Definition df-iplp 7442

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 7482.

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 7267	. 2 class +P
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cnp 7265	. . 3 class P
5		vr	. . . . . . . . . 10 setvar 𝑟
6	5	cv 1352	. . . . . . . . 9 class 𝑟
7	2	cv 1352	. . . . . . . . . 10 class 𝑥
8		c1st 6129	. . . . . . . . . 10 class 1st
9	7, 8	cfv 5208	. . . . . . . . 9 class (1st ‘𝑥)
10	6, 9	wcel 2146	. . . . . . . 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 5208	. . . . . . . . 9 class (1st ‘𝑦)
15	12, 14	wcel 2146	. . . . . . . 8 wff 𝑠 ∈ (1st ‘𝑦)
16		vq	. . . . . . . . . 10 setvar 𝑞
17	16	cv 1352	. . . . . . . . 9 class 𝑞
18		cplq 7256	. . . . . . . . . 10 class +Q
19	6, 12, 18	co 5865	. . . . . . . . 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 7254	. . . . . . 7 class Q
23	21, 11, 22	wrex 2454	. . . . . 6 wff ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
24	23, 5, 22	wrex 2454	. . . . 5 wff ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
25	24, 16, 22	crab 2457	. . . 4 class {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}
26		c2nd 6130	. . . . . . . . . 10 class 2nd
27	7, 26	cfv 5208	. . . . . . . . 9 class (2nd ‘𝑥)
28	6, 27	wcel 2146	. . . . . . . 8 wff 𝑟 ∈ (2nd ‘𝑥)
29	13, 26	cfv 5208	. . . . . . . . 9 class (2nd ‘𝑦)
30	12, 29	wcel 2146	. . . . . . . 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 2454	. . . . . 6 wff ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
33	32, 5, 22	wrex 2454	. . . . 5 wff ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
34	33, 16, 22	crab 2457	. . . 4 class {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}
35	25, 34	cop 3592	. . 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 5867	. 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:  addnqprl  7503  addnqpru  7504  addclpr  7511  plpvlu  7512  dmplp  7514  addnqprlemrl  7531  addnqprlemru  7532  addassprg  7553  distrlem1prl  7556  distrlem1pru  7557  distrlem4prl  7558  distrlem4pru  7559  distrlem5prl  7560  distrlem5pru  7561  ltaddpr  7571  ltexprlemfl  7583  ltexprlemrl  7584  ltexprlemfu  7585  ltexprlemru  7586  addcanprleml  7588  addcanprlemu  7589  cauappcvgprlemladdfu  7628  cauappcvgprlemladdfl  7629  caucvgprlemladdfu  7651