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Theorem cauappcvgprlemlim 6633
 Description: Lemma for cauappcvgpr 6634. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-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
cauappcvgprlemlim (φ𝑞 Q 𝑟 Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩))
Distinct variable groups:   A,𝑝   𝐿,𝑝,𝑞   φ,𝑝,𝑞   𝐹,𝑙,𝑝,𝑞,𝑟,u   𝐿,𝑟
Allowed substitution hints:   φ(u,𝑟,𝑙)   A(u,𝑟,𝑞,𝑙)   𝐿(u,𝑙)

Proof of Theorem cauappcvgprlemlim
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cauappcvgpr.f . . . . . 6 (φ𝐹:QQ)
21adantr 261 . . . . 5 ((φ (x Q y Q)) → 𝐹:QQ)
3 cauappcvgpr.app . . . . . 6 (φ𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
43adantr 261 . . . . 5 ((φ (x Q y Q)) → 𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
5 cauappcvgpr.bnd . . . . . 6 (φ𝑝 Q A <Q (𝐹𝑝))
65adantr 261 . . . . 5 ((φ (x Q y Q)) → 𝑝 Q A <Q (𝐹𝑝))
7 cauappcvgpr.lim . . . . 5 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
8 simprl 483 . . . . 5 ((φ (x Q y Q)) → x Q)
9 simprr 484 . . . . 5 ((φ (x Q y Q)) → y Q)
102, 4, 6, 7, 8, 9cauappcvgprlem1 6631 . . . 4 ((φ (x Q y Q)) → ⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩))
112, 4, 6, 7, 8, 9cauappcvgprlem2 6632 . . . 4 ((φ (x Q y Q)) → 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩)
1210, 11jca 290 . . 3 ((φ (x Q y Q)) → (⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩))
1312ralrimivva 2395 . 2 (φx Q y Q (⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩))
14 fveq2 5121 . . . . . . . 8 (x = 𝑞 → (𝐹x) = (𝐹𝑞))
1514breq2d 3767 . . . . . . 7 (x = 𝑞 → (𝑙 <Q (𝐹x) ↔ 𝑙 <Q (𝐹𝑞)))
1615abbidv 2152 . . . . . 6 (x = 𝑞 → {𝑙𝑙 <Q (𝐹x)} = {𝑙𝑙 <Q (𝐹𝑞)})
1714breq1d 3765 . . . . . . 7 (x = 𝑞 → ((𝐹x) <Q u ↔ (𝐹𝑞) <Q u))
1817abbidv 2152 . . . . . 6 (x = 𝑞 → {u ∣ (𝐹x) <Q u} = {u ∣ (𝐹𝑞) <Q u})
1916, 18opeq12d 3548 . . . . 5 (x = 𝑞 → ⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩ = ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩)
20 oveq1 5462 . . . . . . . . 9 (x = 𝑞 → (x +Q y) = (𝑞 +Q y))
2120breq2d 3767 . . . . . . . 8 (x = 𝑞 → (𝑙 <Q (x +Q y) ↔ 𝑙 <Q (𝑞 +Q y)))
2221abbidv 2152 . . . . . . 7 (x = 𝑞 → {𝑙𝑙 <Q (x +Q y)} = {𝑙𝑙 <Q (𝑞 +Q y)})
2320breq1d 3765 . . . . . . . 8 (x = 𝑞 → ((x +Q y) <Q u ↔ (𝑞 +Q y) <Q u))
2423abbidv 2152 . . . . . . 7 (x = 𝑞 → {u ∣ (x +Q y) <Q u} = {u ∣ (𝑞 +Q y) <Q u})
2522, 24opeq12d 3548 . . . . . 6 (x = 𝑞 → ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩ = ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩)
2625oveq2d 5471 . . . . 5 (x = 𝑞 → (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) = (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩))
2719, 26breq12d 3768 . . . 4 (x = 𝑞 → (⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩)))
2814, 20oveq12d 5473 . . . . . . . 8 (x = 𝑞 → ((𝐹x) +Q (x +Q y)) = ((𝐹𝑞) +Q (𝑞 +Q y)))
2928breq2d 3767 . . . . . . 7 (x = 𝑞 → (𝑙 <Q ((𝐹x) +Q (x +Q y)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))))
3029abbidv 2152 . . . . . 6 (x = 𝑞 → {𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))})
3128breq1d 3765 . . . . . . 7 (x = 𝑞 → (((𝐹x) +Q (x +Q y)) <Q u ↔ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u))
3231abbidv 2152 . . . . . 6 (x = 𝑞 → {u ∣ ((𝐹x) +Q (x +Q y)) <Q u} = {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u})
3330, 32opeq12d 3548 . . . . 5 (x = 𝑞 → ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩ = ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩)
3433breq2d 3767 . . . 4 (x = 𝑞 → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩ ↔ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩))
3527, 34anbi12d 442 . . 3 (x = 𝑞 → ((⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩) ↔ (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩)))
36 oveq2 5463 . . . . . . . . 9 (y = 𝑟 → (𝑞 +Q y) = (𝑞 +Q 𝑟))
3736breq2d 3767 . . . . . . . 8 (y = 𝑟 → (𝑙 <Q (𝑞 +Q y) ↔ 𝑙 <Q (𝑞 +Q 𝑟)))
3837abbidv 2152 . . . . . . 7 (y = 𝑟 → {𝑙𝑙 <Q (𝑞 +Q y)} = {𝑙𝑙 <Q (𝑞 +Q 𝑟)})
3936breq1d 3765 . . . . . . . 8 (y = 𝑟 → ((𝑞 +Q y) <Q u ↔ (𝑞 +Q 𝑟) <Q u))
4039abbidv 2152 . . . . . . 7 (y = 𝑟 → {u ∣ (𝑞 +Q y) <Q u} = {u ∣ (𝑞 +Q 𝑟) <Q u})
4138, 40opeq12d 3548 . . . . . 6 (y = 𝑟 → ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩ = ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩)
4241oveq2d 5471 . . . . 5 (y = 𝑟 → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩) = (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩))
4342breq2d 3767 . . . 4 (y = 𝑟 → (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩)))
4436oveq2d 5471 . . . . . . . 8 (y = 𝑟 → ((𝐹𝑞) +Q (𝑞 +Q y)) = ((𝐹𝑞) +Q (𝑞 +Q 𝑟)))
4544breq2d 3767 . . . . . . 7 (y = 𝑟 → (𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))))
4645abbidv 2152 . . . . . 6 (y = 𝑟 → {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))})
4744breq1d 3765 . . . . . . 7 (y = 𝑟 → (((𝐹𝑞) +Q (𝑞 +Q y)) <Q u ↔ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u))
4847abbidv 2152 . . . . . 6 (y = 𝑟 → {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u} = {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u})
4946, 48opeq12d 3548 . . . . 5 (y = 𝑟 → ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩ = ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩)
5049breq2d 3767 . . . 4 (y = 𝑟 → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩ ↔ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩))
5143, 50anbi12d 442 . . 3 (y = 𝑟 → ((⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩) ↔ (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩)))
5235, 51cbvral2v 2535 . 2 (x Q y Q (⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩) ↔ 𝑞 Q 𝑟 Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩))
5313, 52sylib 127 1 (φ𝑞 Q 𝑟 Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  {crab 2304  ⟨cop 3370   class class class wbr 3755  ⟶wf 4841  ‘cfv 4845  (class class class)co 5455  Qcnq 6264   +Q cplq 6266
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