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Theorem ltexprlemopl 6853
Description: The lower cut of our constructed difference is open. Lemma for ltexpri 6865. (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
ltexprlemopl ((𝐴<P 𝐵𝑞Q𝑞 ∈ (1st𝐶)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑞,𝑟,𝐴   𝑥,𝐵,𝑦,𝑞,𝑟   𝑥,𝐶,𝑦,𝑞,𝑟

Proof of Theorem ltexprlemopl
Dummy variable 𝑠 is 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))}⟩
21ltexprlemell 6850 . . . 4 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
32simprbi 269 . . 3 (𝑞 ∈ (1st𝐶) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
4 19.42v 1828 . . . . . . . 8 (∃𝑦(𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ↔ (𝐴<P 𝐵 ∧ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
5 19.42v 1828 . . . . . . . . 9 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
65anbi2i 445 . . . . . . . 8 ((𝐴<P 𝐵 ∧ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ↔ (𝐴<P 𝐵 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
74, 6bitri 182 . . . . . . 7 (∃𝑦(𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ↔ (𝐴<P 𝐵 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
8 ltrelpr 6757 . . . . . . . . . . . . . 14 <P ⊆ (P × P)
98brel 4418 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → (𝐴P𝐵P))
109simprd 112 . . . . . . . . . . . 12 (𝐴<P 𝐵𝐵P)
11 prop 6727 . . . . . . . . . . . . 13 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
12 prnmaxl 6740 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1311, 12sylan 277 . . . . . . . . . . . 12 ((𝐵P ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1410, 13sylan 277 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1514adantrl 462 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1615adantrl 462 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
179simpld 110 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵𝐴P)
1817ad2antrr 472 . . . . . . . . . . . . . 14 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝐴P)
19 simplrr 503 . . . . . . . . . . . . . . 15 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2019simpld 110 . . . . . . . . . . . . . 14 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑦 ∈ (2nd𝐴))
21 prop 6727 . . . . . . . . . . . . . . 15 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
22 elprnqu 6734 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2321, 22sylan 277 . . . . . . . . . . . . . 14 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2418, 20, 23syl2anc 403 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑦Q)
25 simplrl 502 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑞Q)
26 ltaddnq 6659 . . . . . . . . . . . . 13 ((𝑦Q𝑞Q) → 𝑦 <Q (𝑦 +Q 𝑞))
2724, 25, 26syl2anc 403 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑦 <Q (𝑦 +Q 𝑞))
28 simprr 499 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → (𝑦 +Q 𝑞) <Q 𝑠)
29 ltsonq 6650 . . . . . . . . . . . . 13 <Q Or Q
30 ltrelnq 6617 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
3129, 30sotri 4750 . . . . . . . . . . . 12 ((𝑦 <Q (𝑦 +Q 𝑞) ∧ (𝑦 +Q 𝑞) <Q 𝑠) → 𝑦 <Q 𝑠)
3227, 28, 31syl2anc 403 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑦 <Q 𝑠)
3310ad2antrr 472 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝐵P)
34 simprl 498 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑠 ∈ (1st𝐵))
35 elprnql 6733 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑠 ∈ (1st𝐵)) → 𝑠Q)
3611, 35sylan 277 . . . . . . . . . . . . 13 ((𝐵P𝑠 ∈ (1st𝐵)) → 𝑠Q)
3733, 34, 36syl2anc 403 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑠Q)
38 ltexnqq 6660 . . . . . . . . . . . 12 ((𝑦Q𝑠Q) → (𝑦 <Q 𝑠 ↔ ∃𝑟Q (𝑦 +Q 𝑟) = 𝑠))
3924, 37, 38syl2anc 403 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → (𝑦 <Q 𝑠 ↔ ∃𝑟Q (𝑦 +Q 𝑟) = 𝑠))
4032, 39mpbid 145 . . . . . . . . . 10 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → ∃𝑟Q (𝑦 +Q 𝑟) = 𝑠)
41 simplrr 503 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → (𝑦 +Q 𝑞) <Q 𝑠)
42 simprr 499 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → (𝑦 +Q 𝑟) = 𝑠)
4341, 42breqtrrd 3819 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
4425adantr 270 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → 𝑞Q)
45 simprl 498 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → 𝑟Q)
4624adantr 270 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → 𝑦Q)
47 ltanqg 6652 . . . . . . . . . . . . . . 15 ((𝑞Q𝑟Q𝑦Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
4844, 45, 46, 47syl3anc 1170 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
4943, 48mpbird 165 . . . . . . . . . . . . 13 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → 𝑞 <Q 𝑟)
5020adantr 270 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → 𝑦 ∈ (2nd𝐴))
51 simplrl 502 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → 𝑠 ∈ (1st𝐵))
5242, 51eqeltrd 2156 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → (𝑦 +Q 𝑟) ∈ (1st𝐵))
5350, 52jca 300 . . . . . . . . . . . . 13 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))
5449, 45, 53jca32 303 . . . . . . . . . . . 12 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ (𝑟Q ∧ (𝑦 +Q 𝑟) = 𝑠)) → (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
5554expr 367 . . . . . . . . . . 11 ((((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) ∧ 𝑟Q) → ((𝑦 +Q 𝑟) = 𝑠 → (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))))
5655reximdva 2464 . . . . . . . . . 10 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → (∃𝑟Q (𝑦 +Q 𝑟) = 𝑠 → ∃𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))))
5740, 56mpd 13 . . . . . . . . 9 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → ∃𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
5816, 57rexlimddv 2482 . . . . . . . 8 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
5958eximi 1532 . . . . . . 7 (∃𝑦(𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑦𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
607, 59sylbir 133 . . . . . 6 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑦𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
61 rexcom4 2623 . . . . . 6 (∃𝑟Q𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))) ↔ ∃𝑦𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
6260, 61sylibr 132 . . . . 5 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑟Q𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
63 19.42v 1828 . . . . . . 7 (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
64 19.42v 1828 . . . . . . . 8 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
6564anbi2i 445 . . . . . . 7 ((𝑞 <Q 𝑟 ∧ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
6663, 65bitri 182 . . . . . 6 (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
6766rexbii 2374 . . . . 5 (∃𝑟Q𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))) ↔ ∃𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
6862, 67sylib 120 . . . 4 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
691ltexprlemell 6850 . . . . . 6 (𝑟 ∈ (1st𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
7069anbi2i 445 . . . . 5 ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
7170rexbii 2374 . . . 4 (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ ∃𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
7268, 71sylibr 132 . . 3 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
733, 72sylanr2 397 . 2 ((𝐴<P 𝐵 ∧ (𝑞Q𝑞 ∈ (1st𝐶))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
74733impb 1135 1 ((𝐴<P 𝐵𝑞Q𝑞 ∈ (1st𝐶)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wex 1422  wcel 1434  wrex 2350  {crab 2353  cop 3409   class class class wbr 3793  cfv 4932  (class class class)co 5543  1st c1st 5796  2nd c2nd 5797  Qcnq 6532   +Q cplq 6534   <Q cltq 6537  Pcnp 6543  <P cltp 6547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-ltnqqs 6605  df-inp 6718  df-iltp 6722
This theorem is referenced by:  ltexprlemrnd  6857
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