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Definition df-mul 6959
Description: Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.)
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 6952 . 2 class ·
2 vx . . . . . . 7 setvar 𝑥
32cv 1258 . . . . . 6 class 𝑥
4 cc 6945 . . . . . 6 class
53, 4wcel 1409 . . . . 5 wff 𝑥 ∈ ℂ
6 vy . . . . . . 7 setvar 𝑦
76cv 1258 . . . . . 6 class 𝑦
87, 4wcel 1409 . . . . 5 wff 𝑦 ∈ ℂ
95, 8wa 101 . . . 4 wff (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)
10 vw . . . . . . . . . . . . 13 setvar 𝑤
1110cv 1258 . . . . . . . . . . . 12 class 𝑤
12 vv . . . . . . . . . . . . 13 setvar 𝑣
1312cv 1258 . . . . . . . . . . . 12 class 𝑣
1411, 13cop 3406 . . . . . . . . . . 11 class 𝑤, 𝑣
153, 14wceq 1259 . . . . . . . . . 10 wff 𝑥 = ⟨𝑤, 𝑣
16 vu . . . . . . . . . . . . 13 setvar 𝑢
1716cv 1258 . . . . . . . . . . . 12 class 𝑢
18 vf . . . . . . . . . . . . 13 setvar 𝑓
1918cv 1258 . . . . . . . . . . . 12 class 𝑓
2017, 19cop 3406 . . . . . . . . . . 11 class 𝑢, 𝑓
217, 20wceq 1259 . . . . . . . . . 10 wff 𝑦 = ⟨𝑢, 𝑓
2215, 21wa 101 . . . . . . . . 9 wff (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)
23 vz . . . . . . . . . . 11 setvar 𝑧
2423cv 1258 . . . . . . . . . 10 class 𝑧
25 cmr 6458 . . . . . . . . . . . . 13 class ·R
2611, 17, 25co 5540 . . . . . . . . . . . 12 class (𝑤 ·R 𝑢)
27 cm1r 6456 . . . . . . . . . . . . 13 class -1R
2813, 19, 25co 5540 . . . . . . . . . . . . 13 class (𝑣 ·R 𝑓)
2927, 28, 25co 5540 . . . . . . . . . . . 12 class (-1R ·R (𝑣 ·R 𝑓))
30 cplr 6457 . . . . . . . . . . . 12 class +R
3126, 29, 30co 5540 . . . . . . . . . . 11 class ((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓)))
3213, 17, 25co 5540 . . . . . . . . . . . 12 class (𝑣 ·R 𝑢)
3311, 19, 25co 5540 . . . . . . . . . . . 12 class (𝑤 ·R 𝑓)
3432, 33, 30co 5540 . . . . . . . . . . 11 class ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))
3531, 34cop 3406 . . . . . . . . . 10 class ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩
3624, 35wceq 1259 . . . . . . . . 9 wff 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩
3722, 36wa 101 . . . . . . . 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 1397 . . . . . . 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 1397 . . . . . 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 1397 . . . . 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 1397 . . . 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 101 . . 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 5541 . 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 1259 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 set class
This definition is referenced by:  mulcnsr  6969
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