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Theorem cauappcvgprlemlol 6745
 Description: Lemma for cauappcvgpr 6760. The lower cut of the putative limit is lower. (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
cauappcvgprlemlol ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (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 cauappcvgprlemlol
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 6463 . . . . 5 <Q ⊆ (Q × Q)
21brel 4392 . . . 4 (𝑠 <Q 𝑟 → (𝑠Q𝑟Q))
32simpld 105 . . 3 (𝑠 <Q 𝑟𝑠Q)
433ad2ant2 926 . 2 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠Q)
5 oveq1 5519 . . . . . . . 8 (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
65breq1d 3774 . . . . . . 7 (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
76rexbidv 2327 . . . . . 6 (𝑙 = 𝑟 → (∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ ∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
8 cauappcvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
98fveq2i 5181 . . . . . . 7 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
10 nqex 6461 . . . . . . . . 9 Q ∈ V
1110rabex 3901 . . . . . . . 8 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
1210rabex 3901 . . . . . . . 8 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
1311, 12op1st 5773 . . . . . . 7 (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
149, 13eqtri 2060 . . . . . 6 (1st𝐿) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
157, 14elrab2 2700 . . . . 5 (𝑟 ∈ (1st𝐿) ↔ (𝑟Q ∧ ∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞)))
1615simprbi 260 . . . 4 (𝑟 ∈ (1st𝐿) → ∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞))
17163ad2ant3 927 . . 3 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → ∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞))
18 simpll2 944 . . . . . . 7 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑞Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) → 𝑠 <Q 𝑟)
19 ltanqg 6498 . . . . . . . . 9 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
2019adantl 262 . . . . . . . 8 (((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑞Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
214ad2antrr 457 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑞Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) → 𝑠Q)
222simprd 107 . . . . . . . . . 10 (𝑠 <Q 𝑟𝑟Q)
23223ad2ant2 926 . . . . . . . . 9 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑟Q)
2423ad2antrr 457 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑞Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) → 𝑟Q)
25 simplr 482 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑞Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) → 𝑞Q)
26 addcomnqg 6479 . . . . . . . . 9 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
2726adantl 262 . . . . . . . 8 (((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑞Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
2820, 21, 24, 25, 27caovord2d 5670 . . . . . . 7 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑞Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) → (𝑠 <Q 𝑟 ↔ (𝑠 +Q 𝑞) <Q (𝑟 +Q 𝑞)))
2918, 28mpbid 135 . . . . . 6 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑞Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) → (𝑠 +Q 𝑞) <Q (𝑟 +Q 𝑞))
30 ltsonq 6496 . . . . . . 7 <Q Or Q
3130, 1sotri 4720 . . . . . 6 (((𝑠 +Q 𝑞) <Q (𝑟 +Q 𝑞) ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) → (𝑠 +Q 𝑞) <Q (𝐹𝑞))
3229, 31sylancom 397 . . . . 5 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑞Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹𝑞)) → (𝑠 +Q 𝑞) <Q (𝐹𝑞))
3332ex 108 . . . 4 (((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑞Q) → ((𝑟 +Q 𝑞) <Q (𝐹𝑞) → (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
3433reximdva 2421 . . 3 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → (∃𝑞Q (𝑟 +Q 𝑞) <Q (𝐹𝑞) → ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
3517, 34mpd 13 . 2 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
36 oveq1 5519 . . . . 5 (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
3736breq1d 3774 . . . 4 (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
3837rexbidv 2327 . . 3 (𝑙 = 𝑠 → (∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
3938, 14elrab2 2700 . 2 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
404, 35, 39sylanbrc 394 1 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = 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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