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Theorem cru 9740
Description: The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
cru  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) )  =  ( C  +  ( _i  x.  D ) )  <-> 
( A  =  C  /\  B  =  D ) ) )

Proof of Theorem cru
StepHypRef Expression
1 simplrl 736 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  C  e.  RR )
21recnd 8863 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  C  e.  CC )
3 simplll 734 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  A  e.  RR )
43recnd 8863 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  A  e.  CC )
5 simpr 447 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( A  +  ( _i  x.  B
) )  =  ( C  +  ( _i  x.  D ) ) )
6 ax-icn 8798 . . . . . . . . . . 11  |-  _i  e.  CC
76a1i 10 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  _i  e.  CC )
8 simpllr 735 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  B  e.  RR )
98recnd 8863 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  B  e.  CC )
107, 9mulcld 8857 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( _i  x.  B )  e.  CC )
11 simplrr 737 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  D  e.  RR )
1211recnd 8863 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  D  e.  CC )
137, 12mulcld 8857 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( _i  x.  D )  e.  CC )
144, 10, 2, 13addsubeq4d 9210 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( ( A  +  ( _i  x.  B ) )  =  ( C  +  ( _i  x.  D ) )  <->  ( C  -  A )  =  ( ( _i  x.  B
)  -  ( _i  x.  D ) ) ) )
155, 14mpbid 201 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( C  -  A )  =  ( ( _i  x.  B
)  -  ( _i  x.  D ) ) )
168, 11resubcld 9213 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( B  -  D )  e.  RR )
177, 9, 12subdid 9237 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( _i  x.  ( B  -  D
) )  =  ( ( _i  x.  B
)  -  ( _i  x.  D ) ) )
1817, 15eqtr4d 2320 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( _i  x.  ( B  -  D
) )  =  ( C  -  A ) )
191, 3resubcld 9213 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( C  -  A )  e.  RR )
2018, 19eqeltrd 2359 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( _i  x.  ( B  -  D
) )  e.  RR )
21 rimul 9739 . . . . . . . . . . 11  |-  ( ( ( B  -  D
)  e.  RR  /\  ( _i  x.  ( B  -  D )
)  e.  RR )  ->  ( B  -  D )  =  0 )
2216, 20, 21syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( B  -  D )  =  0 )
239, 12, 22subeq0d 9167 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  B  =  D )
2423oveq2d 5876 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( _i  x.  B )  =  ( _i  x.  D ) )
2524oveq1d 5875 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( ( _i  x.  B )  -  ( _i  x.  D
) )  =  ( ( _i  x.  D
)  -  ( _i  x.  D ) ) )
2613subidd 9147 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( ( _i  x.  D )  -  ( _i  x.  D
) )  =  0 )
2715, 25, 263eqtrd 2321 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( C  -  A )  =  0 )
282, 4, 27subeq0d 9167 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  C  =  A )
2928eqcomd 2290 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  A  =  C )
3029, 23jca 518 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  ( A  =  C  /\  B  =  D ) )
3130ex 423 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) )  =  ( C  +  ( _i  x.  D ) )  ->  ( A  =  C  /\  B  =  D ) ) )
32 oveq2 5868 . . 3  |-  ( B  =  D  ->  (
_i  x.  B )  =  ( _i  x.  D ) )
33 oveq12 5869 . . 3  |-  ( ( A  =  C  /\  ( _i  x.  B
)  =  ( _i  x.  D ) )  ->  ( A  +  ( _i  x.  B
) )  =  ( C  +  ( _i  x.  D ) ) )
3432, 33sylan2 460 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +  ( _i  x.  B ) )  =  ( C  +  ( _i  x.  D ) ) )
3531, 34impbid1 194 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) )  =  ( C  +  ( _i  x.  D ) )  <-> 
( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   _ici 8741    + caddc 8742    x. cmul 8744    - cmin 9039
This theorem is referenced by:  crne0  9741  creur  9742  creui  9743  cnref1o  10351  efieq  12445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426
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