Theorem cru 8677

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 8103	. . . . . 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 8103	. . . . . 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 8022	. . . . . . . . . . 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 8103	. . . . . . . . . 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 8095	. . . . . . . . 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 8103	. . . . . . . . . 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 8095	. . . . . . . . 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 8436	. . . . . . . 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 8455	. . . . . . . . . . 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 8488	. . . . . . . . . . . . 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 2241	. . . . . . . . . . . 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 8455	. . . . . . . . . . . 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 2282	. . . . . . . . . . 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 8660	. . . . . . . . . . 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 8393	. . . . . . . . 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 5962	. . . . . . . 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 5961	. . . . . . 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 8373	. . . . . . 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 2242	. . . . . 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 8393	. . . . 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 2211	. . . 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 5954	. . 3 |- ( B = D -> ( _i x. B ) = ( _i x. D ) )
33		oveq12 5955	. . 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 1373   e. wcel 2176  (class class class)co 5946   CCcc 7925   RRcr 7926   0cc0 7927   _ici 7929   + caddc 7930   x. cmul 7932   - cmin 8245

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-ltxr 8114  df-sub 8247  df-neg 8248  df-reap 8650

This theorem is referenced by:  apreim  8678  apti  8697  creur  9034  creui  9035  cnref1o  9774  efieq  12079