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

Theorem crim 11594
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 8822 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 8791 . . . . 5  |-  _i  e.  CC
3 recn 8822 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 8816 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
52, 3, 4sylancr 646 . . . 4  |-  ( B  e.  RR  ->  (
_i  x.  B )  e.  CC )
6 addcl 8814 . . . 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 11586 . . 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 653 . . . . . 6  |-  ( B  e.  CC  ->  (
_i  x.  B )  e.  CC )
11 ine0 9210 . . . . . . 7  |-  _i  =/=  0
12 divdir 9442 . . . . . . . 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 668 . . . . . 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 9436 . . . . . . . 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 11196 . . . . . . . . 9  |-  ( 1  /  _i )  = 
-u _i
1918oveq1i 5829 . . . . . . . 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 9213 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A
) )
222, 21mpan 653 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A ) )
2317, 20, 223eqtrd 2320 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  _i )  =  ( _i  x.  -u A
) )
24 divcan3 9443 . . . . . . 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 5838 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  /  _i )  +  (
( _i  x.  B
)  /  _i ) )  =  ( ( _i  x.  -u A
)  +  B ) )
27 negcl 9047 . . . . . . 7  |-  ( A  e.  CC  ->  -u A  e.  CC )
28 mulcl 8816 . . . . . . 7  |-  ( ( _i  e.  CC  /\  -u A  e.  CC )  ->  ( _i  x.  -u A )  e.  CC )
292, 27, 28sylancr 646 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  e.  CC )
30 addcom 8993 . . . . . 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 2321 . . . 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 5489 . 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 9105 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
37 crre 11593 . . 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 2322 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 1624    e. wcel 1685    =/= wne 2447   ` cfv 5221  (class class class)co 5819   CCcc 8730   RRcr 8731   0cc0 8732   1c1 8733   _ici 8734    + caddc 8735    x. cmul 8737   -ucneg 9033    / cdiv 9418   Recre 11576   Imcim 11577
This theorem is referenced by:  replim  11595  reim0  11597  remullem  11607  imcj  11611  imneg  11612  imadd  11613  imi  11636  crimi  11672  crimd  11711  absreimsq  11771  4sqlem4  12993  logneg  19935  lognegb  19937  basellem3  20314  2sqlem2  20597
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-2 9799  df-cj 11578  df-re 11579  df-im 11580
  Copyright terms: Public domain W3C validator