Theorem mulcnsr 4048

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

Assertion

Ref	Expression
mulcnsr	⊢ (((A ∈ R ∧ B ∈ R) ∧ (C ∈ R ∧ D ∈ R)) → (⟨A, B⟩ · ⟨C, D⟩) = ⟨((A ·R C) +R (-1R ·R (B ·R D))), ((B ·R C) +R (A ·R D))⟩)

Proof of Theorem mulcnsr

Step	Hyp	Ref	Expression
1		opex 1893	. 2 ⊢ ⟨((A ·R C) +R (-1R ·R (B ·R D))), ((B ·R C) +R (A ·R D))⟩ ∈ V
2		opeq12 1878	. . . 4 ⊢ ((((w ·R u) +R (-1R ·R (v ·R f))) = ((A ·R u) +R (-1R ·R (B ·R f))) ∧ ((v ·R u) +R (w ·R f)) = ((B ·R u) +R (A ·R f))) → ⟨((w ·R u) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·R f))⟩ = ⟨((A ·R u) +R (-1R ·R (B ·R f))), ((B ·R u) +R (A ·R f))⟩)
3		opreq1 3006	. . . . 5 ⊢ (w = A → (w ·R u) = (A ·R u))
4		opreq1 3006	. . . . . 6 ⊢ (v = B → (v ·R f) = (B ·R f))
5	4	opreq2d 3013	. . . . 5 ⊢ (v = B → (-1R ·R (v ·R f)) = (-1R ·R (B ·R f)))
6	3, 5	opreqan12d 3015	. . . 4 ⊢ ((w = A ∧ v = B) → ((w ·R u) +R (-1R ·R (v ·R f))) = ((A ·R u) +R (-1R ·R (B ·R f))))
7		opreq1 3006	. . . . 5 ⊢ (v = B → (v ·R u) = (B ·R u))
8		opreq1 3006	. . . . 5 ⊢ (w = A → (w ·R f) = (A ·R f))
9	7, 8	opreqan12rd 3016	. . . 4 ⊢ ((w = A ∧ v = B) → ((v ·R u) +R (w ·R f)) = ((B ·R u) +R (A ·R f)))
10	2, 6, 9	sylanc 361	. . 3 ⊢ ((w = A ∧ v = B) → ⟨((w ·R u) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·R f))⟩ = ⟨((A ·R u) +R (-1R ·R (B ·R f))), ((B ·R u) +R (A ·R f))⟩)
11		opeq12 1878	. . . 4 ⊢ ((((A ·R u) +R (-1R ·R (B ·R f))) = ((A ·R C) +R (-1R ·R (B ·R D))) ∧ ((B ·R u) +R (A ·R f)) = ((B ·R C) +R (A ·R D))) → ⟨((A ·R u) +R (-1R ·R (B ·R f))), ((B ·R u) +R (A ·R f))⟩ = ⟨((A ·R C) +R (-1R ·R (B ·R D))), ((B ·R C) +R (A ·R D))⟩)
12		opreq2 3007	. . . . 5 ⊢ (u = C → (A ·R u) = (A ·R C))
13		opreq2 3007	. . . . . 6 ⊢ (f = D → (B ·R f) = (B ·R D))
14	13	opreq2d 3013	. . . . 5 ⊢ (f = D → (-1R ·R (B ·R f)) = (-1R ·R (B ·R D)))
15	12, 14	opreqan12d 3015	. . . 4 ⊢ ((u = C ∧ f = D) → ((A ·R u) +R (-1R ·R (B ·R f))) = ((A ·R C) +R (-1R ·R (B ·R D))))
16		opreq2 3007	. . . . 5 ⊢ (u = C → (B ·R u) = (B ·R C))
17		opreq2 3007	. . . . 5 ⊢ (f = D → (A ·R f) = (A ·R D))
18	16, 17	opreqan12d 3015	. . . 4 ⊢ ((u = C ∧ f = D) → ((B ·R u) +R (A ·R f)) = ((B ·R C) +R (A ·R D)))
19	11, 15, 18	sylanc 361	. . 3 ⊢ ((u = C ∧ f = D) → ⟨((A ·R u) +R (-1R ·R (B ·R f))), ((B ·R u) +R (A ·R f))⟩ = ⟨((A ·R C) +R (-1R ·R (B ·R D))), ((B ·R C) +R (A ·R D))⟩)
20	10, 19	sylan9eq 1144	. 2 ⊢ (((w = A ∧ v = B) ∧ (u = C ∧ f = D)) → ⟨((w ·R u) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·R f))⟩ = ⟨((A ·R C) +R (-1R ·R (B ·R D))), ((B ·R C) +R (A ·R D))⟩)
21		df-mul 4040	. . 3 ⊢ · = {⟨⟨x, y⟩, z⟩∣((x ∈ ℂ ∧ y ∈ ℂ) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨((w ·R u) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·R f))⟩))}
22		df-c 4034	. . . . . . 7 ⊢ ℂ = (R × R)
23	22	eleq2i 1153	. . . . . 6 ⊢ (x ∈ ℂ ↔ x ∈ (R × R))
24	22	eleq2i 1153	. . . . . 6 ⊢ (y ∈ ℂ ↔ y ∈ (R × R))
25	23, 24	anbi12i 369	. . . . 5 ⊢ ((x ∈ ℂ ∧ y ∈ ℂ) ↔ (x ∈ (R × R) ∧ y ∈ (R × R)))
26	25	anbi1i 368	. . . 4 ⊢ (((x ∈ ℂ ∧ y ∈ ℂ) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨((w ·R u) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·R f))⟩)) ↔ ((x ∈ (R × R) ∧ y ∈ (R × R)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨((w ·R u) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·R f))⟩)))
27	26	bioprabi 3027	. . 3 ⊢ {⟨⟨x, y⟩, z⟩∣((x ∈ ℂ ∧ y ∈ ℂ) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨((w ·R u) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·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) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·R f))⟩))}
28	21, 27	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) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·R f))⟩))}
29	1, 20, 28	oprabval3 3052	1 ⊢ (((A ∈ R ∧ B ∈ R) ∧ (C ∈ R ∧ D ∈ R)) → (⟨A, B⟩ · ⟨C, D⟩) = ⟨((A ·R C) +R (-1R ·R (B ·R D))), ((B ·R C) +R (A ·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  -1_Rcm1r 3790   +_R cplr 3791   ·_R cmr 3792  ℂcc 4026   · cmulc 4032

This theorem is referenced by:  mulresr 4051  mulcnsrec 4058  axmulcl 4068  ax1id 4077  axrecex 4079  axi2m1 4082  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-mul 4040

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