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Theorem ltexprlemloc 7422
 Description: Our constructed difference is located. Lemma for ltexpri 7428. (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 7224 . . . . . 6 (𝑞 <Q 𝑟 → ∃𝑤Q (𝑞 +Q 𝑤) = 𝑟)
21adantl 275 . . . . 5 ((𝐴<P 𝐵𝑞 <Q 𝑟) → ∃𝑤Q (𝑞 +Q 𝑤) = 𝑟)
3 ltrelpr 7320 . . . . . . . . . 10 <P ⊆ (P × P)
43brel 4591 . . . . . . . . 9 (𝐴<P 𝐵 → (𝐴P𝐵P))
54simpld 111 . . . . . . . 8 (𝐴<P 𝐵𝐴P)
6 prop 7290 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 prarloc 7318 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑤Q) → ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)𝑦 <Q (𝑧 +Q 𝑤))
86, 7sylan 281 . . . . . . . 8 ((𝐴P𝑤Q) → ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)𝑦 <Q (𝑧 +Q 𝑤))
95, 8sylan 281 . . . . . . 7 ((𝐴<P 𝐵𝑤Q) → ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)𝑦 <Q (𝑧 +Q 𝑤))
109ad2ant2r 500 . . . . . 6 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)𝑦 <Q (𝑧 +Q 𝑤))
114simprd 113 . . . . . . . . . . . . . 14 (𝐴<P 𝐵𝐵P)
1211ad2antrr 479 . . . . . . . . . . . . 13 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → 𝐵P)
1312ad2antrr 479 . . . . . . . . . . . 12 (((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) ∧ 𝑦 <Q (𝑧 +Q 𝑤)) → 𝐵P)
14 ltanqg 7215 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
1514adantl 275 . . . . . . . . . . . . . . 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 7297 . . . . . . . . . . . . . . . . . . 19 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
176, 16sylan 281 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
185, 17sylan 281 . . . . . . . . . . . . . . . . 17 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
1918adantlr 468 . . . . . . . . . . . . . . . 16 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2019ad2ant2rl 502 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → 𝑦Q)
21 elprnql 7296 . . . . . . . . . . . . . . . . . . . 20 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
226, 21sylan 281 . . . . . . . . . . . . . . . . . . 19 ((𝐴P𝑧 ∈ (1st𝐴)) → 𝑧Q)
235, 22sylan 281 . . . . . . . . . . . . . . . . . 18 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
2423adantlr 468 . . . . . . . . . . . . . . . . 17 (((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ 𝑧 ∈ (1st𝐴)) → 𝑧Q)
2524ad2ant2r 500 . . . . . . . . . . . . . . . 16 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → 𝑧Q)
26 simplrl 524 . . . . . . . . . . . . . . . 16 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → 𝑤Q)
27 addclnq 7190 . . . . . . . . . . . . . . . 16 ((𝑧Q𝑤Q) → (𝑧 +Q 𝑤) ∈ Q)
2825, 26, 27syl2anc 408 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → (𝑧 +Q 𝑤) ∈ Q)
29 ltrelnq 7180 . . . . . . . . . . . . . . . . . . 19 <Q ⊆ (Q × Q)
3029brel 4591 . . . . . . . . . . . . . . . . . 18 (𝑞 <Q 𝑟 → (𝑞Q𝑟Q))
3130simpld 111 . . . . . . . . . . . . . . . . 17 (𝑞 <Q 𝑟𝑞Q)
3231adantl 275 . . . . . . . . . . . . . . . 16 ((𝐴<P 𝐵𝑞 <Q 𝑟) → 𝑞Q)
3332ad2antrr 479 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → 𝑞Q)
34 addcomnqg 7196 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3534adantl 275 . . . . . . . . . . . . . . 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 5940 . . . . . . . . . . . . . 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 7197 . . . . . . . . . . . . . . . . 17 ((𝑧Q𝑤Q𝑞Q) → ((𝑧 +Q 𝑤) +Q 𝑞) = (𝑧 +Q (𝑤 +Q 𝑞)))
3825, 26, 33, 37syl3anc 1216 . . . . . . . . . . . . . . . 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 7196 . . . . . . . . . . . . . . . . . 18 ((𝑤Q𝑞Q) → (𝑤 +Q 𝑞) = (𝑞 +Q 𝑤))
4026, 33, 39syl2anc 408 . . . . . . . . . . . . . . . . 17 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → (𝑤 +Q 𝑞) = (𝑞 +Q 𝑤))
4140oveq2d 5790 . . . . . . . . . . . . . . . 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 525 . . . . . . . . . . . . . . . . 17 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → (𝑞 +Q 𝑤) = 𝑟)
4342oveq2d 5790 . . . . . . . . . . . . . . . 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 2176 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵𝑞 <Q 𝑟) ∧ (𝑤Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴))) → ((𝑧 +Q 𝑤) +Q 𝑞) = (𝑧 +Q 𝑟))
4544breq2d 3941 . . . . . . . . . . . . . 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 187 . . . . . . . . . . . . 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 294 . . . . . . . . . . . 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 7290 . . . . . . . . . . . . 13 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
49 prloc 7306 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟)) → ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))
5048, 49sylan 281 . . . . . . . . . . . 12 ((𝐵P ∧ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟)) → ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))
5113, 47, 50syl2anc 408 . . . . . . . . . . 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 114 . . . . . . . . . 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 397 . . . . . . . . 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 2534 . . . . . . . 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 2534 . . . . . . 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 7292 . . . . . . . . . . . 12 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑧Q 𝑧 ∈ (1st𝐴))
57 rexex 2479 . . . . . . . . . . . 12 (∃𝑧Q 𝑧 ∈ (1st𝐴) → ∃𝑧 𝑧 ∈ (1st𝐴))
586, 56, 573syl 17 . . . . . . . . . . 11 (𝐴P → ∃𝑧 𝑧 ∈ (1st𝐴))
59 r19.45mv 3456 . . . . . . . . . . 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 274 . . . . . . . . 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 7293 . . . . . . . . . . . . 13 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (2nd𝐴))
63 rexex 2479 . . . . . . . . . . . . 13 (∃𝑥Q 𝑥 ∈ (2nd𝐴) → ∃𝑥 𝑥 ∈ (2nd𝐴))
646, 62, 633syl 17 . . . . . . . . . . . 12 (𝐴P → ∃𝑥 𝑥 ∈ (2nd𝐴))
65 r19.9rmv 3454 . . . . . . . . . . . . . 14 (∃𝑥 𝑥 ∈ (2nd𝐴) → ((𝑧 +Q 𝑟) ∈ (2nd𝐵) ↔ ∃𝑦 ∈ (2nd𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵)))
6665orbi2d 779 . . . . . . . . . . . . 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 2589 . . . . . . . . . . . . 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 198 . . . . . . . . . . . 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 2438 . . . . . . . . . 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 274 . . . . . . . . 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 299 . . . . . . . . . . . . . . 15 (𝑞Q → (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
7372adantr 274 . . . . . . . . . . . . . 14 ((𝑞Q𝑟Q) → (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
74 ibar 299 . . . . . . . . . . . . . . 15 (𝑟Q → (∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))))
7574adantl 275 . . . . . . . . . . . . . 14 ((𝑞Q𝑟Q) → (∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)) ↔ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))))
7673, 75orbi12d 782 . . . . . . . . . . . . 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 7413 . . . . . . . . . . . . 13 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
8078ltexprlemelu 7414 . . . . . . . . . . . . . 14 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
81 eleq1 2202 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑦 ∈ (1st𝐴) ↔ 𝑧 ∈ (1st𝐴)))
82 oveq1 5781 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝑦 +Q 𝑟) = (𝑧 +Q 𝑟))
8382eleq1d 2208 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → ((𝑦 +Q 𝑟) ∈ (2nd𝐵) ↔ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))
8481, 83anbi12d 464 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) ↔ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
8584cbvexv 1890 . . . . . . . . . . . . . . 15 (∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) ↔ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))
8685anbi2i 452 . . . . . . . . . . . . . 14 ((𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))) ↔ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
8780, 86bitri 183 . . . . . . . . . . . . 13 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
8879, 87orbi12i 753 . . . . . . . . . . . 12 ((𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)) ↔ ((𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ∨ (𝑟Q ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))))
8977, 88syl6rbbr 198 . . . . . . . . . . 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 2422 . . . . . . . . . . . 12 (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ↔ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
91 df-rex 2422 . . . . . . . . . . . 12 (∃𝑧 ∈ (1st𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵) ↔ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd𝐵)))
9290, 91orbi12i 753 . . . . . . . . . . 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 197 . . . . . . . . . 10 (𝑞 <Q 𝑟 → ((𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)) ↔ (∃𝑦 ∈ (2nd𝐴)(𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ ∃𝑧 ∈ (1st𝐴)(𝑧 +Q 𝑟) ∈ (2nd𝐵))))
9493adantl 275 . . . . . . . . 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 220 . . . . . . . 8 ((𝐴<P 𝐵𝑞 <Q 𝑟) → ((𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (2nd𝐴)((𝑦 +Q 𝑞) ∈ (1st𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd𝐵))))
9695adantr 274 . . . . . . 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 168 . . . . . 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 2554 . . . 4 ((𝐴<P 𝐵𝑞 <Q 𝑟) → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)))
10099ex 114 . . 3 (𝐴<P 𝐵 → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶))))
101100ralrimivw 2506 . 2 (𝐴<P 𝐵 → ∀𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶))))
102101ralrimivw 2506 1 (𝐴<P 𝐵 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420  ⟨cop 3530   class class class wbr 3929  ‘cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7095   +Q cplq 7097
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