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Theorem caucvgprlemlol 7485
 Description: Lemma for caucvgpr 7497. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f (𝜑𝐹:NQ)
caucvgpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
Assertion
Ref Expression
caucvgprlemlol ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))
Distinct variable groups:   𝐴,𝑗   𝐹,𝑙,𝑟,𝑠   𝑢,𝐹   𝑗,𝐿,𝑟,𝑠   𝑗,𝑙,𝑠   𝜑,𝑗,𝑟,𝑠   𝑢,𝑗,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑠,𝑟,𝑙)   𝐹(𝑗,𝑘,𝑛)   𝐿(𝑢,𝑘,𝑛,𝑙)

Proof of Theorem caucvgprlemlol
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 7180 . . . . 5 <Q ⊆ (Q × Q)
21brel 4591 . . . 4 (𝑠 <Q 𝑟 → (𝑠Q𝑟Q))
32simpld 111 . . 3 (𝑠 <Q 𝑟𝑠Q)
433ad2ant2 1003 . 2 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠Q)
5 oveq1 5781 . . . . . . . 8 (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
65breq1d 3939 . . . . . . 7 (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
76rexbidv 2438 . . . . . 6 (𝑙 = 𝑟 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
8 caucvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
98fveq2i 5424 . . . . . . 7 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
10 nqex 7178 . . . . . . . . 9 Q ∈ V
1110rabex 4072 . . . . . . . 8 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
1210rabex 4072 . . . . . . . 8 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
1311, 12op1st 6044 . . . . . . 7 (1st ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
149, 13eqtri 2160 . . . . . 6 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
157, 14elrab2 2843 . . . . 5 (𝑟 ∈ (1st𝐿) ↔ (𝑟Q ∧ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
1615simprbi 273 . . . 4 (𝑟 ∈ (1st𝐿) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
17163ad2ant3 1004 . . 3 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
18 simpll2 1021 . . . . . . 7 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → 𝑠 <Q 𝑟)
19 ltanqg 7215 . . . . . . . . 9 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
2019adantl 275 . . . . . . . 8 (((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
214ad2antrr 479 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → 𝑠Q)
222simprd 113 . . . . . . . . . 10 (𝑠 <Q 𝑟𝑟Q)
23223ad2ant2 1003 . . . . . . . . 9 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑟Q)
2423ad2antrr 479 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → 𝑟Q)
25 simplr 519 . . . . . . . . 9 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → 𝑗N)
26 nnnq 7237 . . . . . . . . 9 (𝑗N → [⟨𝑗, 1o⟩] ~QQ)
27 recclnq 7207 . . . . . . . . 9 ([⟨𝑗, 1o⟩] ~QQ → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
2825, 26, 273syl 17 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
29 addcomnqg 7196 . . . . . . . . 9 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3029adantl 275 . . . . . . . 8 (((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3120, 21, 24, 28, 30caovord2d 5940 . . . . . . 7 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → (𝑠 <Q 𝑟 ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q ))))
3218, 31mpbid 146 . . . . . 6 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
33 ltsonq 7213 . . . . . . 7 <Q Or Q
3433, 1sotri 4934 . . . . . 6 (((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
3532, 34sylancom 416 . . . . 5 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
3635ex 114 . . . 4 (((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) → ((𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
3736reximdva 2534 . . 3 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → (∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
3817, 37mpd 13 . 2 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
39 oveq1 5781 . . . . 5 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
4039breq1d 3939 . . . 4 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
4140rexbidv 2438 . . 3 (𝑙 = 𝑠 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
4241, 14elrab2 2843 . 2 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
434, 38, 42sylanbrc 413 1 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420  ⟨cop 3530   class class class wbr 3929  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  1st c1st 6036  1oc1o 6306  [cec 6427  Ncnpi 7087
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