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

Theorem ltexprlemrnd 6578
 Description: Our constructed difference is rounded. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
Assertion
Ref Expression
ltexprlemrnd (A<P B → (𝑞 Q (𝑞 (1st𝐶) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶))) 𝑟 Q (𝑟 (2nd𝐶) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))))
Distinct variable groups:   x,y,𝑞,𝑟,A   x,B,y,𝑞,𝑟   x,𝐶,y,𝑞,𝑟

Proof of Theorem ltexprlemrnd
StepHypRef Expression
1 ltexprlem.1 . . . . . 6 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
21ltexprlemopl 6574 . . . . 5 ((A<P B 𝑞 Q 𝑞 (1st𝐶)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)))
323expia 1105 . . . 4 ((A<P B 𝑞 Q) → (𝑞 (1st𝐶) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶))))
41ltexprlemlol 6575 . . . 4 ((A<P B 𝑞 Q) → (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)) → 𝑞 (1st𝐶)))
53, 4impbid 120 . . 3 ((A<P B 𝑞 Q) → (𝑞 (1st𝐶) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶))))
65ralrimiva 2386 . 2 (A<P B𝑞 Q (𝑞 (1st𝐶) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶))))
71ltexprlemopu 6576 . . . . 5 ((A<P B 𝑟 Q 𝑟 (2nd𝐶)) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))
873expia 1105 . . . 4 ((A<P B 𝑟 Q) → (𝑟 (2nd𝐶) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶))))
91ltexprlemupu 6577 . . . 4 ((A<P B 𝑟 Q) → (𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)) → 𝑟 (2nd𝐶)))
108, 9impbid 120 . . 3 ((A<P B 𝑟 Q) → (𝑟 (2nd𝐶) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶))))
1110ralrimiva 2386 . 2 (A<P B𝑟 Q (𝑟 (2nd𝐶) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶))))
126, 11jca 290 1 (A<P B → (𝑞 Q (𝑞 (1st𝐶) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶))) 𝑟 Q (𝑟 (2nd𝐶) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  {crab 2304  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +Q cplq 6266
 Copyright terms: Public domain W3C validator