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Theorem cru 9956
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 737 . . . . . . 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 9078 . . . . . 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 735 . . . . . . 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 9078 . . . . . 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 448 . . . . . . . 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 9013 . . . . . . . . . . 11  |-  _i  e.  CC
76a1i 11 . . . . . . . . . 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 736 . . . . . . . . . . 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 9078 . . . . . . . . . 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 9072 . . . . . . . . 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 738 . . . . . . . . . . 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 9078 . . . . . . . . . 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 9072 . . . . . . . . 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 9426 . . . . . . . 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 202 . . . . . . 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 9429 . . . . . . . . . . 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 9453 . . . . . . . . . . . . 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 2447 . . . . . . . . . . . 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 9429 . . . . . . . . . . . 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 2486 . . . . . . . . . . 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 9955 . . . . . . . . . . 11  |-  ( ( ( B  -  D
)  e.  RR  /\  ( _i  x.  ( B  -  D )
)  e.  RR )  ->  ( B  -  D )  =  0 )
2216, 20, 21syl2anc 643 . . . . . . . . . 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 9383 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  B  =  D )
2423oveq2d 6064 . . . . . . . 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 6063 . . . . . . 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 9363 . . . . . . 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 2448 . . . . . 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 9383 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  C  =  A )
2928eqcomd 2417 . . . 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 519 . . 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 424 . 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 6056 . . 3  |-  ( B  =  D  ->  (
_i  x.  B )  =  ( _i  x.  D ) )
33 oveq12 6057 . . 3  |-  ( ( A  =  C  /\  ( _i  x.  B
)  =  ( _i  x.  D ) )  ->  ( A  +  ( _i  x.  B
) )  =  ( C  +  ( _i  x.  D ) ) )
3432, 33sylan2 461 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +  ( _i  x.  B ) )  =  ( C  +  ( _i  x.  D ) ) )
3531, 34impbid1 195 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 177    /\ wa 359    = wceq 1649    e. wcel 1721  (class class class)co 6048   CCcc 8952   RRcr 8953   0cc0 8954   _ici 8956    + caddc 8957    x. cmul 8959    - cmin 9255
This theorem is referenced by:  crne0  9957  creur  9958  creui  9959  cnref1o  10571  efieq  12727
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642
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