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Theorem ltexprlemopu 7379
Description: The upper cut of our constructed difference is open. Lemma for ltexpri 7389. (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
ltexprlemopu ((𝐴<P 𝐵𝑟Q𝑟 ∈ (2nd𝐶)) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑞,𝑟,𝐴   𝑥,𝐵,𝑦,𝑞,𝑟   𝑥,𝐶,𝑦,𝑞,𝑟

Proof of Theorem ltexprlemopu
Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexprlem.1 . . . . 5 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
21ltexprlemelu 7375 . . . 4 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
32simprbi 273 . . 3 (𝑟 ∈ (2nd𝐶) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
4 19.42v 1862 . . . . . . . 8 (∃𝑦(𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ↔ (𝐴<P 𝐵 ∧ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))))
5 19.42v 1862 . . . . . . . . 9 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
65anbi2i 452 . . . . . . . 8 ((𝐴<P 𝐵 ∧ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ↔ (𝐴<P 𝐵 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))))
74, 6bitri 183 . . . . . . 7 (∃𝑦(𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ↔ (𝐴<P 𝐵 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))))
8 ltrelpr 7281 . . . . . . . . . . . . . . 15 <P ⊆ (P × P)
98brel 4561 . . . . . . . . . . . . . 14 (𝐴<P 𝐵 → (𝐴P𝐵P))
109simprd 113 . . . . . . . . . . . . 13 (𝐴<P 𝐵𝐵P)
11 prop 7251 . . . . . . . . . . . . 13 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
1210, 11syl 14 . . . . . . . . . . . 12 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
13 prnminu 7265 . . . . . . . . . . . 12 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
1412, 13sylan 281 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
1514adantrl 469 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
1615adantrl 469 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
17 ltdfpr 7282 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑡Q (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵))))
1817biimpd 143 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴<P 𝐵 → ∃𝑡Q (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵))))
199, 18mpcom 36 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ∃𝑡Q (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))
2019ad2antrr 479 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → ∃𝑡Q (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))
219simpld 111 . . . . . . . . . . . . . . . 16 (𝐴<P 𝐵𝐴P)
2221ad2antrr 479 . . . . . . . . . . . . . . 15 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝐴P)
2322adantr 274 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡Q ∧ (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))) → 𝐴P)
24 simplrr 510 . . . . . . . . . . . . . . . 16 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
2524simpld 111 . . . . . . . . . . . . . . 15 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑦 ∈ (1st𝐴))
2625adantr 274 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡Q ∧ (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))) → 𝑦 ∈ (1st𝐴))
27 simprrl 513 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡Q ∧ (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))) → 𝑡 ∈ (2nd𝐴))
28 prop 7251 . . . . . . . . . . . . . . 15 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
29 prltlu 7263 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴) ∧ 𝑡 ∈ (2nd𝐴)) → 𝑦 <Q 𝑡)
3028, 29syl3an1 1234 . . . . . . . . . . . . . 14 ((𝐴P𝑦 ∈ (1st𝐴) ∧ 𝑡 ∈ (2nd𝐴)) → 𝑦 <Q 𝑡)
3123, 26, 27, 30syl3anc 1201 . . . . . . . . . . . . 13 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡Q ∧ (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))) → 𝑦 <Q 𝑡)
32 simplll 507 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡Q ∧ (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))) → 𝐴<P 𝐵)
33 simprrr 514 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡Q ∧ (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))) → 𝑡 ∈ (1st𝐵))
34 simplrl 509 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡Q ∧ (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))) → 𝑠 ∈ (2nd𝐵))
35 prltlu 7263 . . . . . . . . . . . . . . 15 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑡 ∈ (1st𝐵) ∧ 𝑠 ∈ (2nd𝐵)) → 𝑡 <Q 𝑠)
3612, 35syl3an1 1234 . . . . . . . . . . . . . 14 ((𝐴<P 𝐵𝑡 ∈ (1st𝐵) ∧ 𝑠 ∈ (2nd𝐵)) → 𝑡 <Q 𝑠)
3732, 33, 34, 36syl3anc 1201 . . . . . . . . . . . . 13 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡Q ∧ (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))) → 𝑡 <Q 𝑠)
38 ltsonq 7174 . . . . . . . . . . . . . 14 <Q Or Q
39 ltrelnq 7141 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4038, 39sotri 4904 . . . . . . . . . . . . 13 ((𝑦 <Q 𝑡𝑡 <Q 𝑠) → 𝑦 <Q 𝑠)
4131, 37, 40syl2anc 408 . . . . . . . . . . . 12 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡Q ∧ (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))) → 𝑦 <Q 𝑠)
4220, 41rexlimddv 2531 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑦 <Q 𝑠)
43 elprnql 7257 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
4428, 43sylan 281 . . . . . . . . . . . . 13 ((𝐴P𝑦 ∈ (1st𝐴)) → 𝑦Q)
4522, 25, 44syl2anc 408 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑦Q)
46 elprnqu 7258 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑠 ∈ (2nd𝐵)) → 𝑠Q)
4712, 46sylan 281 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑠 ∈ (2nd𝐵)) → 𝑠Q)
4847ad2ant2r 500 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑠Q)
49 ltexnqq 7184 . . . . . . . . . . . 12 ((𝑦Q𝑠Q) → (𝑦 <Q 𝑠 ↔ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑠))
5045, 48, 49syl2anc 408 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → (𝑦 <Q 𝑠 ↔ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑠))
5142, 50mpbid 146 . . . . . . . . . 10 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → ∃𝑞Q (𝑦 +Q 𝑞) = 𝑠)
52 simprr 506 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑦 +Q 𝑞) = 𝑠)
53 simplrr 510 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑠 <Q (𝑦 +Q 𝑟))
5452, 53eqbrtrd 3920 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
55 simprl 505 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑞Q)
56 simplrl 509 . . . . . . . . . . . . . . . 16 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑟Q)
5756adantr 274 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑟Q)
5845adantr 274 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑦Q)
59 ltanqg 7176 . . . . . . . . . . . . . . 15 ((𝑞Q𝑟Q𝑦Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
6055, 57, 58, 59syl3anc 1201 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
6154, 60mpbird 166 . . . . . . . . . . . . 13 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑞 <Q 𝑟)
6225adantr 274 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑦 ∈ (1st𝐴))
63 simplrl 509 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑠 ∈ (2nd𝐵))
6452, 63eqeltrd 2194 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑦 +Q 𝑞) ∈ (2nd𝐵))
6562, 64jca 304 . . . . . . . . . . . . 13 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))
6661, 55, 65jca32 308 . . . . . . . . . . . 12 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
6766expr 372 . . . . . . . . . . 11 ((((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ 𝑞Q) → ((𝑦 +Q 𝑞) = 𝑠 → (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))))
6867reximdva 2511 . . . . . . . . . 10 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → (∃𝑞Q (𝑦 +Q 𝑞) = 𝑠 → ∃𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))))
6951, 68mpd 13 . . . . . . . . 9 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → ∃𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
7016, 69rexlimddv 2531 . . . . . . . 8 ((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
7170eximi 1564 . . . . . . 7 (∃𝑦(𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑦𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
727, 71sylbir 134 . . . . . 6 ((𝐴<P 𝐵 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑦𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
73 rexcom4 2683 . . . . . 6 (∃𝑞Q𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) ↔ ∃𝑦𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
7472, 73sylibr 133 . . . . 5 ((𝐴<P 𝐵 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑞Q𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
75 19.42v 1862 . . . . . . 7 (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
76 19.42v 1862 . . . . . . . 8 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
7776anbi2i 452 . . . . . . 7 ((𝑞 <Q 𝑟 ∧ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
7875, 77bitri 183 . . . . . 6 (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
7978rexbii 2419 . . . . 5 (∃𝑞Q𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) ↔ ∃𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
8074, 79sylib 121 . . . 4 ((𝐴<P 𝐵 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
811ltexprlemelu 7375 . . . . . 6 (𝑞 ∈ (2nd𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
8281anbi2i 452 . . . . 5 ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) ↔ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
8382rexbii 2419 . . . 4 (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) ↔ ∃𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
8480, 83sylibr 133 . . 3 ((𝐴<P 𝐵 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))
853, 84sylanr2 402 . 2 ((𝐴<P 𝐵 ∧ (𝑟Q𝑟 ∈ (2nd𝐶))) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))
86853impb 1162 1 ((𝐴<P 𝐵𝑟Q𝑟 ∈ (2nd𝐶)) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 947   = wceq 1316  wex 1453  wcel 1465  wrex 2394  {crab 2397  cop 3500   class class class wbr 3899  cfv 5093  (class class class)co 5742  1st c1st 6004  2nd c2nd 6005  Qcnq 7056   +Q cplq 7058   <Q cltq 7061  Pcnp 7067  <P cltp 7071
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-ltnqqs 7129  df-inp 7242  df-iltp 7246
This theorem is referenced by:  ltexprlemrnd  7381
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