Theorem mulcnsr 10892

Description: Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

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

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5379	. 2 ⊢ ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩ ∈ V
2		oveq1 7282	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝑤 ·R 𝑢) = (𝐴 ·R 𝑢))
3		oveq1 7282	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑣 ·R 𝑓) = (𝐵 ·R 𝑓))
4	3	oveq2d 7291	. . . . 5 ⊢ (𝑣 = 𝐵 → (-1R ·R (𝑣 ·R 𝑓)) = (-1R ·R (𝐵 ·R 𝑓)))
5	2, 4	oveqan12d 7294	. . . 4 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))) = ((𝐴 ·R 𝑢) +R (-1R ·R (𝐵 ·R 𝑓))))
6		oveq1 7282	. . . . 5 ⊢ (𝑣 = 𝐵 → (𝑣 ·R 𝑢) = (𝐵 ·R 𝑢))
7		oveq1 7282	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝑤 ·R 𝑓) = (𝐴 ·R 𝑓))
8	6, 7	oveqan12rd 7295	. . . 4 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓)) = ((𝐵 ·R 𝑢) +R (𝐴 ·R 𝑓)))
9	5, 8	opeq12d 4812	. . 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))⟩)
10		oveq2 7283	. . . . 5 ⊢ (𝑢 = 𝐶 → (𝐴 ·R 𝑢) = (𝐴 ·R 𝐶))
11		oveq2 7283	. . . . . 6 ⊢ (𝑓 = 𝐷 → (𝐵 ·R 𝑓) = (𝐵 ·R 𝐷))
12	11	oveq2d 7291	. . . . 5 ⊢ (𝑓 = 𝐷 → (-1R ·R (𝐵 ·R 𝑓)) = (-1R ·R (𝐵 ·R 𝐷)))
13	10, 12	oveqan12d 7294	. . . 4 ⊢ ((𝑢 = 𝐶 ∧ 𝑓 = 𝐷) → ((𝐴 ·R 𝑢) +R (-1R ·R (𝐵 ·R 𝑓))) = ((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))))
14		oveq2 7283	. . . . 5 ⊢ (𝑢 = 𝐶 → (𝐵 ·R 𝑢) = (𝐵 ·R 𝐶))
15		oveq2 7283	. . . . 5 ⊢ (𝑓 = 𝐷 → (𝐴 ·R 𝑓) = (𝐴 ·R 𝐷))
16	14, 15	oveqan12d 7294	. . . 4 ⊢ ((𝑢 = 𝐶 ∧ 𝑓 = 𝐷) → ((𝐵 ·R 𝑢) +R (𝐴 ·R 𝑓)) = ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷)))
17	13, 16	opeq12d 4812	. . 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))⟩)
18	9, 17	sylan9eq 2798	. 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))⟩)
19		df-mul 10883	. . 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))⟩))}
20		df-c 10877	. . . . . . 7 ⊢ ℂ = (R × R)
21	20	eleq2i 2830	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R))
22	20	eleq2i 2830	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R))
23	21, 22	anbi12i 627	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)))
24	23	anbi1i 624	. . . 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))⟩)))
25	24	oprabbii 7342	. . 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))⟩))}
26	19, 25	eqtri 2766	. 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))⟩))}
27	1, 18, 26	ov3 7435	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 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ⟨cop 4567   × cxp 5587  (class class class)co 7275  {coprab 7276  Rcnr 10621  -1_Rcm1r 10624   +_R cplr 10625   ·_R cmr 10626  ℂcc 10869   · cmul 10876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-c 10877  df-mul 10883

This theorem is referenced by:  mulresr  10895  mulcnsrec  10900  axmulf  10902  axi2m1  10915  axcnre  10920