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Theorem ltexprlemdisj 6761
 Description: Our constructed difference is disjoint. Lemma for ltexpri 6768. (Contributed by Jim Kingdon, 17-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
ltexprlemdisj (𝐴<P 𝐵 → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑞,𝐴   𝑥,𝐵,𝑦,𝑞   𝑥,𝐶,𝑦,𝑞

Proof of Theorem ltexprlemdisj
Dummy variables 𝑧 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltsonq 6553 . . . . . 6 <Q Or Q
2 ltrelnq 6520 . . . . . 6 <Q ⊆ (Q × Q)
31, 2son2lpi 4748 . . . . 5 ¬ (𝑦 <Q 𝑧𝑧 <Q 𝑦)
4 ltrelpr 6660 . . . . . . . . . . . . . . . 16 <P ⊆ (P × P)
54brel 4419 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵 → (𝐴P𝐵P))
65simprd 111 . . . . . . . . . . . . . 14 (𝐴<P 𝐵𝐵P)
7 prop 6630 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
86, 7syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
9 prltlu 6642 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
108, 9syl3an1 1179 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
11103expb 1116 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1211adantlr 454 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1312adantrll 461 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1413adantrrl 463 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
15 ltanqg 6555 . . . . . . . . . 10 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
1615adantl 266 . . . . . . . . 9 ((((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
175simpld 109 . . . . . . . . . . . . 13 (𝐴<P 𝐵𝐴P)
18 prop 6630 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1917, 18syl 14 . . . . . . . . . . . 12 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
20 elprnqu 6637 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2119, 20sylan 271 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2221ad2ant2r 486 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) → 𝑦Q)
2322adantrr 456 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑦Q)
24 elprnql 6636 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
2519, 24sylan 271 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
2625ad2ant2r 486 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → 𝑧Q)
2726adantrl 455 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑧Q)
28 simplr 490 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑞Q)
29 addcomnqg 6536 . . . . . . . . . 10 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3029adantl 266 . . . . . . . . 9 ((((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3116, 23, 27, 28, 30caovord2d 5697 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → (𝑦 <Q 𝑧 ↔ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞)))
3214, 31mpbird 160 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑦 <Q 𝑧)
33 prltlu 6642 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑧 <Q 𝑦)
3419, 33syl3an1 1179 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑧 <Q 𝑦)
35343com23 1121 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴)) → 𝑧 <Q 𝑦)
36353expb 1116 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3736adantlr 454 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3837adantrlr 462 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3938adantrrr 464 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑧 <Q 𝑦)
4032, 39jca 294 . . . . . 6 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → (𝑦 <Q 𝑧𝑧 <Q 𝑦))
4140ex 112 . . . . 5 ((𝐴<P 𝐵𝑞Q) → (((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 <Q 𝑧𝑧 <Q 𝑦)))
423, 41mtoi 600 . . . 4 ((𝐴<P 𝐵𝑞Q) → ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
4342alrimivv 1771 . . 3 ((𝐴<P 𝐵𝑞Q) → ∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
44 ltexprlem.1 . . . . . . . . . . . 12 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
4544ltexprlemell 6753 . . . . . . . . . . 11 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4644ltexprlemelu 6754 . . . . . . . . . . 11 (𝑞 ∈ (2nd𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
4745, 46anbi12i 441 . . . . . . . . . 10 ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ((𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
48 anandi 532 . . . . . . . . . 10 ((𝑞Q ∧ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) ↔ ((𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
4947, 48bitr4i 180 . . . . . . . . 9 ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ (𝑞Q ∧ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
5049baib 839 . . . . . . . 8 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
51 eleq1 2116 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 ∈ (1st𝐴) ↔ 𝑧 ∈ (1st𝐴)))
52 oveq1 5546 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦 +Q 𝑞) = (𝑧 +Q 𝑞))
5352eleq1d 2122 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑦 +Q 𝑞) ∈ (2nd𝐵) ↔ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))
5451, 53anbi12d 450 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) ↔ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
5554cbvexv 1811 . . . . . . . . 9 (∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) ↔ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))
5655anbi2i 438 . . . . . . . 8 ((∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
5750, 56syl6bb 189 . . . . . . 7 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
58 eeanv 1823 . . . . . . 7 (∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
5957, 58syl6bbr 191 . . . . . 6 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
6059notbid 602 . . . . 5 (𝑞Q → (¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ¬ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
61 alnex 1404 . . . . . . 7 (∀𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) ↔ ¬ ∃𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
6261albii 1375 . . . . . 6 (∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) ↔ ∀𝑦 ¬ ∃𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
63 alnex 1404 . . . . . 6 (∀𝑦 ¬ ∃𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) ↔ ¬ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
6462, 63bitri 177 . . . . 5 (∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) ↔ ¬ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
6560, 64syl6bbr 191 . . . 4 (𝑞Q → (¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
6665adantl 266 . . 3 ((𝐴<P 𝐵𝑞Q) → (¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
6743, 66mpbird 160 . 2 ((𝐴<P 𝐵𝑞Q) → ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)))
6867ralrimiva 2409 1 (𝐴<P 𝐵 → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ↔ wb 102   ∧ w3a 896  ∀wal 1257   = wceq 1259  ∃wex 1397   ∈ wcel 1409  ∀wral 2323  {crab 2327  ⟨cop 3405   class class class wbr 3791  ‘cfv 4929  (class class class)co 5539  1st c1st 5792  2nd c2nd 5793  Qcnq 6435   +Q cplq 6437
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