Theorem addcnsr 4047

Description: Addition of complex numbers in terms of signed reals.

Assertion

Ref	Expression
addcnsr	⊢ (((A ∈ R ∧ B ∈ R) ∧ (C ∈ R ∧ D ∈ R)) → (⟨A, B⟩ + ⟨C, D⟩) = ⟨(A +R C), (B +R D)⟩)

Proof of Theorem addcnsr

Step	Hyp	Ref	Expression
1		opex 1893	. 2 ⊢ ⟨(A +R C), (B +R D)⟩ ∈ V
2		opeq12 1878	. . . 4 ⊢ (((w +R u) = (A +R u) ∧ (v +R f) = (B +R f)) → ⟨(w +R u), (v +R f)⟩ = ⟨(A +R u), (B +R f)⟩)
3		opreq1 3006	. . . 4 ⊢ (w = A → (w +R u) = (A +R u))
4		opreq1 3006	. . . 4 ⊢ (v = B → (v +R f) = (B +R f))
5	2, 3, 4	syl2an 349	. . 3 ⊢ ((w = A ∧ v = B) → ⟨(w +R u), (v +R f)⟩ = ⟨(A +R u), (B +R f)⟩)
6		opeq12 1878	. . . 4 ⊢ (((A +R u) = (A +R C) ∧ (B +R f) = (B +R D)) → ⟨(A +R u), (B +R f)⟩ = ⟨(A +R C), (B +R D)⟩)
7		opreq2 3007	. . . 4 ⊢ (u = C → (A +R u) = (A +R C))
8		opreq2 3007	. . . 4 ⊢ (f = D → (B +R f) = (B +R D))
9	6, 7, 8	syl2an 349	. . 3 ⊢ ((u = C ∧ f = D) → ⟨(A +R u), (B +R f)⟩ = ⟨(A +R C), (B +R D)⟩)
10	5, 9	sylan9eq 1144	. 2 ⊢ (((w = A ∧ v = B) ∧ (u = C ∧ f = D)) → ⟨(w +R u), (v +R f)⟩ = ⟨(A +R C), (B +R D)⟩)
11		df-plus 4039	. . 3 ⊢ + = {⟨⟨x, y⟩, z⟩∣((x ∈ ℂ ∧ y ∈ ℂ) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨(w +R u), (v +R f)⟩))}
12		df-c 4034	. . . . . . 7 ⊢ ℂ = (R × R)
13	12	eleq2i 1153	. . . . . 6 ⊢ (x ∈ ℂ ↔ x ∈ (R × R))
14	12	eleq2i 1153	. . . . . 6 ⊢ (y ∈ ℂ ↔ y ∈ (R × R))
15	13, 14	anbi12i 369	. . . . 5 ⊢ ((x ∈ ℂ ∧ y ∈ ℂ) ↔ (x ∈ (R × R) ∧ y ∈ (R × R)))
16	15	anbi1i 368	. . . 4 ⊢ (((x ∈ ℂ ∧ y ∈ ℂ) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨(w +R u), (v +R f)⟩)) ↔ ((x ∈ (R × R) ∧ y ∈ (R × R)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨(w +R u), (v +R f)⟩)))
17	16	bioprabi 3027	. . 3 ⊢ {⟨⟨x, y⟩, z⟩∣((x ∈ ℂ ∧ y ∈ ℂ) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨(w +R u), (v +R f)⟩))} = {⟨⟨x, y⟩, z⟩∣((x ∈ (R × R) ∧ y ∈ (R × R)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨(w +R u), (v +R f)⟩))}
18	11, 17	eqtr 1119	. 2 ⊢ + = {⟨⟨x, y⟩, z⟩∣((x ∈ (R × R) ∧ y ∈ (R × R)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨(w +R u), (v +R f)⟩))}
19	1, 10, 18	oprabval3 3052	1 ⊢ (((A ∈ R ∧ B ∈ R) ∧ (C ∈ R ∧ D ∈ R)) → (⟨A, B⟩ + ⟨C, D⟩) = ⟨(A +R C), (B +R D)⟩)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   × cxp 2408  (class class class)co 3001  {copab2 3002  Rcnr 3787   +_R cplr 3791  ℂcc 4026   + caddc 4031

This theorem is referenced by:  addresr 4050  addcnsrec 4057  axaddcl 4066  ax0id 4076  axnegex 4078  axcnre 4087

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  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-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  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-fv 2438  df-opr 3003  df-oprab 3004  df-c 4034  df-plus 4039

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