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Theorem mulcnsr 10638
Description: Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
mulcnsr (((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨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.
StepHypRef Expression
1 opex 5322 . 2 ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩ ∈ V
2 oveq1 7179 . . . . 5 (𝑤 = 𝐴 → (𝑤 ·R 𝑢) = (𝐴 ·R 𝑢))
3 oveq1 7179 . . . . . 6 (𝑣 = 𝐵 → (𝑣 ·R 𝑓) = (𝐵 ·R 𝑓))
43oveq2d 7188 . . . . 5 (𝑣 = 𝐵 → (-1R ·R (𝑣 ·R 𝑓)) = (-1R ·R (𝐵 ·R 𝑓)))
52, 4oveqan12d 7191 . . . 4 ((𝑤 = 𝐴𝑣 = 𝐵) → ((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))) = ((𝐴 ·R 𝑢) +R (-1R ·R (𝐵 ·R 𝑓))))
6 oveq1 7179 . . . . 5 (𝑣 = 𝐵 → (𝑣 ·R 𝑢) = (𝐵 ·R 𝑢))
7 oveq1 7179 . . . . 5 (𝑤 = 𝐴 → (𝑤 ·R 𝑓) = (𝐴 ·R 𝑓))
86, 7oveqan12rd 7192 . . . 4 ((𝑤 = 𝐴𝑣 = 𝐵) → ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓)) = ((𝐵 ·R 𝑢) +R (𝐴 ·R 𝑓)))
95, 8opeq12d 4769 . . 3 ((𝑤 = 𝐴𝑣 = 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 7180 . . . . 5 (𝑢 = 𝐶 → (𝐴 ·R 𝑢) = (𝐴 ·R 𝐶))
11 oveq2 7180 . . . . . 6 (𝑓 = 𝐷 → (𝐵 ·R 𝑓) = (𝐵 ·R 𝐷))
1211oveq2d 7188 . . . . 5 (𝑓 = 𝐷 → (-1R ·R (𝐵 ·R 𝑓)) = (-1R ·R (𝐵 ·R 𝐷)))
1310, 12oveqan12d 7191 . . . 4 ((𝑢 = 𝐶𝑓 = 𝐷) → ((𝐴 ·R 𝑢) +R (-1R ·R (𝐵 ·R 𝑓))) = ((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))))
14 oveq2 7180 . . . . 5 (𝑢 = 𝐶 → (𝐵 ·R 𝑢) = (𝐵 ·R 𝐶))
15 oveq2 7180 . . . . 5 (𝑓 = 𝐷 → (𝐴 ·R 𝑓) = (𝐴 ·R 𝐷))
1614, 15oveqan12d 7191 . . . 4 ((𝑢 = 𝐶𝑓 = 𝐷) → ((𝐵 ·R 𝑢) +R (𝐴 ·R 𝑓)) = ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷)))
1713, 16opeq12d 4769 . . 3 ((𝑢 = 𝐶𝑓 = 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))⟩)
189, 17sylan9eq 2793 . 2 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 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 10629 . . 3 · = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((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 10623 . . . . . . 7 ℂ = (R × R)
2120eleq2i 2824 . . . . . 6 (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R))
2220eleq2i 2824 . . . . . 6 (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R))
2321, 22anbi12i 630 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)))
2423anbi1i 627 . . . 4 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((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)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)))
2524oprabbii 7237 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((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)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))}
2619, 25eqtri 2761 . 2 · = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))}
271, 18, 26ov3 7329 1 (((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ · ⟨𝐶, 𝐷⟩) = ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wex 1786  wcel 2114  cop 4522   × cxp 5523  (class class class)co 7172  {coprab 7173  Rcnr 10367  -1Rcm1r 10370   +R cplr 10371   ·R cmr 10372  cc 10615   · cmul 10622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7175  df-oprab 7176  df-c 10623  df-mul 10629
This theorem is referenced by:  mulresr  10641  mulcnsrec  10646  axmulf  10648  axi2m1  10661  axcnre  10666
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