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 Description: Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
Assertion
Ref Expression
addnqpru ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋𝑋 ∈ (2nd ‘(𝐴 +P 𝐵))))

Dummy variables 𝑟 𝑞 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7131 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 addnqprulem 7184 . . . . . 6 (((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (2nd𝐴)) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd𝐴)))
31, 2sylanl1 395 . . . . 5 (((𝐴P𝐺 ∈ (2nd𝐴)) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd𝐴)))
43adantlr 462 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd𝐴)))
5 prop 7131 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
6 addnqprulem 7184 . . . . . 6 (((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐻 ∈ (2nd𝐵)) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd𝐵)))
75, 6sylanl1 395 . . . . 5 (((𝐵P𝐻 ∈ (2nd𝐵)) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd𝐵)))
87adantll 461 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd𝐵)))
94, 8jcad 302 . . 3 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd𝐵))))
10 simpl 108 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))))
11 simpl 108 . . . . 5 ((𝐴P𝐺 ∈ (2nd𝐴)) → 𝐴P)
12 simpl 108 . . . . 5 ((𝐵P𝐻 ∈ (2nd𝐵)) → 𝐵P)
1311, 12anim12i 332 . . . 4 (((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) → (𝐴P𝐵P))
14 df-iplp 7124 . . . . 5 +P = (𝑥P, 𝑦P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩)
15 addclnq 7031 . . . . 5 ((𝑟Q𝑠Q) → (𝑟 +Q 𝑠) ∈ Q)
1614, 15genppreclu 7171 . . . 4 ((𝐴P𝐵P) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd𝐵)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (2nd ‘(𝐴 +P 𝐵))))
1710, 13, 163syl 17 . . 3 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd𝐵)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (2nd ‘(𝐴 +P 𝐵))))
189, 17syld 45 . 2 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (2nd ‘(𝐴 +P 𝐵))))
19 simpr 109 . . . . 5 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → 𝑋Q)
20 elprnqu 7138 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (2nd𝐴)) → 𝐺Q)
211, 20sylan 278 . . . . . . . 8 ((𝐴P𝐺 ∈ (2nd𝐴)) → 𝐺Q)
2221ad2antrr 473 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → 𝐺Q)
23 elprnqu 7138 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐻 ∈ (2nd𝐵)) → 𝐻Q)
245, 23sylan 278 . . . . . . . 8 ((𝐵P𝐻 ∈ (2nd𝐵)) → 𝐻Q)
2524ad2antlr 474 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → 𝐻Q)
26 addclnq 7031 . . . . . . 7 ((𝐺Q𝐻Q) → (𝐺 +Q 𝐻) ∈ Q)
2722, 25, 26syl2anc 404 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝐺 +Q 𝐻) ∈ Q)
28 recclnq 7048 . . . . . 6 ((𝐺 +Q 𝐻) ∈ Q → (*Q‘(𝐺 +Q 𝐻)) ∈ Q)
2927, 28syl 14 . . . . 5 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (*Q‘(𝐺 +Q 𝐻)) ∈ Q)
30 mulassnqg 7040 . . . . 5 ((𝑋Q ∧ (*Q‘(𝐺 +Q 𝐻)) ∈ Q ∧ (𝐺 +Q 𝐻) ∈ Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))))
3119, 29, 27, 30syl3anc 1181 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))))
32 mulclnq 7032 . . . . . 6 ((𝑋Q ∧ (*Q‘(𝐺 +Q 𝐻)) ∈ Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q)
3319, 29, 32syl2anc 404 . . . . 5 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q)
34 distrnqg 7043 . . . . 5 (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q𝐺Q𝐻Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)))
3533, 22, 25, 34syl3anc 1181 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)))
36 mulcomnqg 7039 . . . . . . . 8 (((*Q‘(𝐺 +Q 𝐻)) ∈ Q ∧ (𝐺 +Q 𝐻) ∈ Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))))
3729, 27, 36syl2anc 404 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))))
38 recidnq 7049 . . . . . . . 8 ((𝐺 +Q 𝐻) ∈ Q → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
3927, 38syl 14 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
4037, 39eqtrd 2127 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = 1Q)
4140oveq2d 5706 . . . . 5 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = (𝑋 ·Q 1Q))
42 mulidnq 7045 . . . . . 6 (𝑋Q → (𝑋 ·Q 1Q) = 𝑋)
4342adantl 272 . . . . 5 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝑋 ·Q 1Q) = 𝑋)
4441, 43eqtrd 2127 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = 𝑋)
4531, 35, 443eqtr3d 2135 . . 3 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) = 𝑋)
4645eleq1d 2163 . 2 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ 𝑋 ∈ (2nd ‘(𝐴 +P 𝐵))))
4718, 46sylibd 148 1 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋𝑋 ∈ (2nd ‘(𝐴 +P 𝐵))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1296   ∈ wcel 1445  ⟨cop 3469   class class class wbr 3867  ‘cfv 5049  (class class class)co 5690  1st c1st 5947  2nd c2nd 5948  Qcnq 6936  1Qc1q 6937   +Q cplq 6938   ·Q cmq 6939  *Qcrq 6940
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