Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-mul Structured version   Visualization version   GIF version

Definition df-mul 10541
 Description: Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-mul · = {⟨⟨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))⟩))}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Detailed syntax breakdown of Definition df-mul
StepHypRef Expression
1 cmul 10534 . 2 class ·
2 vx . . . . . . 7 setvar 𝑥
32cv 1529 . . . . . 6 class 𝑥
4 cc 10527 . . . . . 6 class
53, 4wcel 2107 . . . . 5 wff 𝑥 ∈ ℂ
6 vy . . . . . . 7 setvar 𝑦
76cv 1529 . . . . . 6 class 𝑦
87, 4wcel 2107 . . . . 5 wff 𝑦 ∈ ℂ
95, 8wa 398 . . . 4 wff (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)
10 vw . . . . . . . . . . . . 13 setvar 𝑤
1110cv 1529 . . . . . . . . . . . 12 class 𝑤
12 vv . . . . . . . . . . . . 13 setvar 𝑣
1312cv 1529 . . . . . . . . . . . 12 class 𝑣
1411, 13cop 4565 . . . . . . . . . . 11 class 𝑤, 𝑣
153, 14wceq 1530 . . . . . . . . . 10 wff 𝑥 = ⟨𝑤, 𝑣
16 vu . . . . . . . . . . . . 13 setvar 𝑢
1716cv 1529 . . . . . . . . . . . 12 class 𝑢
18 vf . . . . . . . . . . . . 13 setvar 𝑓
1918cv 1529 . . . . . . . . . . . 12 class 𝑓
2017, 19cop 4565 . . . . . . . . . . 11 class 𝑢, 𝑓
217, 20wceq 1530 . . . . . . . . . 10 wff 𝑦 = ⟨𝑢, 𝑓
2215, 21wa 398 . . . . . . . . 9 wff (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)
23 vz . . . . . . . . . . 11 setvar 𝑧
2423cv 1529 . . . . . . . . . 10 class 𝑧
25 cmr 10284 . . . . . . . . . . . . 13 class ·R
2611, 17, 25co 7148 . . . . . . . . . . . 12 class (𝑤 ·R 𝑢)
27 cm1r 10282 . . . . . . . . . . . . 13 class -1R
2813, 19, 25co 7148 . . . . . . . . . . . . 13 class (𝑣 ·R 𝑓)
2927, 28, 25co 7148 . . . . . . . . . . . 12 class (-1R ·R (𝑣 ·R 𝑓))
30 cplr 10283 . . . . . . . . . . . 12 class +R
3126, 29, 30co 7148 . . . . . . . . . . 11 class ((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓)))
3213, 17, 25co 7148 . . . . . . . . . . . 12 class (𝑣 ·R 𝑢)
3311, 19, 25co 7148 . . . . . . . . . . . 12 class (𝑤 ·R 𝑓)
3432, 33, 30co 7148 . . . . . . . . . . 11 class ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))
3531, 34cop 4565 . . . . . . . . . 10 class ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩
3624, 35wceq 1530 . . . . . . . . 9 wff 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩
3722, 36wa 398 . . . . . . . 8 wff ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)
3837, 18wex 1773 . . . . . . 7 wff 𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)
3938, 16wex 1773 . . . . . 6 wff 𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)
4039, 12wex 1773 . . . . 5 wff 𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)
4140, 10wex 1773 . . . 4 wff 𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)
429, 41wa 398 . . 3 wff ((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))⟩))
4342, 2, 6, 23coprab 7149 . 2 class {⟨⟨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))⟩))}
441, 43wceq 1530 1 wff · = {⟨⟨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))⟩))}
 Colors of variables: wff setvar class This definition is referenced by:  mulcnsr  10550  axmulf  10560
 Copyright terms: Public domain W3C validator