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

Theorem prcunqu 6339
 Description: An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
Assertion
Ref Expression
prcunqu ((⟨𝐿, 𝑈 P 𝐶 𝑈) → (𝐶 <Q BB 𝑈))

Proof of Theorem prcunqu
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 6224 . . . . . 6 <Q ⊆ (Q × Q)
21brel 4319 . . . . 5 (𝐶 <Q B → (𝐶 Q B Q))
32simprd 107 . . . 4 (𝐶 <Q BB Q)
43adantl 262 . . 3 (((⟨𝐿, 𝑈 P 𝐶 𝑈) 𝐶 <Q B) → B Q)
5 breq2 3742 . . . . . . 7 (𝑏 = B → (𝐶 <Q 𝑏𝐶 <Q B))
6 eleq1 2082 . . . . . . 7 (𝑏 = B → (𝑏 𝑈B 𝑈))
75, 6imbi12d 223 . . . . . 6 (𝑏 = B → ((𝐶 <Q 𝑏𝑏 𝑈) ↔ (𝐶 <Q BB 𝑈)))
87imbi2d 219 . . . . 5 (𝑏 = B → (((⟨𝐿, 𝑈 P 𝐶 𝑈) → (𝐶 <Q 𝑏𝑏 𝑈)) ↔ ((⟨𝐿, 𝑈 P 𝐶 𝑈) → (𝐶 <Q BB 𝑈))))
91brel 4319 . . . . . . . 8 (𝐶 <Q 𝑏 → (𝐶 Q 𝑏 Q))
10 an42 508 . . . . . . . . 9 (((𝐶 Q 𝑏 Q) (𝐶 𝑈 𝐿, 𝑈 P)) ↔ ((𝐶 Q 𝐶 𝑈) (⟨𝐿, 𝑈 P 𝑏 Q)))
11 breq1 3741 . . . . . . . . . . . . . . . 16 (𝑐 = 𝐶 → (𝑐 <Q 𝑏𝐶 <Q 𝑏))
12 eleq1 2082 . . . . . . . . . . . . . . . 16 (𝑐 = 𝐶 → (𝑐 𝑈𝐶 𝑈))
1311, 12anbi12d 445 . . . . . . . . . . . . . . 15 (𝑐 = 𝐶 → ((𝑐 <Q 𝑏 𝑐 𝑈) ↔ (𝐶 <Q 𝑏 𝐶 𝑈)))
1413rspcev 2633 . . . . . . . . . . . . . 14 ((𝐶 Q (𝐶 <Q 𝑏 𝐶 𝑈)) → 𝑐 Q (𝑐 <Q 𝑏 𝑐 𝑈))
15 elinp 6328 . . . . . . . . . . . . . . . 16 (⟨𝐿, 𝑈 P ↔ (((𝐿Q 𝑈Q) (𝑐 Q 𝑐 𝐿 𝑏 Q 𝑏 𝑈)) ((𝑐 Q (𝑐 𝐿𝑏 Q (𝑐 <Q 𝑏 𝑏 𝐿)) 𝑏 Q (𝑏 𝑈𝑐 Q (𝑐 <Q 𝑏 𝑐 𝑈))) 𝑐 Q ¬ (𝑐 𝐿 𝑐 𝑈) 𝑐 Q 𝑏 Q (𝑐 <Q 𝑏 → (𝑐 𝐿 𝑏 𝑈)))))
16 simpr1r 950 . . . . . . . . . . . . . . . 16 ((((𝐿Q 𝑈Q) (𝑐 Q 𝑐 𝐿 𝑏 Q 𝑏 𝑈)) ((𝑐 Q (𝑐 𝐿𝑏 Q (𝑐 <Q 𝑏 𝑏 𝐿)) 𝑏 Q (𝑏 𝑈𝑐 Q (𝑐 <Q 𝑏 𝑐 𝑈))) 𝑐 Q ¬ (𝑐 𝐿 𝑐 𝑈) 𝑐 Q 𝑏 Q (𝑐 <Q 𝑏 → (𝑐 𝐿 𝑏 𝑈)))) → 𝑏 Q (𝑏 𝑈𝑐 Q (𝑐 <Q 𝑏 𝑐 𝑈)))
1715, 16sylbi 114 . . . . . . . . . . . . . . 15 (⟨𝐿, 𝑈 P𝑏 Q (𝑏 𝑈𝑐 Q (𝑐 <Q 𝑏 𝑐 𝑈)))
1817r19.21bi 2385 . . . . . . . . . . . . . 14 ((⟨𝐿, 𝑈 P 𝑏 Q) → (𝑏 𝑈𝑐 Q (𝑐 <Q 𝑏 𝑐 𝑈)))
1914, 18syl5ibrcom 146 . . . . . . . . . . . . 13 ((𝐶 Q (𝐶 <Q 𝑏 𝐶 𝑈)) → ((⟨𝐿, 𝑈 P 𝑏 Q) → 𝑏 𝑈))
20193impb 1086 . . . . . . . . . . . 12 ((𝐶 Q 𝐶 <Q 𝑏 𝐶 𝑈) → ((⟨𝐿, 𝑈 P 𝑏 Q) → 𝑏 𝑈))
21203com12 1094 . . . . . . . . . . 11 ((𝐶 <Q 𝑏 𝐶 Q 𝐶 𝑈) → ((⟨𝐿, 𝑈 P 𝑏 Q) → 𝑏 𝑈))
22213expib 1093 . . . . . . . . . 10 (𝐶 <Q 𝑏 → ((𝐶 Q 𝐶 𝑈) → ((⟨𝐿, 𝑈 P 𝑏 Q) → 𝑏 𝑈)))
2322impd 242 . . . . . . . . 9 (𝐶 <Q 𝑏 → (((𝐶 Q 𝐶 𝑈) (⟨𝐿, 𝑈 P 𝑏 Q)) → 𝑏 𝑈))
2410, 23syl5bi 141 . . . . . . . 8 (𝐶 <Q 𝑏 → (((𝐶 Q 𝑏 Q) (𝐶 𝑈 𝐿, 𝑈 P)) → 𝑏 𝑈))
259, 24mpand 407 . . . . . . 7 (𝐶 <Q 𝑏 → ((𝐶 𝑈 𝐿, 𝑈 P) → 𝑏 𝑈))
2625com12 27 . . . . . 6 ((𝐶 𝑈 𝐿, 𝑈 P) → (𝐶 <Q 𝑏𝑏 𝑈))
2726ancoms 255 . . . . 5 ((⟨𝐿, 𝑈 P 𝐶 𝑈) → (𝐶 <Q 𝑏𝑏 𝑈))
288, 27vtoclg 2590 . . . 4 (B Q → ((⟨𝐿, 𝑈 P 𝐶 𝑈) → (𝐶 <Q BB 𝑈)))
2928impd 242 . . 3 (B Q → (((⟨𝐿, 𝑈 P 𝐶 𝑈) 𝐶 <Q B) → B 𝑈))
304, 29mpcom 32 . 2 (((⟨𝐿, 𝑈 P 𝐶 𝑈) 𝐶 <Q B) → B 𝑈)
3130ex 108 1 ((⟨𝐿, 𝑈 P 𝐶 𝑈) → (𝐶 <Q BB 𝑈))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616   ∧ w3a 873   = wceq 1228   ∈ wcel 1374  ∀wral 2284  ∃wrex 2285   ⊆ wss 2894  ⟨cop 3353   class class class wbr 3738  Qcnq 6138
 Copyright terms: Public domain W3C validator