Theorem crut 6676

Description: The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130.

Assertion

Ref	Expression
crut	|- (((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 crut

Step	Hyp	Ref	Expression
1		opreq1 3959	. . . 4 |- (A = if(A e. RR, A, 0) -> (A + (i x. B)) = (if(A e. RR, A, 0) + (i x. B)))
2	1	eqeq1d 1480	. . 3 |- (A = if(A e. RR, A, 0) -> ((A + (i x. B)) = (C + (i x. D)) <-> (if(A e. RR, A, 0) + (i x. B)) = (C + (i x. D))))
3		eqeq1 1478	. . . 4 |- (A = if(A e. RR, A, 0) -> (A = C <-> if(A e. RR, A, 0) = C))
4	3	anbi1d 616	. . 3 |- (A = if(A e. RR, A, 0) -> ((A = C /\ B = D) <-> (if(A e. RR, A, 0) = C /\ B = D)))
5	2, 4	bibi12d 628	. 2 |- (A = if(A e. RR, A, 0) -> (((A + (i x. B)) = (C + (i x. D)) <-> (A = C /\ B = D)) <-> ((if(A e. RR, A, 0) + (i x. B)) = (C + (i x. D)) <-> (if(A e. RR, A, 0) = C /\ B = D))))
6		opreq2 3960	. . . . 5 |- (B = if(B e. RR, B, 0) -> (i x. B) = (i x. if(B e. RR, B, 0)))
7	6	opreq2d 3967	. . . 4 |- (B = if(B e. RR, B, 0) -> (if(A e. RR, A, 0) + (i x. B)) = (if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))))
8	7	eqeq1d 1480	. . 3 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) + (i x. B)) = (C + (i x. D)) <-> (if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))) = (C + (i x. D))))
9		eqeq1 1478	. . . 4 |- (B = if(B e. RR, B, 0) -> (B = D <-> if(B e. RR, B, 0) = D))
10	9	anbi2d 615	. . 3 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) = C /\ B = D) <-> (if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D)))
11	8, 10	bibi12d 628	. 2 |- (B = if(B e. RR, B, 0) -> (((if(A e. RR, A, 0) + (i x. B)) = (C + (i x. D)) <-> (if(A e. RR, A, 0) = C /\ B = D)) <-> ((if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))) = (C + (i x. D)) <-> (if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D))))
12		opreq1 3959	. . . 4 |- (C = if(C e. RR, C, 0) -> (C + (i x. D)) = (if(C e. RR, C, 0) + (i x. D)))
13	12	eqeq2d 1483	. . 3 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))) = (C + (i x. D)) <-> (if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (i x. D))))
14		eqeq2 1481	. . . 4 |- (C = if(C e. RR, C, 0) -> (if(A e. RR, A, 0) = C <-> if(A e. RR, A, 0) = if(C e. RR, C, 0)))
15	14	anbi1d 616	. . 3 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D)))
16	13, 15	bibi12d 628	. 2 |- (C = if(C e. RR, C, 0) -> (((if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))) = (C + (i x. D)) <-> (if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D)) <-> ((if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (i x. D)) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D))))
17		opreq2 3960	. . . . 5 |- (D = if(D e. RR, D, 0) -> (i x. D) = (i x. if(D e. RR, D, 0)))
18	17	opreq2d 3967	. . . 4 |- (D = if(D e. RR, D, 0) -> (if(C e. RR, C, 0) + (i x. D)) = (if(C e. RR, C, 0) + (i x. if(D e. RR, D, 0))))
19	18	eqeq2d 1483	. . 3 |- (D = if(D e. RR, D, 0) -> ((if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (i x. D)) <-> (if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (i x. if(D e. RR, D, 0)))))
20		eqeq2 1481	. . . 4 |- (D = if(D e. RR, D, 0) -> (if(B e. RR, B, 0) = D <-> if(B e. RR, B, 0) = if(D e. RR, D, 0)))
21	20	anbi2d 615	. . 3 |- (D = if(D e. RR, D, 0) -> ((if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = if(D e. RR, D, 0))))
22	19, 21	bibi12d 628	. 2 |- (D = if(D e. RR, D, 0) -> (((if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (i x. D)) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D)) <-> ((if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (i x. if(D e. RR, D, 0))) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = if(D e. RR, D, 0)))))
23		0re 5420	. . . 4 |- 0 e. RR
24	23	elimel 2390	. . 3 |- if(A e. RR, A, 0) e. RR
25	23	elimel 2390	. . 3 |- if(B e. RR, B, 0) e. RR
26	23	elimel 2390	. . 3 |- if(C e. RR, C, 0) e. RR
27	23	elimel 2390	. . 3 |- if(D e. RR, D, 0) e. RR
28	24, 25, 26, 27	cru 6675	. 2 |- ((if(A e. RR, A, 0) + (i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (i x. if(D e. RR, D, 0))) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = if(D e. RR, D, 0)))
29	5, 11, 16, 22, 28	dedth4h 2385	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 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  ifcif 2357  (class class class)co 3954  RRcr 5213  0cc0 5214  ici 5216   + caddc 5217   x. cmul 5219

This theorem is referenced by:  crutOLD 6677  creur 6681  creui 6682  rimul 6683  replimt 6700  crret 6710  crimt 6712  cj11t 6773  efieq 7400  sinperlem1 8624  efifolem7 8662  efif1lem3 8666  eff1i 8683

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358