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Theorem crim 11696
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 8917 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 8886 . . . . 5  |-  _i  e.  CC
3 recn 8917 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 8911 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
52, 3, 4sylancr 644 . . . 4  |-  ( B  e.  RR  ->  (
_i  x.  B )  e.  CC )
6 addcl 8909 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  +  ( _i  x.  B
) )  e.  CC )
71, 5, 6syl2an 463 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  ( _i  x.  B ) )  e.  CC )
8 imval 11688 . . 3  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
Im `  ( A  +  ( _i  x.  B ) ) )  =  ( Re `  ( ( A  +  ( _i  x.  B
) )  /  _i ) ) )
97, 8syl 15 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  ( Re `  ( ( A  +  ( _i  x.  B
) )  /  _i ) ) )
102, 4mpan 651 . . . . . 6  |-  ( B  e.  CC  ->  (
_i  x.  B )  e.  CC )
11 ine0 9305 . . . . . . 7  |-  _i  =/=  0
12 divdir 9537 . . . . . . . 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 1151 . . . . . . 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 666 . . . . . 6  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( ( A  +  ( _i  x.  B ) )  /  _i )  =  (
( A  /  _i )  +  ( (
_i  x.  B )  /  _i ) ) )
1510, 14sylan2 460 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( _i  x.  B
) )  /  _i )  =  ( ( A  /  _i )  +  ( ( _i  x.  B )  /  _i ) ) )
16 divrec2 9531 . . . . . . . 8  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  ( A  /  _i )  =  ( ( 1  /  _i )  x.  A
) )
172, 11, 16mp3an23 1269 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  /  _i )  =  ( ( 1  /  _i )  x.  A
) )
18 irec 11295 . . . . . . . . 9  |-  ( 1  /  _i )  = 
-u _i
1918oveq1i 5955 . . . . . . . 8  |-  ( ( 1  /  _i )  x.  A )  =  ( -u _i  x.  A )
2019a1i 10 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  /  _i )  x.  A )  =  ( -u _i  x.  A ) )
21 mulneg12 9308 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A
) )
222, 21mpan 651 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A ) )
2317, 20, 223eqtrd 2394 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  _i )  =  ( _i  x.  -u A
) )
24 divcan3 9538 . . . . . . 7  |-  ( ( B  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  B
)  /  _i )  =  B )
252, 11, 24mp3an23 1269 . . . . . 6  |-  ( B  e.  CC  ->  (
( _i  x.  B
)  /  _i )  =  B )
2623, 25oveqan12d 5964 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  /  _i )  +  (
( _i  x.  B
)  /  _i ) )  =  ( ( _i  x.  -u A
)  +  B ) )
27 negcl 9142 . . . . . . 7  |-  ( A  e.  CC  ->  -u A  e.  CC )
28 mulcl 8911 . . . . . . 7  |-  ( ( _i  e.  CC  /\  -u A  e.  CC )  ->  ( _i  x.  -u A )  e.  CC )
292, 27, 28sylancr 644 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  e.  CC )
30 addcom 9088 . . . . . 6  |-  ( ( ( _i  x.  -u A
)  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  -u A )  +  B
)  =  ( B  +  ( _i  x.  -u A ) ) )
3129, 30sylan 457 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  -u A )  +  B
)  =  ( B  +  ( _i  x.  -u A ) ) )
3215, 26, 313eqtrrd 2395 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  ( _i  x.  -u A
) )  =  ( ( A  +  ( _i  x.  B ) )  /  _i ) )
331, 3, 32syl2an 463 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  ( _i  x.  -u A
) )  =  ( ( A  +  ( _i  x.  B ) )  /  _i ) )
3433fveq2d 5612 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( B  +  ( _i  x.  -u A ) ) )  =  ( Re
`  ( ( A  +  ( _i  x.  B ) )  /  _i ) ) )
35 id 19 . . 3  |-  ( B  e.  RR  ->  B  e.  RR )
36 renegcl 9200 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
37 crre 11695 . . 3  |-  ( ( B  e.  RR  /\  -u A  e.  RR )  ->  ( Re `  ( B  +  (
_i  x.  -u A ) ) )  =  B )
3835, 36, 37syl2anr 464 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( B  +  ( _i  x.  -u A ) ) )  =  B )
399, 34, 383eqtr2d 2396 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   ` cfv 5337  (class class class)co 5945   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828   _ici 8829    + caddc 8830    x. cmul 8832   -ucneg 9128    / cdiv 9513   Recre 11678   Imcim 11679
This theorem is referenced by:  replim  11697  reim0  11699  remullem  11709  imcj  11713  imneg  11714  imadd  11715  imi  11738  crimi  11774  crimd  11813  absreimsq  11873  4sqlem4  13096  logneg  20049  lognegb  20051  basellem3  20432  2sqlem2  20715  cnre2csqima  23465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-2 9894  df-cj 11680  df-re 11681  df-im 11682
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