Theorem cru 8561

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 535	. . . . . . 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 7988	. . . . . 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 533	. . . . . . 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 7988	. . . . . 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 7908	. . . . . . . . . . 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 534	. . . . . . . . . . 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 7988	. . . . . . . . . 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 7980	. . . . . . . . 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 536	. . . . . . . . . . 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 7988	. . . . . . . . . 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 7980	. . . . . . . . 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 8321	. . . . . . . 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 8340	. . . . . . . . . . 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 8373	. . . . . . . . . . . . 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 2213	. . . . . . . . . . . 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 8340	. . . . . . . . . . . 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 2254	. . . . . . . . . . 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 8544	. . . . . . . . . . 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 8278	. . . . . . . . 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 5893	. . . . . . . 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 5892	. . . . . . 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 8258	. . . . . . 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 2214	. . . . . 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 8278	. . . . 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 2183	. . . 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 5885	. . 3 |- ( B = D -> ( _i x. B ) = ( _i x. D ) )
33		oveq12 5886	. . 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 1353   e. wcel 2148  (class class class)co 5877   CCcc 7811   RRcr 7812   0cc0 7813   _ici 7815   + caddc 7816   x. cmul 7818   - cmin 8130

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-sub 8132  df-neg 8133  df-reap 8534

This theorem is referenced by:  apreim  8562  apti  8581  creur  8918  creui  8919  cnref1o  9652  efieq  11745