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

Theorem ltexprlemlol 6728
 Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 6739. (Contributed by Jim Kingdon, 21-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
Assertion
Ref Expression
ltexprlemlol ((𝐴<P 𝐵𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑞,𝑟,𝐴   𝑥,𝐵,𝑦,𝑞,𝑟   𝑥,𝐶,𝑦,𝑞,𝑟

Proof of Theorem ltexprlemlol
StepHypRef Expression
1 simplr 490 . . . . . 6 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑞Q)
2 simprrr 500 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))
32simpld 109 . . . . . 6 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑦 ∈ (2nd𝐴))
4 simprl 491 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑞 <Q 𝑟)
5 simpll 489 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝐴<P 𝐵)
6 ltrelpr 6631 . . . . . . . . . . . 12 <P ⊆ (P × P)
76brel 4417 . . . . . . . . . . 11 (𝐴<P 𝐵 → (𝐴P𝐵P))
87simpld 109 . . . . . . . . . 10 (𝐴<P 𝐵𝐴P)
9 prop 6601 . . . . . . . . . . 11 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
10 elprnqu 6608 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
119, 10sylan 271 . . . . . . . . . 10 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
128, 11sylan 271 . . . . . . . . 9 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
135, 3, 12syl2anc 397 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑦Q)
14 ltanqi 6528 . . . . . . . 8 ((𝑞 <Q 𝑟𝑦Q) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
154, 13, 14syl2anc 397 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
167simprd 111 . . . . . . . . 9 (𝐴<P 𝐵𝐵P)
175, 16syl 14 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝐵P)
182simprd 111 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑦 +Q 𝑟) ∈ (1st𝐵))
19 prop 6601 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
20 prcdnql 6610 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2119, 20sylan 271 . . . . . . . 8 ((𝐵P ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2217, 18, 21syl2anc 397 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2315, 22mpd 13 . . . . . 6 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑦 +Q 𝑞) ∈ (1st𝐵))
241, 3, 23jca32 297 . . . . 5 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
2524eximi 1505 . . . 4 (∃𝑦((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → ∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
26 ltexprlem.1 . . . . . . . . . 10 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
2726ltexprlemell 6724 . . . . . . . . 9 (𝑟 ∈ (1st𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
28 19.42v 1800 . . . . . . . . 9 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
2927, 28bitr4i 180 . . . . . . . 8 (𝑟 ∈ (1st𝐶) ↔ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
3029anbi2i 438 . . . . . . 7 ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
31 19.42v 1800 . . . . . . 7 (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
3230, 31bitr4i 180 . . . . . 6 ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
3332anbi2i 438 . . . . 5 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) ↔ ((𝐴<P 𝐵𝑞Q) ∧ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))))
34 19.42v 1800 . . . . 5 (∃𝑦((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) ↔ ((𝐴<P 𝐵𝑞Q) ∧ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))))
3533, 34bitr4i 180 . . . 4 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) ↔ ∃𝑦((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))))
3626ltexprlemell 6724 . . . . 5 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
37 19.42v 1800 . . . . 5 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
3836, 37bitr4i 180 . . . 4 (𝑞 ∈ (1st𝐶) ↔ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
3925, 35, 383imtr4i 194 . . 3 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) → 𝑞 ∈ (1st𝐶))
4039ex 112 . 2 ((𝐴<P 𝐵𝑞Q) → ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))
4140rexlimdvw 2451 1 ((𝐴<P 𝐵𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1257  ∃wex 1395   ∈ wcel 1407  ∃wrex 2322  {crab 2325  ⟨cop 3403   class class class wbr 3789  ‘cfv 4927  (class class class)co 5537  1st c1st 5790  2nd c2nd 5791  Qcnq 6406   +Q cplq 6408
 Copyright terms: Public domain W3C validator