Theorem mulcnsr 11127

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 5454	. 2 ⊢ ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩ ∈ V
2		oveq1 7408	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝑤 ·R 𝑢) = (𝐴 ·R 𝑢))
3		oveq1 7408	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑣 ·R 𝑓) = (𝐵 ·R 𝑓))
4	3	oveq2d 7417	. . . . 5 ⊢ (𝑣 = 𝐵 → (-1R ·R (𝑣 ·R 𝑓)) = (-1R ·R (𝐵 ·R 𝑓)))
5	2, 4	oveqan12d 7420	. . . 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 7408	. . . . 5 ⊢ (𝑣 = 𝐵 → (𝑣 ·R 𝑢) = (𝐵 ·R 𝑢))
7		oveq1 7408	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝑤 ·R 𝑓) = (𝐴 ·R 𝑓))
8	6, 7	oveqan12rd 7421	. . . 4 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓)) = ((𝐵 ·R 𝑢) +R (𝐴 ·R 𝑓)))
9	5, 8	opeq12d 4873	. . 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 7409	. . . . 5 ⊢ (𝑢 = 𝐶 → (𝐴 ·R 𝑢) = (𝐴 ·R 𝐶))
11		oveq2 7409	. . . . . 6 ⊢ (𝑓 = 𝐷 → (𝐵 ·R 𝑓) = (𝐵 ·R 𝐷))
12	11	oveq2d 7417	. . . . 5 ⊢ (𝑓 = 𝐷 → (-1R ·R (𝐵 ·R 𝑓)) = (-1R ·R (𝐵 ·R 𝐷)))
13	10, 12	oveqan12d 7420	. . . 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 7409	. . . . 5 ⊢ (𝑢 = 𝐶 → (𝐵 ·R 𝑢) = (𝐵 ·R 𝐶))
15		oveq2 7409	. . . . 5 ⊢ (𝑓 = 𝐷 → (𝐴 ·R 𝑓) = (𝐴 ·R 𝐷))
16	14, 15	oveqan12d 7420	. . . 4 ⊢ ((𝑢 = 𝐶 ∧ 𝑓 = 𝐷) → ((𝐵 ·R 𝑢) +R (𝐴 ·R 𝑓)) = ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷)))
17	13, 16	opeq12d 4873	. . 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 2784	. 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 11118	. . 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 11112	. . . . . . 7 ⊢ ℂ = (R × R)
21	20	eleq2i 2817	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R))
22	20	eleq2i 2817	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R))
23	21, 22	anbi12i 626	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)))
24	23	anbi1i 623	. . . 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 7468	. . 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 2752	. 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 7563	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 1533  ∃wex 1773   ∈ wcel 2098  ⟨cop 4626   × cxp 5664  (class class class)co 7401  {coprab 7402  Rcnr 10856  -1_Rcm1r 10859   +_R cplr 10860   ·_R cmr 10861  ℂcc 11104   · cmul 11111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  df-oprab 7405  df-c 11112  df-mul 11118

This theorem is referenced by:  mulresr  11130  mulcnsrec  11135  axmulf  11137  axi2m1  11150  axcnre  11155