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Theorem ltexprlemloc 6762
Description: Our constructed difference is located. 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
ltexprlemloc (𝐴<P 𝐵 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶))))
Distinct variable groups:   𝑥,𝑦,𝑞,𝑟,𝐴   𝑥,𝐵,𝑦,𝑞,𝑟   𝑥,𝐶,𝑦,𝑞,𝑟

Proof of Theorem ltexprlemloc
Dummy variables 𝑧 𝑤 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqi 6564 . . . . . 6 (𝑞 <Q 𝑟 → ∃𝑤Q (𝑞 +Q 𝑤) = 𝑟)
21adantl 266 . . . . 5 ((𝐴<P 𝐵𝑞 <Q 𝑟) → ∃𝑤Q (𝑞 +Q 𝑤) = 𝑟)
3 ltrelpr 6660 . . . . . . . . . 10 <P ⊆ (P × P)
43brel 4419 . . . . . . . . 9 (𝐴<P 𝐵 → (𝐴P𝐵P))
54simpld 109 . . . . . . . 8 (𝐴<P 𝐵𝐴P)
6 prop 6630 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 prarloc 6658 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑤Q) → ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)𝑦 <Q (𝑧 +Q 𝑤))
86, 7sylan 271 . . . . . . . 8 ((𝐴P𝑤Q) → ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)𝑦 <Q (𝑧 +Q 𝑤))
95, 8sylan 271 . . . . . . 7 ((𝐴<P 𝐵𝑤Q) → ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)𝑦 <Q (𝑧 +Q 𝑤))
109ad2ant2r 486 . . . . . 6 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)𝑦 <Q (𝑧 +Q 𝑤))
114simprd 111 . . . . . . . . . . . . . 14 (𝐴<P 𝐵𝐵P)
1211ad2antrr 465 . . . . . . . . . . . . 13 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → 𝐵P)
1312ad2antrr 465 . . . . . . . . . . . 12 (((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) ∧ 𝑦 <Q (𝑧 +Q 𝑤)) → 𝐵P)
14 ltanqg 6555 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
1514adantl 266 . . . . . . . . . . . . . . 15 (((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
16 elprnqu 6637 . . . . . . . . . . . . . . . . . . 19 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
176, 16sylan 271 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
185, 17sylan 271 . . . . . . . . . . . . . . . . 17 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
1918adantlr 454 . . . . . . . . . . . . . . . 16 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2019ad2ant2rl 488 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → 𝑦Q)
21 elprnql 6636 . . . . . . . . . . . . . . . . . . . 20 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
226, 21sylan 271 . . . . . . . . . . . . . . . . . . 19 ((𝐴P𝑧 ∈ (1st𝐴)) → 𝑧Q)
235, 22sylan 271 . . . . . . . . . . . . . . . . . 18 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
2423adantlr 454 . . . . . . . . . . . . . . . . 17 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ 𝑧 ∈ (1st𝐴)) → 𝑧Q)
2524ad2ant2r 486 . . . . . . . . . . . . . . . 16 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → 𝑧Q)
26 simplrl 495 . . . . . . . . . . . . . . . 16 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → 𝑤Q)
27 addclnq 6530 . . . . . . . . . . . . . . . 16 ((𝑧Q𝑤Q) → (𝑧 +Q 𝑤) ∈ Q)
2825, 26, 27syl2anc 397 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → (𝑧 +Q 𝑤) ∈ Q)
29 ltrelnq 6520 . . . . . . . . . . . . . . . . . . 19 <Q ⊆ (Q × Q)
3029brel 4419 . . . . . . . . . . . . . . . . . 18 (𝑞 <Q 𝑟 → (𝑞Q𝑟Q))
3130simpld 109 . . . . . . . . . . . . . . . . 17 (𝑞 <Q 𝑟𝑞Q)
3231adantl 266 . . . . . . . . . . . . . . . 16 ((𝐴<P 𝐵𝑞 <Q 𝑟) → 𝑞Q)
3332ad2antrr 465 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → 𝑞Q)
34 addcomnqg 6536 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3534adantl 266 . . . . . . . . . . . . . . 15 (((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3615, 20, 28, 33, 35caovord2d 5697 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → (𝑦 <Q (𝑧 +Q 𝑤) ↔ (𝑦 +Q 𝑞) <Q ((𝑧 +Q 𝑤) +Q 𝑞)))
37 addassnqg 6537 . . . . . . . . . . . . . . . . 17 ((𝑧Q𝑤Q𝑞Q) → ((𝑧 +Q 𝑤) +Q 𝑞) = (𝑧 +Q (𝑤 +Q 𝑞)))
3825, 26, 33, 37syl3anc 1146 . . . . . . . . . . . . . . . 16 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → ((𝑧 +Q 𝑤) +Q 𝑞) = (𝑧 +Q (𝑤 +Q 𝑞)))
39 addcomnqg 6536 . . . . . . . . . . . . . . . . . 18 ((𝑤Q𝑞Q) → (𝑤 +Q 𝑞) = (𝑞 +Q 𝑤))
4026, 33, 39syl2anc 397 . . . . . . . . . . . . . . . . 17 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → (𝑤 +Q 𝑞) = (𝑞 +Q 𝑤))
4140oveq2d 5555 . . . . . . . . . . . . . . . 16 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → (𝑧 +Q (𝑤 +Q 𝑞)) = (𝑧 +Q (𝑞 +Q 𝑤)))
42 simplrr 496 . . . . . . . . . . . . . . . . 17 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → (𝑞 +Q 𝑤) = 𝑟)
4342oveq2d 5555 . . . . . . . . . . . . . . . 16 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → (𝑧 +Q (𝑞 +Q 𝑤)) = (𝑧 +Q 𝑟))
4438, 41, 433eqtrd 2092 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → ((𝑧 +Q 𝑤) +Q 𝑞) = (𝑧 +Q 𝑟))
4544breq2d 3803 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → ((𝑦 +Q 𝑞) <Q ((𝑧 +Q 𝑤) +Q 𝑞) ↔ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟)))
4636, 45bitrd 181 . . . . . . . . . . . . 13 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → (𝑦 <Q (𝑧 +Q 𝑤) ↔ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟)))
4746biimpa 284 . . . . . . . . . . . 12 (((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) ∧ 𝑦 <Q (𝑧 +Q 𝑤)) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟))
48 prop 6630 . . . . . . . . . . . . 13 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
49 prloc 6646 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟)) → ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))
5048, 49sylan 271 . . . . . . . . . . . 12 ((𝐵P ∧ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟)) → ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))
5113, 47, 50syl2anc 397 . . . . . . . . . . 11 (((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) ∧ 𝑦 <Q (𝑧 +Q 𝑤)) → ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))
5251ex 112 . . . . . . . . . 10 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → (𝑦 <Q (𝑧 +Q 𝑤) → ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
5352anassrs 386 . . . . . . . . 9 (((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ 𝑧 ∈ (1st𝐴)) ∧ 𝑦 ∈ (2nd𝐴)) → (𝑦 <Q (𝑧 +Q 𝑤) → ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
5453reximdva 2438 . . . . . . . 8 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ 𝑧 ∈ (1st𝐴)) → (∃𝑦 ∈ (2nd𝐴)𝑦 <Q (𝑧 +Q 𝑤) → ∃𝑦 ∈ (2nd𝐴)((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
5554reximdva 2438 . . . . . . 7 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → (∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)𝑦 <Q (𝑧 +Q 𝑤) → ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
56 prml 6632 . . . . . . . . . . . 12 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑧Q 𝑧 ∈ (1st𝐴))
57 rexex 2385 . . . . . . . . . . . 12 (∃𝑧Q 𝑧 ∈ (1st𝐴) → ∃𝑧 𝑧 ∈ (1st𝐴))
586, 56, 573syl 17 . . . . . . . . . . 11 (𝐴P → ∃𝑧 𝑧 ∈ (1st𝐴))
59 r19.45mv 3342 . . . . . . . . . . 11 (∃𝑧 𝑧 ∈ (1st𝐴) → (∃𝑧 ∈ (1st𝐴)(∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ ∃𝑧 ∈ (1st𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵))))
605, 58, 593syl 17 . . . . . . . . . 10 (𝐴<P 𝐵 → (∃𝑧 ∈ (1st𝐴)(∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ ∃𝑧 ∈ (1st𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵))))
6160adantr 265 . . . . . . . . 9 ((𝐴<P 𝐵𝑞 <Q 𝑟) → (∃𝑧 ∈ (1st𝐴)(∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ ∃𝑧 ∈ (1st𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵))))
62 prmu 6633 . . . . . . . . . . . . 13 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (2nd𝐴))
63 rexex 2385 . . . . . . . . . . . . 13 (∃𝑥Q 𝑥 ∈ (2nd𝐴) → ∃𝑥 𝑥 ∈ (2nd𝐴))
646, 62, 633syl 17 . . . . . . . . . . . 12 (𝐴P → ∃𝑥 𝑥 ∈ (2nd𝐴))
65 r19.9rmv 3340 . . . . . . . . . . . . . 14 (∃𝑥 𝑥 ∈ (2nd𝐴) → ((𝑧 +Q 𝑟) ∈ (2nd𝐵) ↔ ∃𝑦 ∈ (2nd𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵)))
6665orbi2d 714 . . . . . . . . . . . . 13 (∃𝑥 𝑥 ∈ (2nd𝐴) → ((∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ ∃𝑦 ∈ (2nd𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵))))
67 r19.43 2485 . . . . . . . . . . . . 13 (∃𝑦 ∈ (2nd𝐴)((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ ∃𝑦 ∈ (2nd𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵)))
6866, 67syl6rbbr 192 . . . . . . . . . . . 12 (∃𝑥 𝑥 ∈ (2nd𝐴) → (∃𝑦 ∈ (2nd𝐴)((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
695, 64, 683syl 17 . . . . . . . . . . 11 (𝐴<P 𝐵 → (∃𝑦 ∈ (2nd𝐴)((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
7069rexbidv 2344 . . . . . . . . . 10 (𝐴<P 𝐵 → (∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)(∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
7170adantr 265 . . . . . . . . 9 ((𝐴<P 𝐵𝑞 <Q 𝑟) → (∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)(∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
72 ibar 289 . . . . . . . . . . . . . . 15 (𝑞Q → (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
7372adantr 265 . . . . . . . . . . . . . 14 ((𝑞Q𝑟Q) → (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
74 ibar 289 . . . . . . . . . . . . . . 15 (𝑟Q → (∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))))
7574adantl 266 . . . . . . . . . . . . . 14 ((𝑞Q𝑟Q) → (∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))))
7673, 75orbi12d 717 . . . . . . . . . . . . 13 ((𝑞Q𝑟Q) → ((∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∨ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵))) ↔ ((𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ∨ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))))
7730, 76syl 14 . . . . . . . . . . . 12 (𝑞 <Q 𝑟 → ((∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∨ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵))) ↔ ((𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ∨ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))))
78 ltexprlem.1 . . . . . . . . . . . . . 14 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
7978ltexprlemell 6753 . . . . . . . . . . . . 13 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
8078ltexprlemelu 6754 . . . . . . . . . . . . . 14 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
81 eleq1 2116 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑦 ∈ (1st𝐴) ↔ 𝑧 ∈ (1st𝐴)))
82 oveq1 5546 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝑦 +Q 𝑟) = (𝑧 +Q 𝑟))
8382eleq1d 2122 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → ((𝑦 +Q 𝑟) ∈ (2nd𝐵) ↔ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))
8481, 83anbi12d 450 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) ↔ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
8584cbvexv 1811 . . . . . . . . . . . . . . 15 (∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) ↔ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))
8685anbi2i 438 . . . . . . . . . . . . . 14 ((𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))) ↔ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
8780, 86bitri 177 . . . . . . . . . . . . 13 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
8879, 87orbi12i 691 . . . . . . . . . . . 12 ((𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)) ↔ ((𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ∨ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))))
8977, 88syl6rbbr 192 . . . . . . . . . . 11 (𝑞 <Q 𝑟 → ((𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∨ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))))
90 df-rex 2329 . . . . . . . . . . . 12 (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ↔ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
91 df-rex 2329 . . . . . . . . . . . 12 (∃𝑧 ∈ (1st𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵) ↔ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))
9290, 91orbi12i 691 . . . . . . . . . . 11 ((∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ ∃𝑧 ∈ (1st𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∨ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
9389, 92syl6bbr 191 . . . . . . . . . 10 (𝑞 <Q 𝑟 → ((𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)) ↔ (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ ∃𝑧 ∈ (1st𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵))))
9493adantl 266 . . . . . . . . 9 ((𝐴<P 𝐵𝑞 <Q 𝑟) → ((𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)) ↔ (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ ∃𝑧 ∈ (1st𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵))))
9561, 71, 943bitr4rd 214 . . . . . . . 8 ((𝐴<P 𝐵𝑞 <Q 𝑟) → ((𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
9695adantr 265 . . . . . . 7 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → ((𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
9755, 96sylibrd 162 . . . . . 6 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → (∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)𝑦 <Q (𝑧 +Q 𝑤) → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶))))
9810, 97mpd 13 . . . . 5 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)))
992, 98rexlimddv 2454 . . . 4 ((𝐴<P 𝐵𝑞 <Q 𝑟) → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)))
10099ex 112 . . 3 (𝐴<P 𝐵 → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶))))
101100ralrimivw 2410 . 2 (𝐴<P 𝐵 → ∀𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶))))
102101ralrimivw 2410 1 (𝐴<P 𝐵 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 639  w3a 896   = wceq 1259  wex 1397  wcel 1409  wral 2323  wrex 2324  {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   <Q cltq 6440  Pcnp 6446  <P cltp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-iltp 6625
This theorem is referenced by:  ltexprlempr  6763
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