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Theorem ltexprlemopl 7591
Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7603. (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 7588 . . . 4 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
32simprbi 275 . . 3 (𝑞 ∈ (1st𝐶) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
4 19.42v 1906 . . . . . . . 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 1906 . . . . . . . . 9 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
65anbi2i 457 . . . . . . . 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 184 . . . . . . 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 7495 . . . . . . . . . . . . . 14 <P ⊆ (P × P)
98brel 4675 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → (𝐴P𝐵P))
109simprd 114 . . . . . . . . . . . 12 (𝐴<P 𝐵𝐵P)
11 prop 7465 . . . . . . . . . . . . 13 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
12 prnmaxl 7478 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1311, 12sylan 283 . . . . . . . . . . . 12 ((𝐵P ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1410, 13sylan 283 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1514adantrl 478 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1615adantrl 478 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
179simpld 112 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵𝐴P)
1817ad2antrr 488 . . . . . . . . . . . . . 14 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝐴P)
19 simplrr 536 . . . . . . . . . . . . . . 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 112 . . . . . . . . . . . . . 14 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑦 ∈ (2nd𝐴))
21 prop 7465 . . . . . . . . . . . . . . 15 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
22 elprnqu 7472 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2321, 22sylan 283 . . . . . . . . . . . . . 14 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2418, 20, 23syl2anc 411 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑦Q)
25 simplrl 535 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑞Q)
26 ltaddnq 7397 . . . . . . . . . . . . 13 ((𝑦Q𝑞Q) → 𝑦 <Q (𝑦 +Q 𝑞))
2724, 25, 26syl2anc 411 . . . . . . . . . . . 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 531 . . . . . . . . . . . 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 7388 . . . . . . . . . . . . 13 <Q Or Q
30 ltrelnq 7355 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
3129, 30sotri 5020 . . . . . . . . . . . 12 ((𝑦 <Q (𝑦 +Q 𝑞) ∧ (𝑦 +Q 𝑞) <Q 𝑠) → 𝑦 <Q 𝑠)
3227, 28, 31syl2anc 411 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑦 <Q 𝑠)
3310ad2antrr 488 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝐵P)
34 simprl 529 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑠 ∈ (1st𝐵))
35 elprnql 7471 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑠 ∈ (1st𝐵)) → 𝑠Q)
3611, 35sylan 283 . . . . . . . . . . . . 13 ((𝐵P𝑠 ∈ (1st𝐵)) → 𝑠Q)
3733, 34, 36syl2anc 411 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑠Q)
38 ltexnqq 7398 . . . . . . . . . . . 12 ((𝑦Q𝑠Q) → (𝑦 <Q 𝑠 ↔ ∃𝑟Q (𝑦 +Q 𝑟) = 𝑠))
3924, 37, 38syl2anc 411 . . . . . . . . . . 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 147 . . . . . . . . . 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 536 . . . . . . . . . . . . . . 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 531 . . . . . . . . . . . . . . 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 4028 . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . . 15 ((𝑞Q𝑟Q𝑦Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
4844, 45, 46, 47syl3anc 1238 . . . . . . . . . . . . . 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 167 . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . 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 2254 . . . . . . . . . . . . . 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 306 . . . . . . . . . . . . 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 310 . . . . . . . . . . . 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 375 . . . . . . . . . . 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 2579 . . . . . . . . . 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 2599 . . . . . . . 8 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
5958eximi 1600 . . . . . . 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 135 . . . . . 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 2760 . . . . . 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 134 . . . . 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 1906 . . . . . . 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 1906 . . . . . . . 8 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
6564anbi2i 457 . . . . . . 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 184 . . . . . 6 (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
6766rexbii 2484 . . . . 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 122 . . . 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 7588 . . . . . 6 (𝑟 ∈ (1st𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
7069anbi2i 457 . . . . 5 ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
7170rexbii 2484 . . . 4 (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ ∃𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
7268, 71sylibr 134 . . 3 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
733, 72sylanr2 405 . 2 ((𝐴<P 𝐵 ∧ (𝑞Q𝑞 ∈ (1st𝐶))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
74733impb 1199 1 ((𝐴<P 𝐵𝑞Q𝑞 ∈ (1st𝐶)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wex 1492  wcel 2148  wrex 2456  {crab 2459  cop 3594   class class class wbr 4000  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  Qcnq 7270   +Q cplq 7272   <Q cltq 7275  Pcnp 7281  <P cltp 7285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-ltnqqs 7343  df-inp 7456  df-iltp 7460
This theorem is referenced by:  ltexprlemrnd  7595
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