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Theorem cauappcvgprlemopl 7676
Description: Lemma for cauappcvgpr 7692. 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 5904 . . . . . . 7 (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
21breq1d 4028 . . . . . 6 (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
32rexbidv 2491 . . . . 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 5537 . . . . . 6 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
6 nqex 7393 . . . . . . . 8 Q ∈ V
76rabex 4162 . . . . . . 7 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
86rabex 4162 . . . . . . 7 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
97, 8op1st 6172 . . . . . 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 2210 . . . . 5 (1st𝐿) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
113, 10elrab2 2911 . . . 4 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
1211simprbi 275 . . 3 (𝑠 ∈ (1st𝐿) → ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
1312adantl 277 . 2 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
14 simprr 531 . . . 4 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) → (𝑠 +Q 𝑞) <Q (𝐹𝑞))
15 ltbtwnnqq 7445 . . . 4 ((𝑠 +Q 𝑞) <Q (𝐹𝑞) ↔ ∃𝑡Q ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))
1614, 15sylib 122 . . 3 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) → ∃𝑡Q ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))
17 simplrl 535 . . . . . . . 8 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑞Q)
1811simplbi 274 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) → 𝑠Q)
1918ad3antlr 493 . . . . . . . 8 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑠Q)
20 ltaddnq 7437 . . . . . . . 8 ((𝑞Q𝑠Q) → 𝑞 <Q (𝑞 +Q 𝑠))
2117, 19, 20syl2anc 411 . . . . . . 7 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑞 <Q (𝑞 +Q 𝑠))
22 addcomnqg 7411 . . . . . . . 8 ((𝑞Q𝑠Q) → (𝑞 +Q 𝑠) = (𝑠 +Q 𝑞))
2317, 19, 22syl2anc 411 . . . . . . 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 4044 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑞 <Q (𝑠 +Q 𝑞))
25 simprrl 539 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → (𝑠 +Q 𝑞) <Q 𝑡)
26 ltsonq 7428 . . . . . . 7 <Q Or Q
27 ltrelnq 7395 . . . . . . 7 <Q ⊆ (Q × Q)
2826, 27sotri 5042 . . . . . 6 ((𝑞 <Q (𝑠 +Q 𝑞) ∧ (𝑠 +Q 𝑞) <Q 𝑡) → 𝑞 <Q 𝑡)
2924, 25, 28syl2anc 411 . . . . 5 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑞 <Q 𝑡)
30 simprl 529 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑡Q)
31 ltexnqq 7438 . . . . . 6 ((𝑞Q𝑡Q) → (𝑞 <Q 𝑡 ↔ ∃𝑟Q (𝑞 +Q 𝑟) = 𝑡))
3217, 30, 31syl2anc 411 . . . . 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 147 . . . 4 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → ∃𝑟Q (𝑞 +Q 𝑟) = 𝑡)
3425ad2antrr 488 . . . . . . . . . 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 488 . . . . . . . . . . . 12 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) ∧ 𝑟Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑠Q)
3617ad2antrr 488 . . . . . . . . . . . 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 7411 . . . . . . . . . . . 12 ((𝑠Q𝑞Q) → (𝑠 +Q 𝑞) = (𝑞 +Q 𝑠))
3835, 36, 37syl2anc 411 . . . . . . . . . . 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 4028 . . . . . . . . . 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 147 . . . . . . . . 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 110 . . . . . . . . 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 4046 . . . . . . . 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 528 . . . . . . . . 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 7430 . . . . . . . . 9 ((𝑠Q𝑟Q𝑞Q) → (𝑠 <Q 𝑟 ↔ (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟)))
4535, 43, 36, 44syl3anc 1249 . . . . . . . 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 167 . . . . . . 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 540 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) ∧ (𝑡Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡𝑡 <Q (𝐹𝑞)))) → 𝑡 <Q (𝐹𝑞))
4847ad2antrr 488 . . . . . . . . . 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 7411 . . . . . . . . . . . . 13 ((𝑞Q𝑟Q) → (𝑞 +Q 𝑟) = (𝑟 +Q 𝑞))
5036, 43, 49syl2anc 411 . . . . . . . . . . . 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 2224 . . . . . . . . . . 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 4028 . . . . . . . . . 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 167 . . . . . . . . 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 2539 . . . . . . . . 9 ((𝑞Q ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) → ∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞))
5536, 53, 54syl2anc 411 . . . . . . . 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 5904 . . . . . . . . . . 11 (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
5756breq1d 4028 . . . . . . . . . 10 (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
5857rexbidv 2491 . . . . . . . . 9 (𝑙 = 𝑟 → (∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ ∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
5958, 10elrab2 2911 . . . . . . . 8 (𝑟 ∈ (1st𝐿) ↔ (𝑟Q ∧ ∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
6043, 55, 59sylanbrc 417 . . . . . . 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 306 . . . . . 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 115 . . . . 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 2592 . . . 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 2612 . 2 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑞Q ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞))) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
6613, 65rexlimddv 2612 1 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  wral 2468  wrex 2469  {crab 2472  cop 3610   class class class wbr 4018  wf 5231  cfv 5235  (class class class)co 5897  1st c1st 6164  Qcnq 7310   +Q cplq 7312   <Q cltq 7315
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383
This theorem is referenced by:  cauappcvgprlemrnd  7680
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