ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltexprlemopl GIF version

Theorem ltexprlemopl 7096
Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7108. (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 7093 . . . 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 1831 . . . . . . . 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 1831 . . . . . . . . 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 7000 . . . . . . . . . . . . . 14 <P ⊆ (P × P)
98brel 4456 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → (𝐴P𝐵P))
109simprd 112 . . . . . . . . . . . 12 (𝐴<P 𝐵𝐵P)
11 prop 6970 . . . . . . . . . . . . 13 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
12 prnmaxl 6983 . . . . . . . . . . . . 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 6970 . . . . . . . . . . . . . . 15 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
22 elprnqu 6977 . . . . . . . . . . . . . . 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 6902 . . . . . . . . . . . . 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 6893 . . . . . . . . . . . . 13 <Q Or Q
30 ltrelnq 6860 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
3129, 30sotri 4790 . . . . . . . . . . . 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 6976 . . . . . . . . . . . . . 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 6903 . . . . . . . . . . . 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 3845 . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . . 15 ((𝑞Q𝑟Q𝑦Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
4844, 45, 46, 47syl3anc 1172 . . . . . . . . . . . . . 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 2161 . . . . . . . . . . . . . 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 2471 . . . . . . . . . 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 2489 . . . . . . . 8 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
5958eximi 1534 . . . . . . 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 2636 . . . . . 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 1831 . . . . . . 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 1831 . . . . . . . 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 2381 . . . . 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 7093 . . . . . 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 2381 . . . 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 1137 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 922   = wceq 1287  wex 1424  wcel 1436  wrex 2356  {crab 2359  cop 3433   class class class wbr 3819  cfv 4977  (class class class)co 5606  1st c1st 5859  2nd c2nd 5860  Qcnq 6775   +Q cplq 6777   <Q cltq 6780  Pcnp 6786  <P cltp 6790
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3927  ax-sep 3930  ax-nul 3938  ax-pow 3982  ax-pr 4008  ax-un 4232  ax-setind 4324  ax-iinf 4374
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439  df-uni 3636  df-int 3671  df-iun 3714  df-br 3820  df-opab 3874  df-mpt 3875  df-tr 3910  df-eprel 4088  df-id 4092  df-po 4095  df-iso 4096  df-iord 4165  df-on 4167  df-suc 4170  df-iom 4377  df-xp 4415  df-rel 4416  df-cnv 4417  df-co 4418  df-dm 4419  df-rn 4420  df-res 4421  df-ima 4422  df-iota 4942  df-fun 4979  df-fn 4980  df-f 4981  df-f1 4982  df-fo 4983  df-f1o 4984  df-fv 4985  df-ov 5609  df-oprab 5610  df-mpt2 5611  df-1st 5861  df-2nd 5862  df-recs 6017  df-irdg 6082  df-1o 6128  df-oadd 6132  df-omul 6133  df-er 6237  df-ec 6239  df-qs 6243  df-ni 6799  df-pli 6800  df-mi 6801  df-lti 6802  df-plpq 6839  df-mpq 6840  df-enq 6842  df-nqqs 6843  df-plqqs 6844  df-mqqs 6845  df-1nqqs 6846  df-ltnqqs 6848  df-inp 6961  df-iltp 6965
This theorem is referenced by:  ltexprlemrnd  7100
  Copyright terms: Public domain W3C validator