Theorem cru 8521

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 530	. . . . . . 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 7948	. . . . . 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 528	. . . . . . 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 7948	. . . . . 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 109	. . . . . . . 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 7869	. . . . . . . . . . 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 529	. . . . . . . . . . 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 7948	. . . . . . . . . 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 7940	. . . . . . . . 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 531	. . . . . . . . . . 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 7948	. . . . . . . . . 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 7940	. . . . . . . . 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 8281	. . . . . . . 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 146	. . . . . . 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 8300	. . . . . . . . . . 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 8333	. . . . . . . . . . . . 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 2206	. . . . . . . . . . . 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 8300	. . . . . . . . . . . 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 2247	. . . . . . . . . . 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 8504	. . . . . . . . . . 11 |- ( ( ( B - D ) e. RR /\ ( _i x. ( B - D ) ) e. RR ) -> ( B - D ) = 0 )
22	16, 20, 21	syl2anc 409	. . . . . . . . . 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 8238	. . . . . . . . 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 5869	. . . . . . . 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 5868	. . . . . . 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 8218	. . . . . . 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 2207	. . . . . 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 8238	. . . . 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 2176	. . . 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 304	. . 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 114	. 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 5861	. . 3 |- ( B = D -> ( _i x. B ) = ( _i x. D ) )
33		oveq12 5862	. . 3 |- ( ( A = C /\ ( _i x. B ) = ( _i x. D ) ) -> ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) )
34	32, 33	sylan2 284	. 2 |- ( ( A = C /\ B = D ) -> ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) )
35	31, 34	impbid1 141	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 103   <-> wb 104   = wceq 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   _ici 7776   + caddc 7777   x. cmul 7779   - cmin 8090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093  df-reap 8494

This theorem is referenced by:  apreim  8522  apti  8541  creur  8875  creui  8876  cnref1o  9609  efieq  11698