Theorem cru 8893

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

Step	Hyp	Ref	Expression
1		simplrl 537	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> C e. RR )
2	1	recnd 8318	. . . . . 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 535	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> A e. RR )
4	3	recnd 8318	. . . . . 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 110	. . . . . . . 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 8238	. . . . . . . . . . 11 |- _i e. CC
7	6	a1i 9	. . . . . . . . . 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 536	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> B e. RR )
9	8	recnd 8318	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> B e. CC )
10	7, 9	mulcld 8310	. . . . . . . . 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 538	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> D e. RR )
12	11	recnd 8318	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> D e. CC )
13	7, 12	mulcld 8310	. . . . . . . . 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 )
14	4, 10, 2, 13	addsubeq4d 8651	. . . . . . . 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 ) ) ) )
15	5, 14	mpbid 147	. . . . . . 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 ) ) )
16	8, 11	resubcld 8671	. . . . . . . . . . 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 )
17	7, 9, 12	subdid 8704	. . . . . . . . . . . . 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 ) ) )
18	17, 15	eqtr4d 2270	. . . . . . . . . . . 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 ) )
19	1, 3	resubcld 8671	. . . . . . . . . . . 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 )
20	18, 19	eqeltrd 2311	. . . . . . . . . . 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 8876	. . . . . . . . . . 11 |- ( ( ( B - D ) e. RR /\ ( _i x. ( B - D ) ) e. RR ) -> ( B - D ) = 0 )
22	16, 20, 21	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( B - D ) = 0 )
23	9, 12, 22	subeq0d 8608	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> B = D )
24	23	oveq2d 6074	. . . . . . . 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 ) )
25	24	oveq1d 6073	. . . . . . 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 ) ) )
26	13	subidd 8588	. . . . . . 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 )
27	15, 25, 26	3eqtrd 2271	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( C - A ) = 0 )
28	2, 4, 27	subeq0d 8608	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> C = A )
29	28	eqcomd 2240	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> A = C )
30	29, 23	jca 306	. . 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 ) )
31	30	ex 115	. 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 6066	. . 3 |- ( B = D -> ( _i x. B ) = ( _i x. D ) )
33		oveq12 6067	. . 3 |- ( ( A = C /\ ( _i x. B ) = ( _i x. D ) ) -> ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) )
34	32, 33	sylan2 286	. 2 |- ( ( A = C /\ B = D ) -> ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) )
35	31, 34	impbid1 142	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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   _ici 8145   + caddc 8146   x. cmul 8148   - cmin 8460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-sub 8462  df-neg 8463  df-reap 8866

This theorem is referenced by:  apreim  8894  apti  8913  creur  9250  creui  9251  cnref1o  10001  efieq  12446