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

Theorem cru 9734
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 8857 . . . . . 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 8857 . . . . . 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 8792 . . . . . . . . . . 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 8857 . . . . . . . . . 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 8851 . . . . . . . . 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 8857 . . . . . . . . . 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 8851 . . . . . . . . 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 9204 . . . . . . . 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 9207 . . . . . . . . . . 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 9231 . . . . . . . . . . . . 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 2319 . . . . . . . . . . . 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 9207 . . . . . . . . . . . 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 2358 . . . . . . . . . . 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 9733 . . . . . . . . . . 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 9161 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  B  =  D )
2423oveq2d 5836 . . . . . . . 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 5835 . . . . . . 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 9141 . . . . . . 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 2320 . . . . . 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 9161 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A  +  (
_i  x.  B )
)  =  ( C  +  ( _i  x.  D ) ) )  ->  C  =  A )
2928eqcomd 2289 . . . 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 5828 . . 3  |-  ( B  =  D  ->  (
_i  x.  B )  =  ( _i  x.  D ) )
33 oveq12 5829 . . 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 1623    e. wcel 1685  (class class class)co 5820   CCcc 8731   RRcr 8732   0cc0 8733   _ici 8735    + caddc 8736    x. cmul 8738    - cmin 9033
This theorem is referenced by:  crne0  9735  creur  9736  creui  9737  cnref1o  10345  efieq  12439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
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 1529  df-nf 1532  df-sb 1631  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 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420
  Copyright terms: Public domain W3C validator