Definition df-mp 10907

Description: Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 11044, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
df-mp	⊢ ·P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·Q 𝑢)})

Distinct variable group:   𝑥,𝑦,𝑤,𝑣,𝑢

Detailed syntax breakdown of Definition df-mp

Step	Hyp	Ref	Expression
1		cmp 10785	. 2 class ·P
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cnp 10782	. . 3 class P
5		vw	. . . . . . . 8 setvar 𝑤
6	5	cv 1541	. . . . . . 7 class 𝑤
7		vv	. . . . . . . . 9 setvar 𝑣
8	7	cv 1541	. . . . . . . 8 class 𝑣
9		vu	. . . . . . . . 9 setvar 𝑢
10	9	cv 1541	. . . . . . . 8 class 𝑢
11		cmq 10779	. . . . . . . 8 class ·Q
12	8, 10, 11	co 7368	. . . . . . 7 class (𝑣 ·Q 𝑢)
13	6, 12	wceq 1542	. . . . . 6 wff 𝑤 = (𝑣 ·Q 𝑢)
14	3	cv 1541	. . . . . 6 class 𝑦
15	13, 9, 14	wrex 3062	. . . . 5 wff ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·Q 𝑢)
16	2	cv 1541	. . . . 5 class 𝑥
17	15, 7, 16	wrex 3062	. . . 4 wff ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·Q 𝑢)
18	17, 5	cab 2715	. . 3 class {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·Q 𝑢)}
19	2, 3, 4, 4, 18	cmpo 7370	. 2 class (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·Q 𝑢)})
20	1, 19	wceq 1542	1 wff ·P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·Q 𝑢)})

This definition is referenced by:  mpv  10934  dmmp  10936  mulclprlem  10942  mulclpr  10943  mulasspr  10947  distrlem1pr  10948  distrlem4pr  10949  distrlem5pr  10950  1idpr  10952  reclem3pr  10972  reclem4pr  10973