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Theorem prcunqu 6583
 Description: An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
Assertion
Ref Expression
prcunqu ((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) → (𝐶 <Q 𝐵𝐵𝑈))

Proof of Theorem prcunqu
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 6463 . . . . . 6 <Q ⊆ (Q × Q)
21brel 4392 . . . . 5 (𝐶 <Q 𝐵 → (𝐶Q𝐵Q))
32simprd 107 . . . 4 (𝐶 <Q 𝐵𝐵Q)
43adantl 262 . . 3 (((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) ∧ 𝐶 <Q 𝐵) → 𝐵Q)
5 breq2 3768 . . . . . . 7 (𝑏 = 𝐵 → (𝐶 <Q 𝑏𝐶 <Q 𝐵))
6 eleq1 2100 . . . . . . 7 (𝑏 = 𝐵 → (𝑏𝑈𝐵𝑈))
75, 6imbi12d 223 . . . . . 6 (𝑏 = 𝐵 → ((𝐶 <Q 𝑏𝑏𝑈) ↔ (𝐶 <Q 𝐵𝐵𝑈)))
87imbi2d 219 . . . . 5 (𝑏 = 𝐵 → (((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) → (𝐶 <Q 𝑏𝑏𝑈)) ↔ ((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) → (𝐶 <Q 𝐵𝐵𝑈))))
91brel 4392 . . . . . . . 8 (𝐶 <Q 𝑏 → (𝐶Q𝑏Q))
10 an42 521 . . . . . . . . 9 (((𝐶Q𝑏Q) ∧ (𝐶𝑈 ∧ ⟨𝐿, 𝑈⟩ ∈ P)) ↔ ((𝐶Q𝐶𝑈) ∧ (⟨𝐿, 𝑈⟩ ∈ P𝑏Q)))
11 breq1 3767 . . . . . . . . . . . . . . . 16 (𝑐 = 𝐶 → (𝑐 <Q 𝑏𝐶 <Q 𝑏))
12 eleq1 2100 . . . . . . . . . . . . . . . 16 (𝑐 = 𝐶 → (𝑐𝑈𝐶𝑈))
1311, 12anbi12d 442 . . . . . . . . . . . . . . 15 (𝑐 = 𝐶 → ((𝑐 <Q 𝑏𝑐𝑈) ↔ (𝐶 <Q 𝑏𝐶𝑈)))
1413rspcev 2656 . . . . . . . . . . . . . 14 ((𝐶Q ∧ (𝐶 <Q 𝑏𝐶𝑈)) → ∃𝑐Q (𝑐 <Q 𝑏𝑐𝑈))
15 elinp 6572 . . . . . . . . . . . . . . . 16 (⟨𝐿, 𝑈⟩ ∈ P ↔ (((𝐿Q𝑈Q) ∧ (∃𝑐Q 𝑐𝐿 ∧ ∃𝑏Q 𝑏𝑈)) ∧ ((∀𝑐Q (𝑐𝐿 ↔ ∃𝑏Q (𝑐 <Q 𝑏𝑏𝐿)) ∧ ∀𝑏Q (𝑏𝑈 ↔ ∃𝑐Q (𝑐 <Q 𝑏𝑐𝑈))) ∧ ∀𝑐Q ¬ (𝑐𝐿𝑐𝑈) ∧ ∀𝑐Q𝑏Q (𝑐 <Q 𝑏 → (𝑐𝐿𝑏𝑈)))))
16 simpr1r 962 . . . . . . . . . . . . . . . 16 ((((𝐿Q𝑈Q) ∧ (∃𝑐Q 𝑐𝐿 ∧ ∃𝑏Q 𝑏𝑈)) ∧ ((∀𝑐Q (𝑐𝐿 ↔ ∃𝑏Q (𝑐 <Q 𝑏𝑏𝐿)) ∧ ∀𝑏Q (𝑏𝑈 ↔ ∃𝑐Q (𝑐 <Q 𝑏𝑐𝑈))) ∧ ∀𝑐Q ¬ (𝑐𝐿𝑐𝑈) ∧ ∀𝑐Q𝑏Q (𝑐 <Q 𝑏 → (𝑐𝐿𝑏𝑈)))) → ∀𝑏Q (𝑏𝑈 ↔ ∃𝑐Q (𝑐 <Q 𝑏𝑐𝑈)))
1715, 16sylbi 114 . . . . . . . . . . . . . . 15 (⟨𝐿, 𝑈⟩ ∈ P → ∀𝑏Q (𝑏𝑈 ↔ ∃𝑐Q (𝑐 <Q 𝑏𝑐𝑈)))
1817r19.21bi 2407 . . . . . . . . . . . . . 14 ((⟨𝐿, 𝑈⟩ ∈ P𝑏Q) → (𝑏𝑈 ↔ ∃𝑐Q (𝑐 <Q 𝑏𝑐𝑈)))
1914, 18syl5ibrcom 146 . . . . . . . . . . . . 13 ((𝐶Q ∧ (𝐶 <Q 𝑏𝐶𝑈)) → ((⟨𝐿, 𝑈⟩ ∈ P𝑏Q) → 𝑏𝑈))
20193impb 1100 . . . . . . . . . . . 12 ((𝐶Q𝐶 <Q 𝑏𝐶𝑈) → ((⟨𝐿, 𝑈⟩ ∈ P𝑏Q) → 𝑏𝑈))
21203com12 1108 . . . . . . . . . . 11 ((𝐶 <Q 𝑏𝐶Q𝐶𝑈) → ((⟨𝐿, 𝑈⟩ ∈ P𝑏Q) → 𝑏𝑈))
22213expib 1107 . . . . . . . . . 10 (𝐶 <Q 𝑏 → ((𝐶Q𝐶𝑈) → ((⟨𝐿, 𝑈⟩ ∈ P𝑏Q) → 𝑏𝑈)))
2322impd 242 . . . . . . . . 9 (𝐶 <Q 𝑏 → (((𝐶Q𝐶𝑈) ∧ (⟨𝐿, 𝑈⟩ ∈ P𝑏Q)) → 𝑏𝑈))
2410, 23syl5bi 141 . . . . . . . 8 (𝐶 <Q 𝑏 → (((𝐶Q𝑏Q) ∧ (𝐶𝑈 ∧ ⟨𝐿, 𝑈⟩ ∈ P)) → 𝑏𝑈))
259, 24mpand 405 . . . . . . 7 (𝐶 <Q 𝑏 → ((𝐶𝑈 ∧ ⟨𝐿, 𝑈⟩ ∈ P) → 𝑏𝑈))
2625com12 27 . . . . . 6 ((𝐶𝑈 ∧ ⟨𝐿, 𝑈⟩ ∈ P) → (𝐶 <Q 𝑏𝑏𝑈))
2726ancoms 255 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) → (𝐶 <Q 𝑏𝑏𝑈))
288, 27vtoclg 2613 . . . 4 (𝐵Q → ((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) → (𝐶 <Q 𝐵𝐵𝑈)))
2928impd 242 . . 3 (𝐵Q → (((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) ∧ 𝐶 <Q 𝐵) → 𝐵𝑈))
304, 29mpcom 32 . 2 (((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) ∧ 𝐶 <Q 𝐵) → 𝐵𝑈)
3130ex 108 1 ((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) → (𝐶 <Q 𝐵𝐵𝑈))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764  Qcnq 6378
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