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Theorem cauappcvgprlemopl 6618
 Description: Lemma for cauappcvgpr 6634. 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 (φ𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd (φ𝑝 Q A <Q (𝐹𝑝))
cauappcvgpr.lim 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
Assertion
Ref Expression
cauappcvgprlemopl ((φ 𝑠 (1st𝐿)) → 𝑟 Q (𝑠 <Q 𝑟 𝑟 (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 5462 . . . . . . 7 (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
21breq1d 3765 . . . . . 6 (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
32rexbidv 2321 . . . . 5 (𝑙 = 𝑠 → (𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
4 cauappcvgpr.lim . . . . . . 7 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
54fveq2i 5124 . . . . . 6 (1st𝐿) = (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩)
6 nqex 6347 . . . . . . . 8 Q V
76rabex 3892 . . . . . . 7 {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} V
86rabex 3892 . . . . . . 7 {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u} V
97, 8op1st 5715 . . . . . 6 (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩) = {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
105, 9eqtri 2057 . . . . 5 (1st𝐿) = {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
113, 10elrab2 2694 . . . 4 (𝑠 (1st𝐿) ↔ (𝑠 Q 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
1211simprbi 260 . . 3 (𝑠 (1st𝐿) → 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
1312adantl 262 . 2 ((φ 𝑠 (1st𝐿)) → 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
14 simprr 484 . . . 4 (((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) → (𝑠 +Q 𝑞) <Q (𝐹𝑞))
15 ltbtwnnqq 6398 . . . 4 ((𝑠 +Q 𝑞) <Q (𝐹𝑞) ↔ 𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))
1614, 15sylib 127 . . 3 (((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) → 𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))
17 simplrl 487 . . . . . . . 8 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → 𝑞 Q)
1811simplbi 259 . . . . . . . . 9 (𝑠 (1st𝐿) → 𝑠 Q)
1918ad3antlr 462 . . . . . . . 8 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → 𝑠 Q)
20 ltaddnq 6390 . . . . . . . 8 ((𝑞 Q 𝑠 Q) → 𝑞 <Q (𝑞 +Q 𝑠))
2117, 19, 20syl2anc 391 . . . . . . 7 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → 𝑞 <Q (𝑞 +Q 𝑠))
22 addcomnqg 6365 . . . . . . . 8 ((𝑞 Q 𝑠 Q) → (𝑞 +Q 𝑠) = (𝑠 +Q 𝑞))
2317, 19, 22syl2anc 391 . . . . . . 7 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → (𝑞 +Q 𝑠) = (𝑠 +Q 𝑞))
2421, 23breqtrd 3779 . . . . . 6 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → 𝑞 <Q (𝑠 +Q 𝑞))
25 simprrl 491 . . . . . 6 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → (𝑠 +Q 𝑞) <Q 𝑡)
26 ltsonq 6382 . . . . . . 7 <Q Or Q
27 ltrelnq 6349 . . . . . . 7 <Q ⊆ (Q × Q)
2826, 27sotri 4663 . . . . . 6 ((𝑞 <Q (𝑠 +Q 𝑞) (𝑠 +Q 𝑞) <Q 𝑡) → 𝑞 <Q 𝑡)
2924, 25, 28syl2anc 391 . . . . 5 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → 𝑞 <Q 𝑡)
30 simprl 483 . . . . . 6 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → 𝑡 Q)
31 ltexnqq 6391 . . . . . 6 ((𝑞 Q 𝑡 Q) → (𝑞 <Q 𝑡𝑟 Q (𝑞 +Q 𝑟) = 𝑡))
3217, 30, 31syl2anc 391 . . . . 5 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → (𝑞 <Q 𝑡𝑟 Q (𝑞 +Q 𝑟) = 𝑡))
3329, 32mpbid 135 . . . 4 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → 𝑟 Q (𝑞 +Q 𝑟) = 𝑡)
3425ad2antrr 457 . . . . . . . . . 10 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → (𝑠 +Q 𝑞) <Q 𝑡)
3519ad2antrr 457 . . . . . . . . . . . 12 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → 𝑠 Q)
3617ad2antrr 457 . . . . . . . . . . . 12 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → 𝑞 Q)
37 addcomnqg 6365 . . . . . . . . . . . 12 ((𝑠 Q 𝑞 Q) → (𝑠 +Q 𝑞) = (𝑞 +Q 𝑠))
3835, 36, 37syl2anc 391 . . . . . . . . . . 11 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → (𝑠 +Q 𝑞) = (𝑞 +Q 𝑠))
3938breq1d 3765 . . . . . . . . . 10 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → ((𝑠 +Q 𝑞) <Q 𝑡 ↔ (𝑞 +Q 𝑠) <Q 𝑡))
4034, 39mpbid 135 . . . . . . . . 9 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑠) <Q 𝑡)
41 simpr 103 . . . . . . . . 9 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑟) = 𝑡)
4240, 41breqtrrd 3781 . . . . . . . 8 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟))
43 simplr 482 . . . . . . . . 9 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → 𝑟 Q)
44 ltanqg 6384 . . . . . . . . 9 ((𝑠 Q 𝑟 Q 𝑞 Q) → (𝑠 <Q 𝑟 ↔ (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟)))
4535, 43, 36, 44syl3anc 1134 . . . . . . . 8 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → (𝑠 <Q 𝑟 ↔ (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟)))
4642, 45mpbird 156 . . . . . . 7 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → 𝑠 <Q 𝑟)
47 simprrr 492 . . . . . . . . . . 11 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → 𝑡 <Q (𝐹𝑞))
4847ad2antrr 457 . . . . . . . . . 10 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → 𝑡 <Q (𝐹𝑞))
49 addcomnqg 6365 . . . . . . . . . . . . 13 ((𝑞 Q 𝑟 Q) → (𝑞 +Q 𝑟) = (𝑟 +Q 𝑞))
5036, 43, 49syl2anc 391 . . . . . . . . . . . 12 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑟) = (𝑟 +Q 𝑞))
5150, 41eqtr3d 2071 . . . . . . . . . . 11 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → (𝑟 +Q 𝑞) = 𝑡)
5251breq1d 3765 . . . . . . . . . 10 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → ((𝑟 +Q 𝑞) <Q (𝐹𝑞) ↔ 𝑡 <Q (𝐹𝑞)))
5348, 52mpbird 156 . . . . . . . . 9 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → (𝑟 +Q 𝑞) <Q (𝐹𝑞))
54 rspe 2364 . . . . . . . . 9 ((𝑞 Q (𝑟 +Q 𝑞) <Q (𝐹𝑞)) → 𝑞 Q (𝑟 +Q 𝑞) <Q (𝐹𝑞))
5536, 53, 54syl2anc 391 . . . . . . . 8 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → 𝑞 Q (𝑟 +Q 𝑞) <Q (𝐹𝑞))
56 oveq1 5462 . . . . . . . . . . 11 (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
5756breq1d 3765 . . . . . . . . . 10 (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
5857rexbidv 2321 . . . . . . . . 9 (𝑙 = 𝑟 → (𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ 𝑞 Q (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
5958, 10elrab2 2694 . . . . . . . 8 (𝑟 (1st𝐿) ↔ (𝑟 Q 𝑞 Q (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
6043, 55, 59sylanbrc 394 . . . . . . 7 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → 𝑟 (1st𝐿))
6146, 60jca 290 . . . . . 6 ((((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) (𝑞 +Q 𝑟) = 𝑡) → (𝑠 <Q 𝑟 𝑟 (1st𝐿)))
6261ex 108 . . . . 5 (((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) 𝑟 Q) → ((𝑞 +Q 𝑟) = 𝑡 → (𝑠 <Q 𝑟 𝑟 (1st𝐿))))
6362reximdva 2415 . . . 4 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → (𝑟 Q (𝑞 +Q 𝑟) = 𝑡𝑟 Q (𝑠 <Q 𝑟 𝑟 (1st𝐿))))
6433, 63mpd 13 . . 3 ((((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) (𝑡 Q ((𝑠 +Q 𝑞) <Q 𝑡 𝑡 <Q (𝐹𝑞)))) → 𝑟 Q (𝑠 <Q 𝑟 𝑟 (1st𝐿)))
6516, 64rexlimddv 2431 . 2 (((φ 𝑠 (1st𝐿)) (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))) → 𝑟 Q (𝑠 <Q 𝑟 𝑟 (1st𝐿)))
6613, 65rexlimddv 2431 1 ((φ 𝑠 (1st𝐿)) → 𝑟 Q (𝑠 <Q 𝑟 𝑟 (1st𝐿)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  {crab 2304  ⟨cop 3370   class class class wbr 3755  ⟶wf 4841  ‘cfv 4845  (class class class)co 5455  1st c1st 5707  Qcnq 6264   +Q cplq 6266
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