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Theorem prcdnql 6332
 Description: A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
Assertion
Ref Expression
prcdnql ((⟨𝐿, 𝑈 P B 𝐿) → (𝐶 <Q B𝐶 𝐿))

Proof of Theorem prcdnql
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 6218 . . . . . 6 <Q ⊆ (Q × Q)
21brel 4315 . . . . 5 (𝐶 <Q B → (𝐶 Q B Q))
32simpld 105 . . . 4 (𝐶 <Q B𝐶 Q)
43adantl 262 . . 3 (((⟨𝐿, 𝑈 P B 𝐿) 𝐶 <Q B) → 𝐶 Q)
5 breq1 3737 . . . . . . 7 (𝑐 = 𝐶 → (𝑐 <Q B𝐶 <Q B))
6 eleq1 2078 . . . . . . 7 (𝑐 = 𝐶 → (𝑐 𝐿𝐶 𝐿))
75, 6imbi12d 223 . . . . . 6 (𝑐 = 𝐶 → ((𝑐 <Q B𝑐 𝐿) ↔ (𝐶 <Q B𝐶 𝐿)))
87imbi2d 219 . . . . 5 (𝑐 = 𝐶 → (((⟨𝐿, 𝑈 P B 𝐿) → (𝑐 <Q B𝑐 𝐿)) ↔ ((⟨𝐿, 𝑈 P B 𝐿) → (𝐶 <Q B𝐶 𝐿))))
91brel 4315 . . . . . . . . 9 (𝑐 <Q B → (𝑐 Q B Q))
109ancomd 254 . . . . . . . 8 (𝑐 <Q B → (B Q 𝑐 Q))
11 an42 508 . . . . . . . . 9 (((B Q 𝑐 Q) (B 𝐿 𝐿, 𝑈 P)) ↔ ((B Q B 𝐿) (⟨𝐿, 𝑈 P 𝑐 Q)))
12 breq2 3738 . . . . . . . . . . . . . . . 16 (𝑏 = B → (𝑐 <Q 𝑏𝑐 <Q B))
13 eleq1 2078 . . . . . . . . . . . . . . . 16 (𝑏 = B → (𝑏 𝐿B 𝐿))
1412, 13anbi12d 445 . . . . . . . . . . . . . . 15 (𝑏 = B → ((𝑐 <Q 𝑏 𝑏 𝐿) ↔ (𝑐 <Q B B 𝐿)))
1514rspcev 2629 . . . . . . . . . . . . . 14 ((B Q (𝑐 <Q B B 𝐿)) → 𝑏 Q (𝑐 <Q 𝑏 𝑏 𝐿))
16 elinp 6322 . . . . . . . . . . . . . . . 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)))))
17 simpr1l 947 . . . . . . . . . . . . . . . 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)))) → 𝑐 Q (𝑐 𝐿𝑏 Q (𝑐 <Q 𝑏 𝑏 𝐿)))
1816, 17sylbi 114 . . . . . . . . . . . . . . 15 (⟨𝐿, 𝑈 P𝑐 Q (𝑐 𝐿𝑏 Q (𝑐 <Q 𝑏 𝑏 𝐿)))
1918r19.21bi 2381 . . . . . . . . . . . . . 14 ((⟨𝐿, 𝑈 P 𝑐 Q) → (𝑐 𝐿𝑏 Q (𝑐 <Q 𝑏 𝑏 𝐿)))
2015, 19syl5ibrcom 146 . . . . . . . . . . . . 13 ((B Q (𝑐 <Q B B 𝐿)) → ((⟨𝐿, 𝑈 P 𝑐 Q) → 𝑐 𝐿))
21203impb 1084 . . . . . . . . . . . 12 ((B Q 𝑐 <Q B B 𝐿) → ((⟨𝐿, 𝑈 P 𝑐 Q) → 𝑐 𝐿))
22213com12 1092 . . . . . . . . . . 11 ((𝑐 <Q B B Q B 𝐿) → ((⟨𝐿, 𝑈 P 𝑐 Q) → 𝑐 𝐿))
23223expib 1091 . . . . . . . . . 10 (𝑐 <Q B → ((B Q B 𝐿) → ((⟨𝐿, 𝑈 P 𝑐 Q) → 𝑐 𝐿)))
2423impd 242 . . . . . . . . 9 (𝑐 <Q B → (((B Q B 𝐿) (⟨𝐿, 𝑈 P 𝑐 Q)) → 𝑐 𝐿))
2511, 24syl5bi 141 . . . . . . . 8 (𝑐 <Q B → (((B Q 𝑐 Q) (B 𝐿 𝐿, 𝑈 P)) → 𝑐 𝐿))
2610, 25mpand 407 . . . . . . 7 (𝑐 <Q B → ((B 𝐿 𝐿, 𝑈 P) → 𝑐 𝐿))
2726com12 27 . . . . . 6 ((B 𝐿 𝐿, 𝑈 P) → (𝑐 <Q B𝑐 𝐿))
2827ancoms 255 . . . . 5 ((⟨𝐿, 𝑈 P B 𝐿) → (𝑐 <Q B𝑐 𝐿))
298, 28vtoclg 2586 . . . 4 (𝐶 Q → ((⟨𝐿, 𝑈 P B 𝐿) → (𝐶 <Q B𝐶 𝐿)))
3029impd 242 . . 3 (𝐶 Q → (((⟨𝐿, 𝑈 P B 𝐿) 𝐶 <Q B) → 𝐶 𝐿))
314, 30mpcom 32 . 2 (((⟨𝐿, 𝑈 P B 𝐿) 𝐶 <Q B) → 𝐶 𝐿)
3231ex 108 1 ((⟨𝐿, 𝑈 P B 𝐿) → (𝐶 <Q B𝐶 𝐿))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616   ∧ w3a 871   = wceq 1226   ∈ wcel 1370  ∀wral 2280  ∃wrex 2281   ⊆ wss 2890  ⟨cop 3349   class class class wbr 3734  Qcnq 6134
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