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Theorem cauappcvgprlemupu 6804
 Description: Lemma for cauappcvgpr 6817. The upper cut of the putative limit is upper. (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
cauappcvgprlemupu ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿))
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 cauappcvgprlemupu
StepHypRef Expression
1 ltrelnq 6520 . . . . 5 <Q ⊆ (Q × Q)
21brel 4419 . . . 4 (𝑠 <Q 𝑟 → (𝑠Q𝑟Q))
32simprd 111 . . 3 (𝑠 <Q 𝑟𝑟Q)
433ad2ant2 937 . 2 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟Q)
5 breq2 3795 . . . . . . 7 (𝑢 = 𝑠 → (((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
65rexbidv 2344 . . . . . 6 (𝑢 = 𝑠 → (∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
7 cauappcvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
87fveq2i 5208 . . . . . . 7 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
9 nqex 6518 . . . . . . . . 9 Q ∈ V
109rabex 3928 . . . . . . . 8 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
119rabex 3928 . . . . . . . 8 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
1210, 11op2nd 5801 . . . . . . 7 (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
138, 12eqtri 2076 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
146, 13elrab2 2722 . . . . 5 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
1514simprbi 264 . . . 4 (𝑠 ∈ (2nd𝐿) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠)
16153ad2ant3 938 . . 3 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠)
17 ltsonq 6553 . . . . . . 7 <Q Or Q
1817, 1sotri 4747 . . . . . 6 ((((𝐹𝑞) +Q 𝑞) <Q 𝑠𝑠 <Q 𝑟) → ((𝐹𝑞) +Q 𝑞) <Q 𝑟)
1918expcom 113 . . . . 5 (𝑠 <Q 𝑟 → (((𝐹𝑞) +Q 𝑞) <Q 𝑠 → ((𝐹𝑞) +Q 𝑞) <Q 𝑟))
20193ad2ant2 937 . . . 4 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → (((𝐹𝑞) +Q 𝑞) <Q 𝑠 → ((𝐹𝑞) +Q 𝑞) <Q 𝑟))
2120reximdv 2437 . . 3 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → (∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠 → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑟))
2216, 21mpd 13 . 2 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑟)
23 breq2 3795 . . . 4 (𝑢 = 𝑟 → (((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹𝑞) +Q 𝑞) <Q 𝑟))
2423rexbidv 2344 . . 3 (𝑢 = 𝑟 → (∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑟))
2524, 13elrab2 2722 . 2 (𝑟 ∈ (2nd𝐿) ↔ (𝑟Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑟))
264, 22, 25sylanbrc 402 1 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ∧ w3a 896   = wceq 1259   ∈ wcel 1409  ∀wral 2323  ∃wrex 2324  {crab 2327  ⟨cop 3405   class class class wbr 3791  ⟶wf 4925  ‘cfv 4929  (class class class)co 5539  2nd c2nd 5793  Qcnq 6435   +Q cplq 6437
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