11, 24mp3an23 1271 . . . . . 6  |-  ( B  e.  CC  ->  (
( _i  x.  B
)  /  _i )  =  B )
2623, 25oveqan12d 6059 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  /  _i )  +  (
( _i  x.  B
)  /  _i ) )  =  ( ( _i  x.  -u A
)  +  B ) )
27 negcl 9262 . . . . . . 7  |-  ( A  e.  CC  ->  -u A  e.  CC )
28 mulcl 9030 . . . . . . 7  |-  ( ( _i  e.  CC  /\  -u A  e.  CC )  ->  ( _i  x.  -u A )  e.  CC )
292, 27, 28sylancr 645 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  e.  CC )
30 addcom 9208 . . . . . 6  |-  ( ( ( _i  x.  -u A
)  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  -u A )  +  B
)  =  ( B  +  ( _i  x.  -u A ) ) )
3129, 30sylan 458 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  -u A )  +  B
)  =  ( B  +  ( _i  x.  -u A ) ) )
3215, 26, 313eqtrrd 2441 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  ( _i  x.  -u A
) )  =  ( ( A  +  ( _i  x.  B ) )  /  _i ) )
331, 3, 32syl2an 464 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  ( _i  x.  -u A
) )  =  ( ( A  +  ( _i  x.  B ) )  /  _i ) )
3433fveq2d 5691 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( B  +  ( _i  x.  -u A ) ) )  =  ( Re
`  ( ( A  +  ( _i  x.  B ) )  /  _i ) ) )
35 id 20 . . 3  |-  ( B  e.  RR  ->  B  e.  RR )
36 renegcl 9320 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
37 crre 11874 . . 3  |-  ( ( B  e.  RR  /\  -u A  e.  RR )  ->  ( Re `  ( B  +  (
_i  x.  -u A ) ) )  =  B )
3835, 36, 37syl2anr 465 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( B  +  ( _i  x.  -u A ) ) )  =  B )
399, 34, 383eqtr2d 2442 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   _ici 8948    + caddc 8949    x. cmul 8951   -ucneg 9248    / cdiv 9633   Recre 11857   Imcim 11858
This theorem is referenced by:  replim  11876  reim0  11878  remullem  11888  imcj  11892  imneg  11893  imadd  11894  imi  11917  crimi  11953  crimd  11992  absreimsq  12052  4sqlem4  13275  logneg  20435  lognegb  20437  basellem3  20818  2sqlem2  21101  cnre2csqima  24262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-cj 11859  df-re 11860  df-im 11861
  Copyright terms: Public domain W3C validator