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Theorem cru 9671
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 739 . . . . . . 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 8794 . . . . . 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 737 . . . . . . 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 8794 . . . . . 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 449 . . . . . . . 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 8729 . . . . . . . . . . 11  |-  _i  e.  CC
76a1i 12 . . . . . . . . . 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 738 . . . . . . . . . . 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 8794 . . . . . . . . . 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 8788 . . . . . . . . 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 740 . . . . . . . . . . 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 8794 . . . . . . . . . 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 8788 . . . . . . . . 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 9141 . . . . . . . 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 203 . . . . . . 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 9144 . . . . . . . . . . 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 9168 . . . . . . . . . . . . 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 2291 . . . . . . . . . . . 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 9144 . . . . . . . . . . . 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 2330 . . . . . . . . . . 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 9670 . . . . . . . . . . 11  |-  ( ( ( B  -  D
)  e.  RR  /\  ( _i  x.  ( B  -  D )
)  e.  RR )  ->  ( B  -  D )  =  0 )
2216, 20, 21syl2anc 645 . . . . . . . . . 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 9098 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  B  =  D )
2423oveq2d 5773 . . . . . . . 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 5772 . . . . . . 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 9078 . . . . . . 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 2292 . . . . . 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 9098 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  C  =  A )
2928eqcomd 2261 . . . 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 520 . . 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 425 . 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 5765 . . 3  |-  ( B  =  D  ->  (
_i  x.  B )  =  ( _i  x.  D ) )
33 oveq12 5766 . . 3  |-  ( ( A  =  C  /\  ( _i  x.  B
)  =  ( _i  x.  D ) )  ->  ( A  +  ( _i  x.  B
) )  =  ( C  +  ( _i  x.  D ) ) )
3432, 33sylan2 462 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +  ( _i  x.  B ) )  =  ( C  +  ( _i  x.  D ) ) )
3531, 34impbid1 196 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670   _ici 8672    + caddc 8673    x. cmul 8675    - cmin 8970
This theorem is referenced by:  crne0  9672  creur  9673  creui  9674  cnref1o  10281  efieq  12370
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-po 4251  df-so 4252  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-iota 6190  df-riota 6237  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357
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