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Theorem cauappcvgprlemopl 6744
 Description: Lemma for cauappcvgpr 6760. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f (𝜑𝐹:QQ)
cauappcvgpr.app (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
cauappcvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
Assertion
Ref Expression
cauappcvgprlemopl ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐿,𝑟,𝑠   𝐴,𝑠,𝑝   𝐹,𝑙,𝑢,𝑝,𝑞,𝑟,𝑠   𝜑,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐴(𝑢,𝑟,𝑞,𝑙)   𝐿(𝑢,𝑙)

Proof of Theorem cauappcvgprlemopl
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 oveq1 5519 . . . . . . 7 (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
21breq1d 3774 . . . . . 6 (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
32rexbidv 2327 . . . . 5 (𝑙 = 𝑠 → (∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
4 cauappcvgpr.lim . . . . . . 7 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
54fveq2i 5181 . . . . . 6 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
6 nqex 6461 . . . . . . . 8 Q ∈ V
76rabex 3901 . . . . . . 7 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
86rabex 3901 . . . . . . 7 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
97, 8op1st 5773 . . . . . 6 (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
105, 9eqtri 2060 . . . . 5 (1st𝐿) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
113, 10elrab2 2700 . . . 4 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
1211simprbi 260 . . 3 (𝑠 ∈ (1st𝐿) → ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
1312adantl 262 . 2 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
14 simprr 484 . . . 4 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) → (𝑠 +Q 𝑞) <Q (𝐹𝑞))
15 ltbtwnnqq 6513 . . . 4 ((𝑠 +Q 𝑞) <Q (𝐹𝑞) ↔ ∃𝑡Q ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))
1614, 15sylib 127 . . 3 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) → ∃𝑡Q ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))
17 simplrl 487 . . . . . . . 8 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑞Q)
1811simplbi 259 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) → 𝑠Q)
1918ad3antlr 462 . . . . . . . 8 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑠Q)
20 ltaddnq 6505 . . . . . . . 8 ((𝑞Q𝑠Q) → 𝑞 <Q (𝑞 +Q 𝑠))
2117, 19, 20syl2anc 391 . . . . . . 7 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑞 <Q (𝑞 +Q 𝑠))
22 addcomnqg 6479 . . . . . . . 8 ((𝑞Q𝑠Q) → (𝑞 +Q 𝑠) = (𝑠 +Q 𝑞))
2317, 19, 22syl2anc 391 . . . . . . 7 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → (𝑞 +Q 𝑠) = (𝑠 +Q 𝑞))
2421, 23breqtrd 3788 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑞 <Q (𝑠 +Q 𝑞))
25 simprrl 491 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → (𝑠 +Q 𝑞) <Q 𝑡)
26 ltsonq 6496 . . . . . . 7 <Q Or Q
27 ltrelnq 6463 . . . . . . 7 <Q ⊆ (Q × Q)
2826, 27sotri 4720 . . . . . 6 ((𝑞 <Q (𝑠 +Q 𝑞) ∧ (𝑠 +Q 𝑞) <Q 𝑡) → 𝑞 <Q 𝑡)
2924, 25, 28syl2anc 391 . . . . 5 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑞 <Q 𝑡)
30 simprl 483 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑡Q)
31 ltexnqq 6506 . . . . . 6 ((𝑞Q𝑡Q) → (𝑞 <Q 𝑡 ↔ ∃𝑟Q (𝑞 +Q 𝑟) = 𝑡))
3217, 30, 31syl2anc 391 . . . . 5 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → (𝑞 <Q 𝑡 ↔ ∃𝑟Q (𝑞 +Q 𝑟) = 𝑡))
3329, 32mpbid 135 . . . 4 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → ∃𝑟Q (𝑞 +Q 𝑟) = 𝑡)
3425ad2antrr 457 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑠 +Q 𝑞) <Q 𝑡)
3519ad2antrr 457 . . . . . . . . . . . 12 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑠Q)
3617ad2antrr 457 . . . . . . . . . . . 12 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑞Q)
37 addcomnqg 6479 . . . . . . . . . . . 12 ((𝑠Q𝑞Q) → (𝑠 +Q 𝑞) = (𝑞 +Q 𝑠))
3835, 36, 37syl2anc 391 . . . . . . . . . . 11 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑠 +Q 𝑞) = (𝑞 +Q 𝑠))
3938breq1d 3774 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → ((𝑠 +Q 𝑞) <Q 𝑡 ↔ (𝑞 +Q 𝑠) <Q 𝑡))
4034, 39mpbid 135 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑠) <Q 𝑡)
41 simpr 103 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑟) = 𝑡)
4240, 41breqtrrd 3790 . . . . . . . 8 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟))
43 simplr 482 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑟Q)
44 ltanqg 6498 . . . . . . . . 9 ((𝑠Q𝑟Q𝑞Q) → (𝑠 <Q 𝑟 ↔ (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟)))
4535, 43, 36, 44syl3anc 1135 . . . . . . . 8 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑠 <Q 𝑟 ↔ (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟)))
4642, 45mpbird 156 . . . . . . 7 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑠 <Q 𝑟)
47 simprrr 492 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑡 <Q (𝐹𝑞))
4847ad2antrr 457 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑡 <Q (𝐹𝑞))
49 addcomnqg 6479 . . . . . . . . . . . . 13 ((𝑞Q𝑟Q) → (𝑞 +Q 𝑟) = (𝑟 +Q 𝑞))
5036, 43, 49syl2anc 391 . . . . . . . . . . . 12 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑟) = (𝑟 +Q 𝑞))
5150, 41eqtr3d 2074 . . . . . . . . . . 11 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑟 +Q 𝑞) = 𝑡)
5251breq1d 3774 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → ((𝑟 +Q 𝑞) <Q (𝐹𝑞) ↔ 𝑡 <Q (𝐹𝑞)))
5348, 52mpbird 156 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑟 +Q 𝑞) <Q (𝐹𝑞))
54 rspe 2370 . . . . . . . . 9 ((𝑞Q ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) → ∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞))
5536, 53, 54syl2anc 391 . . . . . . . 8 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → ∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞))
56 oveq1 5519 . . . . . . . . . . 11 (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
5756breq1d 3774 . . . . . . . . . 10 (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
5857rexbidv 2327 . . . . . . . . 9 (𝑙 = 𝑟 → (∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ ∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
5958, 10elrab2 2700 . . . . . . . 8 (𝑟 ∈ (1st𝐿) ↔ (𝑟Q ∧ ∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
6043, 55, 59sylanbrc 394 . . . . . . 7 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑟 ∈ (1st𝐿))
6146, 60jca 290 . . . . . 6 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
6261ex 108 . . . . 5 (((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) → ((𝑞 +Q 𝑟) = 𝑡 → (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))))
6362reximdva 2421 . . . 4 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → (∃𝑟Q (𝑞 +Q 𝑟) = 𝑡 → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))))
6433, 63mpd 13 . . 3 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
6516, 64rexlimddv 2437 . 2 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
6613, 65rexlimddv 2437 1 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  {crab 2310  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1st c1st 5765  Qcnq 6378   +Q cplq 6380
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