Theorem mulcnsr 11027

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 5402	. 2 ⊢ ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩ ∈ V
2		oveq1 7353	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝑤 ·R 𝑢) = (𝐴 ·R 𝑢))
3		oveq1 7353	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑣 ·R 𝑓) = (𝐵 ·R 𝑓))
4	3	oveq2d 7362	. . . . 5 ⊢ (𝑣 = 𝐵 → (-1R ·R (𝑣 ·R 𝑓)) = (-1R ·R (𝐵 ·R 𝑓)))
5	2, 4	oveqan12d 7365	. . . 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 7353	. . . . 5 ⊢ (𝑣 = 𝐵 → (𝑣 ·R 𝑢) = (𝐵 ·R 𝑢))
7		oveq1 7353	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝑤 ·R 𝑓) = (𝐴 ·R 𝑓))
8	6, 7	oveqan12rd 7366	. . . 4 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓)) = ((𝐵 ·R 𝑢) +R (𝐴 ·R 𝑓)))
9	5, 8	opeq12d 4830	. . 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 7354	. . . . 5 ⊢ (𝑢 = 𝐶 → (𝐴 ·R 𝑢) = (𝐴 ·R 𝐶))
11		oveq2 7354	. . . . . 6 ⊢ (𝑓 = 𝐷 → (𝐵 ·R 𝑓) = (𝐵 ·R 𝐷))
12	11	oveq2d 7362	. . . . 5 ⊢ (𝑓 = 𝐷 → (-1R ·R (𝐵 ·R 𝑓)) = (-1R ·R (𝐵 ·R 𝐷)))
13	10, 12	oveqan12d 7365	. . . 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 7354	. . . . 5 ⊢ (𝑢 = 𝐶 → (𝐵 ·R 𝑢) = (𝐵 ·R 𝐶))
15		oveq2 7354	. . . . 5 ⊢ (𝑓 = 𝐷 → (𝐴 ·R 𝑓) = (𝐴 ·R 𝐷))
16	14, 15	oveqan12d 7365	. . . 4 ⊢ ((𝑢 = 𝐶 ∧ 𝑓 = 𝐷) → ((𝐵 ·R 𝑢) +R (𝐴 ·R 𝑓)) = ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷)))
17	13, 16	opeq12d 4830	. . 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 2786	. 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 11018	. . 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 11012	. . . . . . 7 ⊢ ℂ = (R × R)
21	20	eleq2i 2823	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R))
22	20	eleq2i 2823	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R))
23	21, 22	anbi12i 628	. . . . 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 7413	. . 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 2754	. 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 7509	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 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ⟨cop 4579   × cxp 5612  (class class class)co 7346  {coprab 7347  Rcnr 10756  -1_Rcm1r 10759   +_R cplr 10760   ·_R cmr 10761  ℂcc 11004   · cmul 11011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-c 11012  df-mul 11018

This theorem is referenced by:  mulresr  11030  mulcnsrec  11035  axmulf  11037  axi2m1  11050  axcnre  11055