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Theorem caucvgprprlemlol 6853
Description: Lemma for caucvgprpr 6867. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprpr.bnd (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
caucvgprpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
Assertion
Ref Expression
caucvgprprlemlol ((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐹,𝑙   𝑢,𝐹,𝑟   𝑝,𝑙,𝑠   𝑞,𝑙,𝑠,𝑟   𝑡,𝑙,𝑝   𝑢,𝑞,𝑠,𝑟   𝑢,𝑝,𝑡,𝑟   𝜑,𝑟   𝑟,𝑞,𝑡
Allowed substitution hints:   𝜑(𝑢,𝑡,𝑘,𝑚,𝑛,𝑠,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑡,𝑘,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑡,𝑘,𝑛,𝑠,𝑞,𝑝)   𝐿(𝑢,𝑡,𝑘,𝑚,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemlol
Dummy variables 𝑏 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 6520 . . . . 5 <Q ⊆ (Q × Q)
21brel 4419 . . . 4 (𝑠 <Q 𝑡 → (𝑠Q𝑡Q))
32simpld 109 . . 3 (𝑠 <Q 𝑡𝑠Q)
433ad2ant2 937 . 2 ((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) → 𝑠Q)
5 caucvgprpr.lim . . . . . . 7 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
65caucvgprprlemell 6840 . . . . . 6 (𝑡 ∈ (1st𝐿) ↔ (𝑡Q ∧ ∃𝑏N ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
76simprbi 264 . . . . 5 (𝑡 ∈ (1st𝐿) → ∃𝑏N ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
873ad2ant3 938 . . . 4 ((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) → ∃𝑏N ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
9 simpll2 955 . . . . . . . . 9 ((((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) → 𝑠 <Q 𝑡)
10 ltanqg 6555 . . . . . . . . . . 11 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧 +Q 𝑥) <Q (𝑧 +Q 𝑦)))
1110adantl 266 . . . . . . . . . 10 (((((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) ∧ (𝑥Q𝑦Q𝑧Q)) → (𝑥 <Q 𝑦 ↔ (𝑧 +Q 𝑥) <Q (𝑧 +Q 𝑦)))
124ad2antrr 465 . . . . . . . . . 10 ((((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) → 𝑠Q)
132simprd 111 . . . . . . . . . . . 12 (𝑠 <Q 𝑡𝑡Q)
14133ad2ant2 937 . . . . . . . . . . 11 ((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) → 𝑡Q)
1514ad2antrr 465 . . . . . . . . . 10 ((((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) → 𝑡Q)
16 simplr 490 . . . . . . . . . . 11 ((((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) → 𝑏N)
17 nnnq 6577 . . . . . . . . . . 11 (𝑏N → [⟨𝑏, 1𝑜⟩] ~QQ)
18 recclnq 6547 . . . . . . . . . . 11 ([⟨𝑏, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q)
1916, 17, 183syl 17 . . . . . . . . . 10 ((((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q)
20 addcomnqg 6536 . . . . . . . . . . 11 ((𝑥Q𝑦Q) → (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥))
2120adantl 266 . . . . . . . . . 10 (((((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) ∧ (𝑥Q𝑦Q)) → (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥))
2211, 12, 15, 19, 21caovord2d 5697 . . . . . . . . 9 ((((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) → (𝑠 <Q 𝑡 ↔ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))))
239, 22mpbid 139 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) → (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
24 ltnqpri 6749 . . . . . . . 8 ((𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩)
2523, 24syl 14 . . . . . . 7 ((((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩)
26 ltsopr 6751 . . . . . . . 8 <P Or P
27 ltrelpr 6660 . . . . . . . 8 <P ⊆ (P × P)
2826, 27sotri 4747 . . . . . . 7 ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
2925, 28sylancom 405 . . . . . 6 ((((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
3029ex 112 . . . . 5 (((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) ∧ 𝑏N) → (⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
3130reximdva 2438 . . . 4 ((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) → (∃𝑏N ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏) → ∃𝑏N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
328, 31mpd 13 . . 3 ((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) → ∃𝑏N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
33 opeq1 3576 . . . . . . . . . . 11 (𝑏 = 𝑟 → ⟨𝑏, 1𝑜⟩ = ⟨𝑟, 1𝑜⟩)
3433eceq1d 6172 . . . . . . . . . 10 (𝑏 = 𝑟 → [⟨𝑏, 1𝑜⟩] ~Q = [⟨𝑟, 1𝑜⟩] ~Q )
3534fveq2d 5209 . . . . . . . . 9 (𝑏 = 𝑟 → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))
3635oveq2d 5555 . . . . . . . 8 (𝑏 = 𝑟 → (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )))
3736breq2d 3803 . . . . . . 7 (𝑏 = 𝑟 → (𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))))
3837abbidv 2171 . . . . . 6 (𝑏 = 𝑟 → {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))})
3936breq1d 3801 . . . . . . 7 (𝑏 = 𝑟 → ((𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞))
4039abbidv 2171 . . . . . 6 (𝑏 = 𝑟 → {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞})
4138, 40opeq12d 3584 . . . . 5 (𝑏 = 𝑟 → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩)
42 fveq2 5205 . . . . 5 (𝑏 = 𝑟 → (𝐹𝑏) = (𝐹𝑟))
4341, 42breq12d 3804 . . . 4 (𝑏 = 𝑟 → (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏) ↔ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
4443cbvrexv 2551 . . 3 (∃𝑏N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏) ↔ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟))
4532, 44sylib 131 . 2 ((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) → ∃𝑟N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟))
465caucvgprprlemell 6840 . 2 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
474, 45, 46sylanbrc 402 1 ((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  {crab 2327  cop 3405   class class class wbr 3791  wf 4925  cfv 4929  (class class class)co 5539  1st c1st 5792  1𝑜c1o 6024  [cec 6134  Ncnpi 6427   <N clti 6430   ~Q ceq 6434  Qcnq 6435   +Q cplq 6437  *Qcrq 6439   <Q cltq 6440  Pcnp 6446   +P cpp 6448  <P cltp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-inp 6621  df-iltp 6625
This theorem is referenced by:  caucvgprprlemrnd  6856
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