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Theorem ltexprlemopu 6758
Description: The upper cut of our constructed difference is open. Lemma for ltexpri 6768. (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 6754 . . . 4 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
32simprbi 264 . . 3 (𝑟 ∈ (2nd𝐶) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
4 19.42v 1802 . . . . . . . 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 1802 . . . . . . . . 9 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
65anbi2i 438 . . . . . . . 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 177 . . . . . . 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 6660 . . . . . . . . . . . . . . 15 <P ⊆ (P × P)
98brel 4419 . . . . . . . . . . . . . 14 (𝐴<P 𝐵 → (𝐴P𝐵P))
109simprd 111 . . . . . . . . . . . . 13 (𝐴<P 𝐵𝐵P)
11 prop 6630 . . . . . . . . . . . . 13 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
1210, 11syl 14 . . . . . . . . . . . 12 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
13 prnminu 6644 . . . . . . . . . . . 12 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
1412, 13sylan 271 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
1514adantrl 455 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
1615adantrl 455 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q (𝑦 +Q 𝑟))
17 ltdfpr 6661 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑡Q (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵))))
1817biimpd 136 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴<P 𝐵 → ∃𝑡Q (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵))))
199, 18mpcom 36 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ∃𝑡Q (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))
2019ad2antrr 465 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → ∃𝑡Q (𝑡 ∈ (2nd𝐴) ∧ 𝑡 ∈ (1st𝐵)))
219simpld 109 . . . . . . . . . . . . . . . 16 (𝐴<P 𝐵𝐴P)
2221ad2antrr 465 . . . . . . . . . . . . . . 15 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝐴P)
2322adantr 265 . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . 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 109 . . . . . . . . . . . . . . 15 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑦 ∈ (1st𝐴))
2625adantr 265 . . . . . . . . . . . . . 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 499 . . . . . . . . . . . . . 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 6630 . . . . . . . . . . . . . . 15 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
29 prltlu 6642 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴) ∧ 𝑡 ∈ (2nd𝐴)) → 𝑦 <Q 𝑡)
3028, 29syl3an1 1179 . . . . . . . . . . . . . 14 ((𝐴P𝑦 ∈ (1st𝐴) ∧ 𝑡 ∈ (2nd𝐴)) → 𝑦 <Q 𝑡)
3123, 26, 27, 30syl3anc 1146 . . . . . . . . . . . . 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 493 . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . 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 6642 . . . . . . . . . . . . . . 15 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑡 ∈ (1st𝐵) ∧ 𝑠 ∈ (2nd𝐵)) → 𝑡 <Q 𝑠)
3612, 35syl3an1 1179 . . . . . . . . . . . . . 14 ((𝐴<P 𝐵𝑡 ∈ (1st𝐵) ∧ 𝑠 ∈ (2nd𝐵)) → 𝑡 <Q 𝑠)
3732, 33, 34, 36syl3anc 1146 . . . . . . . . . . . . 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 6553 . . . . . . . . . . . . . 14 <Q Or Q
39 ltrelnq 6520 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4038, 39sotri 4747 . . . . . . . . . . . . 13 ((𝑦 <Q 𝑡𝑡 <Q 𝑠) → 𝑦 <Q 𝑠)
4131, 37, 40syl2anc 397 . . . . . . . . . . . 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 2454 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑦 <Q 𝑠)
43 elprnql 6636 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
4428, 43sylan 271 . . . . . . . . . . . . 13 ((𝐴P𝑦 ∈ (1st𝐴)) → 𝑦Q)
4522, 25, 44syl2anc 397 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑦Q)
46 elprnqu 6637 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑠 ∈ (2nd𝐵)) → 𝑠Q)
4712, 46sylan 271 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑠 ∈ (2nd𝐵)) → 𝑠Q)
4847ad2ant2r 486 . . . . . . . . . . . 12 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑠Q)
49 ltexnqq 6563 . . . . . . . . . . . 12 ((𝑦Q𝑠Q) → (𝑦 <Q 𝑠 ↔ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑠))
5045, 48, 49syl2anc 397 . . . . . . . . . . 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 139 . . . . . . . . . 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 492 . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . 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 3811 . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . 16 (((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑟Q)
5756adantr 265 . . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . . 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 6555 . . . . . . . . . . . . . . 15 ((𝑞Q𝑟Q𝑦Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
6055, 57, 58, 59syl3anc 1146 . . . . . . . . . . . . . 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 160 . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . 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 2130 . . . . . . . . . . . . . 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 294 . . . . . . . . . . . . 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 297 . . . . . . . . . . . 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 361 . . . . . . . . . . 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 2438 . . . . . . . . . 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 2454 . . . . . . . 8 ((𝐴<P 𝐵 ∧ (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
7170eximi 1507 . . . . . . 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 129 . . . . . 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 2594 . . . . . 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 141 . . . . 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 1802 . . . . . . 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 1802 . . . . . . . 8 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
7776anbi2i 438 . . . . . . 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 177 . . . . . 6 (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
7978rexbii 2348 . . . . 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 131 . . . 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 6754 . . . . . 6 (𝑞 ∈ (2nd𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
8281anbi2i 438 . . . . 5 ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) ↔ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
8382rexbii 2348 . . . 4 (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) ↔ ∃𝑞Q (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
8480, 83sylibr 141 . . 3 ((𝐴<P 𝐵 ∧ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))
853, 84sylanr2 391 . 2 ((𝐴<P 𝐵 ∧ (𝑟Q𝑟 ∈ (2nd𝐶))) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))
86853impb 1111 1 ((𝐴<P 𝐵𝑟Q𝑟 ∈ (2nd𝐶)) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wex 1397  wcel 1409  wrex 2324  {crab 2327  cop 3405   class class class wbr 3791  cfv 4929  (class class class)co 5539  1st c1st 5792  2nd c2nd 5793  Qcnq 6435   +Q cplq 6437   <Q cltq 6440  Pcnp 6446  <P cltp 6450
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 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-ltnqqs 6508  df-inp 6621  df-iltp 6625
This theorem is referenced by:  ltexprlemrnd  6760
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