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

Theorem ltexprlemopl 6727
Description: The lower cut of our constructed difference is open. Lemma for ltexpri 6739. (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 6724 . . . 4 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
32simprbi 264 . . 3 (𝑞 ∈ (1st𝐶) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
4 19.42v 1800 . . . . . . . 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 1800 . . . . . . . . 9 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
65anbi2i 438 . . . . . . . 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 177 . . . . . . 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 6631 . . . . . . . . . . . . . 14 <P ⊆ (P × P)
98brel 4417 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → (𝐴P𝐵P))
109simprd 111 . . . . . . . . . . . 12 (𝐴<P 𝐵𝐵P)
11 prop 6601 . . . . . . . . . . . . 13 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
12 prnmaxl 6614 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1311, 12sylan 271 . . . . . . . . . . . 12 ((𝐵P ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1410, 13sylan 271 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1514adantrl 455 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
1615adantrl 455 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑠 ∈ (1st𝐵)(𝑦 +Q 𝑞) <Q 𝑠)
179simpld 109 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵𝐴P)
1817ad2antrr 465 . . . . . . . . . . . . . 14 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝐴P)
19 simplrr 496 . . . . . . . . . . . . . . 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 109 . . . . . . . . . . . . . 14 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑦 ∈ (2nd𝐴))
21 prop 6601 . . . . . . . . . . . . . . 15 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
22 elprnqu 6608 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2321, 22sylan 271 . . . . . . . . . . . . . 14 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2418, 20, 23syl2anc 397 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑦Q)
25 simplrl 495 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑞Q)
26 ltaddnq 6533 . . . . . . . . . . . . 13 ((𝑦Q𝑞Q) → 𝑦 <Q (𝑦 +Q 𝑞))
2724, 25, 26syl2anc 397 . . . . . . . . . . . 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 492 . . . . . . . . . . . 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 6524 . . . . . . . . . . . . 13 <Q Or Q
30 ltrelnq 6491 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
3129, 30sotri 4745 . . . . . . . . . . . 12 ((𝑦 <Q (𝑦 +Q 𝑞) ∧ (𝑦 +Q 𝑞) <Q 𝑠) → 𝑦 <Q 𝑠)
3227, 28, 31syl2anc 397 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑦 <Q 𝑠)
3310ad2antrr 465 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝐵P)
34 simprl 491 . . . . . . . . . . . . 13 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑠 ∈ (1st𝐵))
35 elprnql 6607 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑠 ∈ (1st𝐵)) → 𝑠Q)
3611, 35sylan 271 . . . . . . . . . . . . 13 ((𝐵P𝑠 ∈ (1st𝐵)) → 𝑠Q)
3733, 34, 36syl2anc 397 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) ∧ (𝑠 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) <Q 𝑠)) → 𝑠Q)
38 ltexnqq 6534 . . . . . . . . . . . 12 ((𝑦Q𝑠Q) → (𝑦 <Q 𝑠 ↔ ∃𝑟Q (𝑦 +Q 𝑟) = 𝑠))
3924, 37, 38syl2anc 397 . . . . . . . . . . 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 139 . . . . . . . . . 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 496 . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . 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 3815 . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . . 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 6526 . . . . . . . . . . . . . . 15 ((𝑞Q𝑟Q𝑦Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
4844, 45, 46, 47syl3anc 1144 . . . . . . . . . . . . . 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 160 . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . 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 2128 . . . . . . . . . . . . . 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 294 . . . . . . . . . . . . 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 297 . . . . . . . . . . . 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 361 . . . . . . . . . . 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 2436 . . . . . . . . . 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 2452 . . . . . . . 8 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
5958eximi 1505 . . . . . . 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 129 . . . . . 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 2592 . . . . . 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 141 . . . . 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 1800 . . . . . . 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 1800 . . . . . . . 8 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
6564anbi2i 438 . . . . . . 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 177 . . . . . 6 (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
6766rexbii 2346 . . . . 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 131 . . . 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 6724 . . . . . 6 (𝑟 ∈ (1st𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
7069anbi2i 438 . . . . 5 ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
7170rexbii 2346 . . . 4 (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ ∃𝑟Q (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
7268, 71sylibr 141 . . 3 ((𝐴<P 𝐵 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
733, 72sylanr2 391 . 2 ((𝐴<P 𝐵 ∧ (𝑞Q𝑞 ∈ (1st𝐶))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
74733impb 1109 1 ((𝐴<P 𝐵𝑞Q𝑞 ∈ (1st𝐶)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 894   = wceq 1257  wex 1395  wcel 1407  wrex 2322  {crab 2325  cop 3403   class class class wbr 3789  cfv 4927  (class class class)co 5537  1st c1st 5790  2nd c2nd 5791  Qcnq 6406   +Q cplq 6408   <Q cltq 6411  Pcnp 6417  <P cltp 6421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-eprel 4051  df-id 4055  df-po 4058  df-iso 4059  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-1o 6029  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-pli 6431  df-mi 6432  df-lti 6433  df-plpq 6470  df-mpq 6471  df-enq 6473  df-nqqs 6474  df-plqqs 6475  df-mqqs 6476  df-1nqqs 6477  df-ltnqqs 6479  df-inp 6592  df-iltp 6596
This theorem is referenced by:  ltexprlemrnd  6731
  Copyright terms: Public domain W3C validator