Definition df-iplp 7841

Description: Define addition on positive reals. From Section 11.2.1 of [HoTT], p. (varies). We write this definition to closely resemble the definition in HoTT although some of the conditions are redundant (for example, 𝑟 ∈ (1^st ‘𝑥) implies 𝑟 ∈ Q) and can be simplified as shown at genpdf 7881.

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, 26-Sep-2019.)

Assertion

Ref	Expression
df-iplp	⊢ +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-iplp

Step	Hyp	Ref	Expression
1		cpp 7666	. 2 class +P
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cnp 7664	. . 3 class P
5		vr	. . . . . . . . . 10 setvar 𝑟
6	5	cv 1401	. . . . . . . . 9 class 𝑟
7	2	cv 1401	. . . . . . . . . 10 class 𝑥
8		c1st 6376	. . . . . . . . . 10 class 1st
9	7, 8	cfv 5380	. . . . . . . . 9 class (1st ‘𝑥)
10	6, 9	wcel 2209	. . . . . . . 8 wff 𝑟 ∈ (1st ‘𝑥)
11		vs	. . . . . . . . . 10 setvar 𝑠
12	11	cv 1401	. . . . . . . . 9 class 𝑠
13	3	cv 1401	. . . . . . . . . 10 class 𝑦
14	13, 8	cfv 5380	. . . . . . . . 9 class (1st ‘𝑦)
15	12, 14	wcel 2209	. . . . . . . 8 wff 𝑠 ∈ (1st ‘𝑦)
16		vq	. . . . . . . . . 10 setvar 𝑞
17	16	cv 1401	. . . . . . . . 9 class 𝑞
18		cplq 7655	. . . . . . . . . 10 class +Q
19	6, 12, 18	co 6088	. . . . . . . . 9 class (𝑟 +Q 𝑠)
20	17, 19	wceq 1402	. . . . . . . 8 wff 𝑞 = (𝑟 +Q 𝑠)
21	10, 15, 20	w3a 1009	. . . . . . 7 wff (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
22		cnq 7653	. . . . . . 7 class Q
23	21, 11, 22	wrex 2529	. . . . . 6 wff ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
24	23, 5, 22	wrex 2529	. . . . 5 wff ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
25	24, 16, 22	crab 2532	. . . 4 class {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}
26		c2nd 6377	. . . . . . . . . 10 class 2nd
27	7, 26	cfv 5380	. . . . . . . . 9 class (2nd ‘𝑥)
28	6, 27	wcel 2209	. . . . . . . 8 wff 𝑟 ∈ (2nd ‘𝑥)
29	13, 26	cfv 5380	. . . . . . . . 9 class (2nd ‘𝑦)
30	12, 29	wcel 2209	. . . . . . . 8 wff 𝑠 ∈ (2nd ‘𝑦)
31	28, 30, 20	w3a 1009	. . . . . . 7 wff (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
32	31, 11, 22	wrex 2529	. . . . . 6 wff ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
33	32, 5, 22	wrex 2529	. . . . 5 wff ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
34	33, 16, 22	crab 2532	. . . 4 class {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}
35	25, 34	cop 3712	. . 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 6090	. 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 1402	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 used by:  addnqprl  7902  addnqpru  7903  addclpr  7910  plpvlu  7911  dmplp  7913  addnqprlemrl  7930  addnqprlemru  7931  addassprg  7952  distrlem1prl  7955  distrlem1pru  7956  distrlem4prl  7957  distrlem4pru  7958  distrlem5prl  7959  distrlem5pru  7960  ltaddpr  7970  ltexprlemfl  7982  ltexprlemrl  7983  ltexprlemfu  7984  ltexprlemru  7985  addcanprleml  7987  addcanprlemu  7988  cauappcvgprlemladdfu  8027  cauappcvgprlemladdfl  8028  caucvgprlemladdfu  8050