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Theorem ltexprlemopu 7259
Description: The upper cut of our constructed difference is open. Lemma for ltexpri 7269. (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 7255 . . . 4 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
32simprbi 270 . . 3 (𝑟 ∈ (2nd𝐶) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
4 19.42v 1841 . . . . . . . 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 1841 . . . . . . . . 9 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
65anbi2i 446 . . . . . . . 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 7161 . . . . . . . . . . . . . . 15 <P ⊆ (P × P)
98brel 4519 . . . . . . . . . . . . . 14 (𝐴<P 𝐵 → (𝐴P𝐵P))
109simprd 113 . . . . . . . . . . . . 13 (𝐴<P 𝐵𝐵P)
11 prop 7131 . . . . . . . . . . . . 13 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
1210, 11syl 14 . . . . . . . . . . . 12 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
13 prnminu 7145 . . . . . . . . . . . 12 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
1412, 13sylan 278 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
1514adantrl 463 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
1615adantrl 463 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
17 ltdfpr 7162 . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . 15 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝐴P)
2322adantr 271 . . . . . . . . . . . . . 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 504 . . . . . . . . . . . . . . . 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 271 . . . . . . . . . . . . . 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 507 . . . . . . . . . . . . . 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 7131 . . . . . . . . . . . . . . 15 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
29 prltlu 7143 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴) ∧ 𝑡 ∈ (2nd𝐴)) → 𝑦 <Q 𝑡)
3028, 29syl3an1 1214 . . . . . . . . . . . . . 14 ((𝐴P𝑦 ∈ (1st𝐴) ∧ 𝑡 ∈ (2nd𝐴)) → 𝑦 <Q 𝑡)
3123, 26, 27, 30syl3anc 1181 . . . . . . . . . . . . 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 501 . . . . . . . . . . . . . 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 508 . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . 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 7143 . . . . . . . . . . . . . . 15 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑡 ∈ (1st𝐵) ∧ 𝑠 ∈ (2nd𝐵)) → 𝑡 <Q 𝑠)
3612, 35syl3an1 1214 . . . . . . . . . . . . . 14 ((𝐴<P 𝐵𝑡 ∈ (1st𝐵) ∧ 𝑠 ∈ (2nd𝐵)) → 𝑡 <Q 𝑠)
3732, 33, 34, 36syl3anc 1181 . . . . . . . . . . . . 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 7054 . . . . . . . . . . . . . 14 <Q Or Q
39 ltrelnq 7021 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4038, 39sotri 4860 . . . . . . . . . . . . 13 ((𝑦 <Q 𝑡𝑡 <Q 𝑠) → 𝑦 <Q 𝑠)
4131, 37, 40syl2anc 404 . . . . . . . . . . . 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 2507 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑦 <Q 𝑠)
43 elprnql 7137 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
4428, 43sylan 278 . . . . . . . . . . . . 13 ((𝐴P𝑦 ∈ (1st𝐴)) → 𝑦Q)
4522, 25, 44syl2anc 404 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑦Q)
46 elprnqu 7138 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑠 ∈ (2nd𝐵)) → 𝑠Q)
4712, 46sylan 278 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑠 ∈ (2nd𝐵)) → 𝑠Q)
4847ad2ant2r 494 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑠Q)
49 ltexnqq 7064 . . . . . . . . . . . 12 ((𝑦Q𝑠Q) → (𝑦 <Q 𝑠 ↔ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑠))
5045, 48, 49syl2anc 404 . . . . . . . . . . 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 500 . . . . . . . . . . . . . . 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 504 . . . . . . . . . . . . . . 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 3887 . . . . . . . . . . . . . 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 499 . . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . 16 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑟Q)
5756adantr 271 . . . . . . . . . . . . . . 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 271 . . . . . . . . . . . . . . 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 7056 . . . . . . . . . . . . . . 15 ((𝑞Q𝑟Q𝑦Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
6055, 57, 58, 59syl3anc 1181 . . . . . . . . . . . . . 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 271 . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . 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 2171 . . . . . . . . . . . . . 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 301 . . . . . . . . . . . . 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 304 . . . . . . . . . . . 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 368 . . . . . . . . . . 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 2487 . . . . . . . . . 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 2507 . . . . . . . 8 ((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
7170eximi 1543 . . . . . . 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 2656 . . . . . 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 1841 . . . . . . 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 1841 . . . . . . . 8 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
7776anbi2i 446 . . . . . . 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 2396 . . . . 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 7255 . . . . . 6 (𝑞 ∈ (2nd𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
8281anbi2i 446 . . . . 5 ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) ↔ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
8382rexbii 2396 . . . 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 398 . 2 ((𝐴<P 𝐵 ∧ (𝑟Q𝑟 ∈ (2nd𝐶))) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))
86853impb 1142 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 927   = wceq 1296  wex 1433  wcel 1445  wrex 2371  {crab 2374  cop 3469   class class class wbr 3867  cfv 5049  (class class class)co 5690  1st c1st 5947  2nd c2nd 5948  Qcnq 6936   +Q cplq 6938   <Q cltq 6941  Pcnp 6947  <P cltp 6951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-ltnqqs 7009  df-inp 7122  df-iltp 7126
This theorem is referenced by:  ltexprlemrnd  7261
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