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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   <Q cltq 7100  Pcnp 7106  <P cltp 7110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iltp 7285
This theorem is referenced by:  ltexprlempr  7423
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