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

 Description: Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
Assertion
Ref Expression
addnqprulem (((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) → (𝑆 <Q 𝑋 → ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) ∈ 𝑈))

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . 5 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → 𝑆 <Q 𝑋)
2 ltrnqi 7077 . . . . . 6 (𝑆 <Q 𝑋 → (*Q𝑋) <Q (*Q𝑆))
3 simplr 498 . . . . . . . . 9 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → 𝑋Q)
4 recclnq 7048 . . . . . . . . 9 (𝑋Q → (*Q𝑋) ∈ Q)
53, 4syl 14 . . . . . . . 8 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → (*Q𝑋) ∈ Q)
6 ltrelnq 7021 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
76brel 4519 . . . . . . . . . . 11 (𝑆 <Q 𝑋 → (𝑆Q𝑋Q))
87adantl 272 . . . . . . . . . 10 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → (𝑆Q𝑋Q))
98simpld 111 . . . . . . . . 9 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → 𝑆Q)
10 recclnq 7048 . . . . . . . . 9 (𝑆Q → (*Q𝑆) ∈ Q)
119, 10syl 14 . . . . . . . 8 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → (*Q𝑆) ∈ Q)
12 ltmnqg 7057 . . . . . . . 8 (((*Q𝑋) ∈ Q ∧ (*Q𝑆) ∈ Q𝑋Q) → ((*Q𝑋) <Q (*Q𝑆) ↔ (𝑋 ·Q (*Q𝑋)) <Q (𝑋 ·Q (*Q𝑆))))
135, 11, 3, 12syl3anc 1181 . . . . . . 7 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → ((*Q𝑋) <Q (*Q𝑆) ↔ (𝑋 ·Q (*Q𝑋)) <Q (𝑋 ·Q (*Q𝑆))))
14 ltmnqg 7057 . . . . . . . . 9 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
1514adantl 272 . . . . . . . 8 (((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
16 mulclnq 7032 . . . . . . . . 9 ((𝑋Q ∧ (*Q𝑋) ∈ Q) → (𝑋 ·Q (*Q𝑋)) ∈ Q)
173, 5, 16syl2anc 404 . . . . . . . 8 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → (𝑋 ·Q (*Q𝑋)) ∈ Q)
18 mulclnq 7032 . . . . . . . . 9 ((𝑋Q ∧ (*Q𝑆) ∈ Q) → (𝑋 ·Q (*Q𝑆)) ∈ Q)
193, 11, 18syl2anc 404 . . . . . . . 8 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → (𝑋 ·Q (*Q𝑆)) ∈ Q)
20 elprnqu 7138 . . . . . . . . 9 ((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) → 𝐺Q)
2120ad2antrr 473 . . . . . . . 8 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → 𝐺Q)
22 mulcomnqg 7039 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
2322adantl 272 . . . . . . . 8 (((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
2415, 17, 19, 21, 23caovord2d 5852 . . . . . . 7 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → ((𝑋 ·Q (*Q𝑋)) <Q (𝑋 ·Q (*Q𝑆)) ↔ ((𝑋 ·Q (*Q𝑋)) ·Q 𝐺) <Q ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺)))
2513, 24bitrd 187 . . . . . 6 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → ((*Q𝑋) <Q (*Q𝑆) ↔ ((𝑋 ·Q (*Q𝑋)) ·Q 𝐺) <Q ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺)))
262, 25syl5ib 153 . . . . 5 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → (𝑆 <Q 𝑋 → ((𝑋 ·Q (*Q𝑋)) ·Q 𝐺) <Q ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺)))
271, 26mpd 13 . . . 4 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → ((𝑋 ·Q (*Q𝑋)) ·Q 𝐺) <Q ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺))
28 recidnq 7049 . . . . . . . 8 (𝑋Q → (𝑋 ·Q (*Q𝑋)) = 1Q)
2928oveq1d 5705 . . . . . . 7 (𝑋Q → ((𝑋 ·Q (*Q𝑋)) ·Q 𝐺) = (1Q ·Q 𝐺))
30 1nq 7022 . . . . . . . . 9 1QQ
31 mulcomnqg 7039 . . . . . . . . 9 ((1QQ𝐺Q) → (1Q ·Q 𝐺) = (𝐺 ·Q 1Q))
3230, 31mpan 416 . . . . . . . 8 (𝐺Q → (1Q ·Q 𝐺) = (𝐺 ·Q 1Q))
33 mulidnq 7045 . . . . . . . 8 (𝐺Q → (𝐺 ·Q 1Q) = 𝐺)
3432, 33eqtrd 2127 . . . . . . 7 (𝐺Q → (1Q ·Q 𝐺) = 𝐺)
3529, 34sylan9eqr 2149 . . . . . 6 ((𝐺Q𝑋Q) → ((𝑋 ·Q (*Q𝑋)) ·Q 𝐺) = 𝐺)
3635breq1d 3877 . . . . 5 ((𝐺Q𝑋Q) → (((𝑋 ·Q (*Q𝑋)) ·Q 𝐺) <Q ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) ↔ 𝐺 <Q ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺)))
3721, 3, 36syl2anc 404 . . . 4 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → (((𝑋 ·Q (*Q𝑋)) ·Q 𝐺) <Q ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) ↔ 𝐺 <Q ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺)))
3827, 37mpbid 146 . . 3 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → 𝐺 <Q ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺))
39 prcunqu 7141 . . . 4 ((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) → (𝐺 <Q ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) → ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) ∈ 𝑈))
4039ad2antrr 473 . . 3 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → (𝐺 <Q ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) → ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) ∈ 𝑈))
4138, 40mpd 13 . 2 ((((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) ∧ 𝑆 <Q 𝑋) → ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) ∈ 𝑈)
4241ex 114 1 (((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) → (𝑆 <Q 𝑋 → ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) ∈ 𝑈))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 927   = wceq 1296   ∈ wcel 1445  ⟨cop 3469   class class class wbr 3867  ‘cfv 5049  (class class class)co 5690  Qcnq 6936  1Qc1q 6937   ·Q cmq 6939  *Qcrq 6940
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