Theorem crut 4531

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

Assertion

Ref	Expression
crut	⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (C ∈ ℝ ∧ D ∈ ℝ)) → ((A + (B · i)) = (C + (D · i)) → (A = C ∧ B = D)))

Proof of Theorem crut

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . 4 ⊢ (A = if(A/I> ∈ ℝ, A, 0) → (A + (B · i)) = (if(A ∈ ℝ, A, 0) + (B · i)))
2	1	cleq1d 1109	. . 3 ⊢ (A = if(A ∈ ℝ, A, 0) → ((A + (B · i)) = (C + (D · i)) ↔ (if(A ∈ ℝ, A, 0) + (B · i)) = (C + (D · i))))
3		cleq1 1107	. . . 4 ⊢ (A = if(A ∈ ℝ, A, 0) → (A = C ↔ if(A ∈ ℝ, A, 0) = C))
4	3	anbi1d 469	. . 3 ⊢ (A = if(A ∈ ℝ, A, 0) → ((A = C ∧ B = D) ↔ (if(A ∈ ℝ, A, 0) = C ∧ B = D)))
5	2, 4	imbi12d 474	. 2 ⊢ (A = if(A ∈ ℝ, A, 0) → (((A + (B · i)) = (C + (D · i)) → (A = C ∧ B = D)) ↔ ((if(A ∈ ℝ, A, 0) + (B · i)) = (C + (D · i)) → (if(A ∈ ℝ, A, 0) = C ∧ B = D))))
6		opreq1 3006	. . . . 5 ⊢ (B = if(B ∈ ℝ, B, 0) → (B · i) = (if(B ∈ ℝ, B, 0) · i))
7	6	opreq2d 3013	. . . 4 ⊢ (B = if(B ∈ ℝ, B, 0) → (if(A ∈ ℝ, A, 0) + (B · i)) = (if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)))
8	7	cleq1d 1109	. . 3 ⊢ (B = if(B ∈ ℝ, B, 0) → ((if(A ∈ ℝ, A, 0) + (B · i)) = (C + (D · i)) ↔ (if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)) = (C + (D · i))))
9		cleq1 1107	. . . 4 ⊢ (B = if(B ∈ ℝ, B, 0) → (B = D ↔ if(B ∈ ℝ, B, 0) = D))
10	9	anbi2d 468	. . 3 ⊢ (B = if(B ∈ ℝ, B, 0) → ((if(A ∈ ℝ, A, 0) = C ∧ B = D) ↔ (if(A ∈ ℝ, A, 0) = C ∧ if(B ∈ ℝ, B, 0) = D)))
11	8, 10	imbi12d 474	. 2 ⊢ (B = if(B ∈ ℝ, B, 0) → (((if(A ∈ ℝ, A, 0) + (O · i)) = (C + (D · i)) → (if(A ∈ ℝ, A, 0) = C ∧ B = D)) ↔ ((if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)) = (C + (D · i)) → (if(A ∈ ℝ, A, 0) = C ∧ if(B ∈ ℝ, B, 0) = D))))
12		opreq1 3006	. . . 4 ⊢ (C = if(C ∈ ℝ, C, 0) → (C + (D · i)) = (if(C ∈ ℝ, C, 0) + (D · i)))
13	12	cleq2d 1112	. . 3 ⊢ (C = if(C ∈ ℝ, C, 0) → ((if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)) = (C + (D · i)) ↔ (if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)) = (if(C ∈ ℝ, C, 0) + (D · i))))
14		cleq2 1110	. . . 4 ⊢ (C = if(C ∈ ℝ, C, 0) → (if(A ∈ ℝ, A, 0) = C ↔ if(A ∈ ℝ, A, 0) = if(C ∈ ℝ, C, 0)))
15	14	anbi1d 469	. . 3 ⊢ (C = if(C ∈ ℝ, C, 0) → ((if(A ∈ ℝ, A, 0) = C ∧ if(B ∈ ℝ, B, 0) = D) ↔ (if(A ∈ ℝ, A, 0) = if(C ∈ ℝ, C, 0) ∧ if(B ∈ ℝ, B, 0) = D)))
16	13, 15	imbi12d 474	. 2 ⊢ (C = if(C ∈ ℝ, C, 0) → (((if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)) = (C + (D · i)) → (if(A ∈ ℝ, A, 0) = C ∧ if(B ∈ ℝ, B, 0) = D)) ↔ ((if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)) = (if(C ∈ ℝ, C, 0) + (D · i)) → (if(A ∈ ℝ, A, 0) = if(C ∈ ℝ, C, 0) ∧ if(B ∈ ℝ, B, 0) = D))))
17		opreq1 3006	. . . . 5 ⊢ (D = if(D ∈ ℝ, D, 0) → (D · i) = (if(D ∈ ℝ, D, 0) · i))
18	17	opreq2d 3013	. . . 4 ⊢ (D = if(D ∈ ℝ, D, 0) → (if(C ∈ ℝ, C, 0) + (D · i)) = (if(C ∈ ℝ, C, 0) + (if(D ∈ ℝ, D, 0) · i)))
19	18	cleq2d 1112	. . 3 ⊢ (D = if(D ∈ ℝ, D, 0) → ((if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)) = (if(C ∈ ℝ, C, 0) + (D · i)) ↔ (if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)) = (if(C ∈ ℝ, C, 0) + (if(D ∈ ℝ, D, 0) · i))))
20		cleq2 1110	. . . 4 ⊢ (D = if(D ∈ ℝ, D, 0) → (if(B ∈ ℝ, B, 0) = D ↔ if(B ∈ ℝ, B, 0) = if(D ∈ ℝ, D, 0)))
21	20	anbi2d 468	. . 3 ⊢ (D = if(D ∈ ℝ, D, 0) → ((if(A ∈ ℝ, A, 0) = if(C ∈ ℝ, C, 0) ∧ if(B ∈ ℝ, B, 0) = D) ↔ (if(A ∈ ℝ, A, 0) = if(C ∈ ℝ, C, 0) ∧ if(B ∈ ℝ, B, 0) = if(D ∈ ℝ, D, 0))))
22	19, 21	imbi12d 474	. 2 ⊢ (D = if(D ∈ ℝ, D, 0) → (((if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)) = (if(C ∈ ℝ, C, 0) + (D · i)) → (if(A ∈ ℝ, A, 0) = if(C ∈ ℝ, C, 0) ∧ if(B ∈ ℝ, B, 0) = D)) ↔ ((if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)) = (if(C ∈ ℝ, C, 0) + (if(D ∈ ℝ, D, 0) · i)) → (if(A ∈ ℝ, A, 0) = if(C ∈ ℝ, C, 0) ∧ if(B ∈ ℝ, B, 0) = if(D ∈ ℝ, D, 0)))))
23		ax0re 4063	. . . 4 ⊢ 0 ∈ ℝ
24	23	elimel 1793	. . 3 ⊢ if(A ∈ ℝ, A, 0) ∈ ℝ
25	23	elimel 1793	. . 3 ⊢ if(B ∈ ℝ, B, 0) ∈ ℝ
26	23	elimel 1793	. . 3 ⊢ if(C ∈ ℝ, C, 0) ∈ ℝ
27	23	elimel 1793	. . 3 ⊢ if(D ∈ ℝ, D, 0) ∈ ℝ
28	24, 25, 26, 27	cru 4529	. 2 ⊢ ((if(A ∈ ℝ, A, 0) + (if(B ∈ ℝ, B, 0) · i)) = (if(C ∈ ℝ, C, 0) + (if(D ∈ ℝ, D, 0) · i)) → (if(A ∈ ℝ, A, 0) = if(C ∈ ℝ, C, 0) ∧ if(B ∈ ℝ, B, 0) = if(D ∈ ℝ, D, 0)))
29	5, 11, 16, 22, 28	dedth4h 1789	1 ⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (C ∈ ℝ ∧ D ∈ ℝ)) → ((A + (B · i)) = (C + (D · i)) → (A = C ∧ B = D)))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ifcif 1776  (class class class)co 3001  ℝcr 4027  0cc0 4028  ici 4030   + caddc 4031   · cmulc 4032

This theorem is referenced by:  creur 4532  creui 4533  rimul 4534  replimt 4798

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c&nsp;4034  df-0 4035  df-1 4036  df-i 4037  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216

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