Definition df-imp 7471

Description: Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 7470 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 7296	. 2 class ·P
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cnp 7293	. . 3 class P
5		vr	. . . . . . . . . 10 setvar 𝑟
6	5	cv 1352	. . . . . . . . 9 class 𝑟
7	2	cv 1352	. . . . . . . . . 10 class 𝑥
8		c1st 6142	. . . . . . . . . 10 class 1st
9	7, 8	cfv 5218	. . . . . . . . 9 class (1st ‘𝑥)
10	6, 9	wcel 2148	. . . . . . . 8 wff 𝑟 ∈ (1st ‘𝑥)
11		vs	. . . . . . . . . 10 setvar 𝑠
12	11	cv 1352	. . . . . . . . 9 class 𝑠
13	3	cv 1352	. . . . . . . . . 10 class 𝑦
14	13, 8	cfv 5218	. . . . . . . . 9 class (1st ‘𝑦)
15	12, 14	wcel 2148	. . . . . . . 8 wff 𝑠 ∈ (1st ‘𝑦)
16		vq	. . . . . . . . . 10 setvar 𝑞
17	16	cv 1352	. . . . . . . . 9 class 𝑞
18		cmq 7285	. . . . . . . . . 10 class ·Q
19	6, 12, 18	co 5878	. . . . . . . . 9 class (𝑟 ·Q 𝑠)
20	17, 19	wceq 1353	. . . . . . . 8 wff 𝑞 = (𝑟 ·Q 𝑠)
21	10, 15, 20	w3a 978	. . . . . . 7 wff (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
22		cnq 7282	. . . . . . 7 class Q
23	21, 11, 22	wrex 2456	. . . . . 6 wff ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
24	23, 5, 22	wrex 2456	. . . . 5 wff ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
25	24, 16, 22	crab 2459	. . . 4 class {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}
26		c2nd 6143	. . . . . . . . . 10 class 2nd
27	7, 26	cfv 5218	. . . . . . . . 9 class (2nd ‘𝑥)
28	6, 27	wcel 2148	. . . . . . . 8 wff 𝑟 ∈ (2nd ‘𝑥)
29	13, 26	cfv 5218	. . . . . . . . 9 class (2nd ‘𝑦)
30	12, 29	wcel 2148	. . . . . . . 8 wff 𝑠 ∈ (2nd ‘𝑦)
31	28, 30, 20	w3a 978	. . . . . . 7 wff (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
32	31, 11, 22	wrex 2456	. . . . . 6 wff ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
33	32, 5, 22	wrex 2456	. . . . 5 wff ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
34	33, 16, 22	crab 2459	. . . 4 class {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}
35	25, 34	cop 3597	. . 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 5880	. 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 1353	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  7541  dmmp  7543  mulnqprl  7570  mulnqpru  7571  mulclpr  7574  mulnqprlemrl  7575  mulnqprlemru  7576  mulassprg  7583  distrlem1prl  7584  distrlem1pru  7585  distrlem4prl  7586  distrlem4pru  7587  distrlem5prl  7588  distrlem5pru  7589  1idprl  7592  1idpru  7593  recexprlem1ssl  7635  recexprlem1ssu  7636  recexprlemss1l  7637  recexprlemss1u  7638