ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-iplp GIF version

Definition df-iplp 7430
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 ∈ (1st𝑥) implies 𝑟Q) and can be simplified as shown at genpdf 7470.

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 = (𝑥P, 𝑦P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩)
Distinct variable group:   𝑥,𝑦,𝑞,𝑟,𝑠

Detailed syntax breakdown of Definition df-iplp
StepHypRef Expression
1 cpp 7255 . 2 class +P
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cnp 7253 . . 3 class P
5 vr . . . . . . . . . 10 setvar 𝑟
65cv 1347 . . . . . . . . 9 class 𝑟
72cv 1347 . . . . . . . . . 10 class 𝑥
8 c1st 6117 . . . . . . . . . 10 class 1st
97, 8cfv 5198 . . . . . . . . 9 class (1st𝑥)
106, 9wcel 2141 . . . . . . . 8 wff 𝑟 ∈ (1st𝑥)
11 vs . . . . . . . . . 10 setvar 𝑠
1211cv 1347 . . . . . . . . 9 class 𝑠
133cv 1347 . . . . . . . . . 10 class 𝑦
1413, 8cfv 5198 . . . . . . . . 9 class (1st𝑦)
1512, 14wcel 2141 . . . . . . . 8 wff 𝑠 ∈ (1st𝑦)
16 vq . . . . . . . . . 10 setvar 𝑞
1716cv 1347 . . . . . . . . 9 class 𝑞
18 cplq 7244 . . . . . . . . . 10 class +Q
196, 12, 18co 5853 . . . . . . . . 9 class (𝑟 +Q 𝑠)
2017, 19wceq 1348 . . . . . . . 8 wff 𝑞 = (𝑟 +Q 𝑠)
2110, 15, 20w3a 973 . . . . . . 7 wff (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
22 cnq 7242 . . . . . . 7 class Q
2321, 11, 22wrex 2449 . . . . . 6 wff 𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
2423, 5, 22wrex 2449 . . . . 5 wff 𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
2524, 16, 22crab 2452 . . . 4 class {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}
26 c2nd 6118 . . . . . . . . . 10 class 2nd
277, 26cfv 5198 . . . . . . . . 9 class (2nd𝑥)
286, 27wcel 2141 . . . . . . . 8 wff 𝑟 ∈ (2nd𝑥)
2913, 26cfv 5198 . . . . . . . . 9 class (2nd𝑦)
3012, 29wcel 2141 . . . . . . . 8 wff 𝑠 ∈ (2nd𝑦)
3128, 30, 20w3a 973 . . . . . . 7 wff (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
3231, 11, 22wrex 2449 . . . . . 6 wff 𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
3332, 5, 22wrex 2449 . . . . 5 wff 𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))
3433, 16, 22crab 2452 . . . 4 class {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}
3525, 34cop 3586 . . 3 class ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩
362, 3, 4, 4, 35cmpo 5855 . 2 class (𝑥P, 𝑦P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩)
371, 36wceq 1348 1 wff +P = (𝑥P, 𝑦P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩)
Colors of variables: wff set class
This definition is referenced by:  addnqprl  7491  addnqpru  7492  addclpr  7499  plpvlu  7500  dmplp  7502  addnqprlemrl  7519  addnqprlemru  7520  addassprg  7541  distrlem1prl  7544  distrlem1pru  7545  distrlem4prl  7546  distrlem4pru  7547  distrlem5prl  7548  distrlem5pru  7549  ltaddpr  7559  ltexprlemfl  7571  ltexprlemrl  7572  ltexprlemfu  7573  ltexprlemru  7574  addcanprleml  7576  addcanprlemu  7577  cauappcvgprlemladdfu  7616  cauappcvgprlemladdfl  7617  caucvgprlemladdfu  7639
  Copyright terms: Public domain W3C validator