Definition df-imp 7682

Description: Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 7681 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 7507	. 2 class ·P
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cnp 7504	. . 3 class P
5		vr	. . . . . . . . . 10 setvar 𝑟
6	5	cv 1394	. . . . . . . . 9 class 𝑟
7	2	cv 1394	. . . . . . . . . 10 class 𝑥
8		c1st 6296	. . . . . . . . . 10 class 1st
9	7, 8	cfv 5324	. . . . . . . . 9 class (1st ‘𝑥)
10	6, 9	wcel 2200	. . . . . . . 8 wff 𝑟 ∈ (1st ‘𝑥)
11		vs	. . . . . . . . . 10 setvar 𝑠
12	11	cv 1394	. . . . . . . . 9 class 𝑠
13	3	cv 1394	. . . . . . . . . 10 class 𝑦
14	13, 8	cfv 5324	. . . . . . . . 9 class (1st ‘𝑦)
15	12, 14	wcel 2200	. . . . . . . 8 wff 𝑠 ∈ (1st ‘𝑦)
16		vq	. . . . . . . . . 10 setvar 𝑞
17	16	cv 1394	. . . . . . . . 9 class 𝑞
18		cmq 7496	. . . . . . . . . 10 class ·Q
19	6, 12, 18	co 6013	. . . . . . . . 9 class (𝑟 ·Q 𝑠)
20	17, 19	wceq 1395	. . . . . . . 8 wff 𝑞 = (𝑟 ·Q 𝑠)
21	10, 15, 20	w3a 1002	. . . . . . 7 wff (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
22		cnq 7493	. . . . . . 7 class Q
23	21, 11, 22	wrex 2509	. . . . . 6 wff ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
24	23, 5, 22	wrex 2509	. . . . 5 wff ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
25	24, 16, 22	crab 2512	. . . 4 class {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}
26		c2nd 6297	. . . . . . . . . 10 class 2nd
27	7, 26	cfv 5324	. . . . . . . . 9 class (2nd ‘𝑥)
28	6, 27	wcel 2200	. . . . . . . 8 wff 𝑟 ∈ (2nd ‘𝑥)
29	13, 26	cfv 5324	. . . . . . . . 9 class (2nd ‘𝑦)
30	12, 29	wcel 2200	. . . . . . . 8 wff 𝑠 ∈ (2nd ‘𝑦)
31	28, 30, 20	w3a 1002	. . . . . . 7 wff (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
32	31, 11, 22	wrex 2509	. . . . . 6 wff ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
33	32, 5, 22	wrex 2509	. . . . 5 wff ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))
34	33, 16, 22	crab 2512	. . . 4 class {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}
35	25, 34	cop 3670	. . 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 6015	. 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 1395	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  7752  dmmp  7754  mulnqprl  7781  mulnqpru  7782  mulclpr  7785  mulnqprlemrl  7786  mulnqprlemru  7787  mulassprg  7794  distrlem1prl  7795  distrlem1pru  7796  distrlem4prl  7797  distrlem4pru  7798  distrlem5prl  7799  distrlem5pru  7800  1idprl  7803  1idpru  7804  recexprlem1ssl  7846  recexprlem1ssu  7847  recexprlemss1l  7848  recexprlemss1u  7849