Definition df-imp 7780

Description: Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 7779 or the definition of multiplication on positive reals in Metamath Proof Explorer. This is as opposed to the more complicated definition of multiplication given in Section 11.2.1 of [HoTT], p. (varies), which appears to be motivated by handling negative numbers or handling modified Dedekind cuts in which locatedness is omitted.

This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.)

Assertion

Ref	Expression
df-imp	⊢ ·P = (𝑥 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}⟩)

Distinct variable group:   𝑥,𝑦,𝑞,𝑟,𝑠

Detailed syntax breakdown of Definition df-imp

Step	Hyp	Ref	Expression
1		cmp 7605	. 2 class ·P
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cnp 7602	. . 3 class P
5		vr	. . . . . . . . . 10 setvar 𝑟
6	5	cv 1397	. . . . . . . . 9 class 𝑟
7	2	cv 1397	. . . . . . . . . 10 class 𝑥
8		c1st 6331	. . . . . . . . . 10 class 1st
9	7, 8	cfv 5351	. . . . . . . . 9 class (1st ‘𝑥)
10	6, 9	wcel 2203	. . . . . . . 8 wff 𝑟 ∈ (1st ‘𝑥)
11		vs	. . . . . . . . . 10 setvar 𝑠
12	11	cv 1397	. . . . . . . . 9 class 𝑠
13	3	cv 1397	. . . . . . . . . 10 class 𝑦
14	13, 8	cfv 5351	. . . . . . . . 9 class (1st ‘𝑦)
15	12, 14	wcel 2203	. . . . . . . 8 wff 𝑠 ∈ (1st ‘𝑦)
16		vq	. . . . . . . . . 10 setvar 𝑞
17	16	cv 1397	. . . . . . . . 9 class 𝑞
18		cmq 7594	. . . . . . . . . 10 class ·Q
19	6, 12, 18	co 6049	. . . . . . . . 9 class (𝑟 ·Q 𝑠)
20	17, 19	wceq 1398	. . . . . . . 8 wff 𝑞 = (𝑟 ·Q 𝑠)
21	10, 15, 20	w3a 1005	. . . . . . 7 wff (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
22		cnq 7591	. . . . . . 7 class Q
23	21, 11, 22	wrex 2521	. . . . . 6 wff ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
24	23, 5, 22	wrex 2521	. . . . 5 wff ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
25	24, 16, 22	crab 2524	. . . 4 class {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}
26		c2nd 6332	. . . . . . . . . 10 class 2nd
27	7, 26	cfv 5351	. . . . . . . . 9 class (2nd ‘𝑥)
28	6, 27	wcel 2203	. . . . . . . 8 wff 𝑟 ∈ (2nd ‘𝑥)
29	13, 26	cfv 5351	. . . . . . . . 9 class (2nd ‘𝑦)
30	12, 29	wcel 2203	. . . . . . . 8 wff 𝑠 ∈ (2nd ‘𝑦)
31	28, 30, 20	w3a 1005	. . . . . . 7 wff (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
32	31, 11, 22	wrex 2521	. . . . . 6 wff ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
33	32, 5, 22	wrex 2521	. . . . 5 wff ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
34	33, 16, 22	crab 2524	. . . 4 class {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}
35	25, 34	cop 3691	. . 3 class ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}⟩
36	2, 3, 4, 4, 35	cmpo 6051	. 2 class (𝑥 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}⟩)
37	1, 36	wceq 1398	1 wff ·P = (𝑥 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}⟩)

This definition is referenced by:  mpvlu  7850  dmmp  7852  mulnqprl  7879  mulnqpru  7880  mulclpr  7883  mulnqprlemrl  7884  mulnqprlemru  7885  mulassprg  7892  distrlem1prl  7893  distrlem1pru  7894  distrlem4prl  7895  distrlem4pru  7896  distrlem5prl  7897  distrlem5pru  7898  1idprl  7901  1idpru  7902  recexprlem1ssl  7944  recexprlem1ssu  7945  recexprlemss1l  7946  recexprlemss1u  7947