ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-mul Structured version   GIF version

Definition df-mul 6703
Description: Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.)
Assertion
Ref Expression
df-mul · = {⟨⟨x, y⟩, z⟩ ∣ ((x 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 6696 . 2 class ·
2 vx . . . . . . 7 setvar x
32cv 1241 . . . . . 6 class x
4 cc 6689 . . . . . 6 class
53, 4wcel 1390 . . . . 5 wff x
6 vy . . . . . . 7 setvar y
76cv 1241 . . . . . 6 class y
87, 4wcel 1390 . . . . 5 wff y
95, 8wa 97 . . . 4 wff (x y ℂ)
10 vw . . . . . . . . . . . . 13 setvar w
1110cv 1241 . . . . . . . . . . . 12 class w
12 vv . . . . . . . . . . . . 13 setvar v
1312cv 1241 . . . . . . . . . . . 12 class v
1411, 13cop 3370 . . . . . . . . . . 11 class w, v
153, 14wceq 1242 . . . . . . . . . 10 wff x = ⟨w, v
16 vu . . . . . . . . . . . . 13 setvar u
1716cv 1241 . . . . . . . . . . . 12 class u
18 vf . . . . . . . . . . . . 13 setvar f
1918cv 1241 . . . . . . . . . . . 12 class f
2017, 19cop 3370 . . . . . . . . . . 11 class u, f
217, 20wceq 1242 . . . . . . . . . 10 wff y = ⟨u, f
2215, 21wa 97 . . . . . . . . 9 wff (x = ⟨w, v y = ⟨u, f⟩)
23 vz . . . . . . . . . . 11 setvar z
2423cv 1241 . . . . . . . . . 10 class z
25 cmr 6286 . . . . . . . . . . . . 13 class ·R
2611, 17, 25co 5455 . . . . . . . . . . . 12 class (w ·R u)
27 cm1r 6284 . . . . . . . . . . . . 13 class -1R
2813, 19, 25co 5455 . . . . . . . . . . . . 13 class (v ·R f)
2927, 28, 25co 5455 . . . . . . . . . . . 12 class (-1R ·R (v ·R f))
30 cplr 6285 . . . . . . . . . . . 12 class +R
3126, 29, 30co 5455 . . . . . . . . . . 11 class ((w ·R u) +R (-1R ·R (v ·R f)))
3213, 17, 25co 5455 . . . . . . . . . . . 12 class (v ·R u)
3311, 19, 25co 5455 . . . . . . . . . . . 12 class (w ·R f)
3432, 33, 30co 5455 . . . . . . . . . . 11 class ((v ·R u) +R (w ·R f))
3531, 34cop 3370 . . . . . . . . . 10 class ⟨((w ·R u) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·R f))⟩
3624, 35wceq 1242 . . . . . . . . 9 wff z = ⟨((w ·R u) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·R f))⟩
3722, 36wa 97 . . . . . . . 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 1378 . . . . . . 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 1378 . . . . . 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 1378 . . . . 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 1378 . . . 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 97 . . 3 wff ((x 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 5456 . 2 class {⟨⟨x, y⟩, z⟩ ∣ ((x 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 1242 1 wff · = {⟨⟨x, y⟩, z⟩ ∣ ((x 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  6712
  Copyright terms: Public domain W3C validator