HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version

Definition df-mul 6972
Description: Define multiplication over complex numbers.
Assertion
Ref Expression
df-mul |- x. = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((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 6965 . 2 class x.
2 vx . . . . . . 7 set x
32cv 1456 . . . . . 6 class x
4 cc 6958 . . . . . 6 class CC
53, 4wcel 1459 . . . . 5 wff x e. CC
6 vy . . . . . . 7 set y
76cv 1456 . . . . . 6 class y
87, 4wcel 1459 . . . . 5 wff y e. CC
95, 8wa 382 . . . 4 wff (x e. CC /\ y e. CC)
10 vw . . . . . . . . . . . . 13 set w
1110cv 1456 . . . . . . . . . . . 12 class w
12 vv . . . . . . . . . . . . 13 set v
1312cv 1456 . . . . . . . . . . . 12 class v
1411, 13cop 3097 . . . . . . . . . . 11 class <.w, v>.
153, 14wceq 1457 . . . . . . . . . 10 wff x = <.w, v>.
16 vu . . . . . . . . . . . . 13 set u
1716cv 1456 . . . . . . . . . . . 12 class u
18 vf . . . . . . . . . . . . 13 set f
1918cv 1456 . . . . . . . . . . . 12 class f
2017, 19cop 3097 . . . . . . . . . . 11 class <.u, f>.
217, 20wceq 1457 . . . . . . . . . 10 wff y = <.u, f>.
2215, 21wa 382 . . . . . . . . 9 wff (x = <.w, v>. /\ y = <.u, f>.)
23 vz . . . . . . . . . . 11 set z
2423cv 1456 . . . . . . . . . 10 class z
25 cmr 6715 . . . . . . . . . . . . 13 class .R
2611, 17, 25co 4924 . . . . . . . . . . . 12 class (w .R u)
27 cm1r 6713 . . . . . . . . . . . . 13 class -1R
2813, 19, 25co 4924 . . . . . . . . . . . . 13 class (v .R f)
2927, 28, 25co 4924 . . . . . . . . . . . 12 class (-1R .R (v .R f))
30 cplr 6714 . . . . . . . . . . . 12 class +R
3126, 29, 30co 4924 . . . . . . . . . . 11 class ((w .R u) +R (-1R .R (v .R f)))
3213, 17, 25co 4924 . . . . . . . . . . . 12 class (v .R u)
3311, 19, 25co 4924 . . . . . . . . . . . 12 class (w .R f)
3432, 33, 30co 4924 . . . . . . . . . . 11 class ((v .R u) +R (w .R f))
3531, 34cop 3097 . . . . . . . . . 10 class <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>.
3624, 35wceq 1457 . . . . . . . . 9 wff z = <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>.
3722, 36wa 382 . . . . . . . 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 1380 . . . . . . 7 wff E.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 1380 . . . . . 6 wff E.uE.f((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 1380 . . . . 5 wff E.vE.uE.f((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 1380 . . . 4 wff E.wE.vE.uE.f((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 382 . . 3 wff ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((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, 23copab2 4925 . 2 class {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((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 1457 1 wff x. = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((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 6981  axmulopr 6991
Copyright terms: Public domain