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Theorem caucvgprlemm 7130
Description: Lemma for caucvgpr 7144. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-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
caucvgprlemm (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))
Distinct variable groups:   𝐴,𝑗,𝑠   𝑗,𝐹,𝑙   𝐹,𝑟   𝑢,𝐹,𝑗   𝐿,𝑟   𝜑,𝑗,𝑠   𝑠,𝑙
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑟,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑟,𝑙)   𝐹(𝑘,𝑛,𝑠)   𝐿(𝑢,𝑗,𝑘,𝑛,𝑠,𝑙)

Proof of Theorem caucvgprlemm
StepHypRef Expression
1 fveq2 5253 . . . . . 6 (𝑗 = 1𝑜 → (𝐹𝑗) = (𝐹‘1𝑜))
21breq2d 3823 . . . . 5 (𝑗 = 1𝑜 → (𝐴 <Q (𝐹𝑗) ↔ 𝐴 <Q (𝐹‘1𝑜)))
3 caucvgpr.bnd . . . . 5 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
4 1pi 6777 . . . . . 6 1𝑜N
54a1i 9 . . . . 5 (𝜑 → 1𝑜N)
62, 3, 5rspcdva 2717 . . . 4 (𝜑𝐴 <Q (𝐹‘1𝑜))
7 ltrelnq 6827 . . . . . 6 <Q ⊆ (Q × Q)
87brel 4448 . . . . 5 (𝐴 <Q (𝐹‘1𝑜) → (𝐴Q ∧ (𝐹‘1𝑜) ∈ Q))
98simpld 110 . . . 4 (𝐴 <Q (𝐹‘1𝑜) → 𝐴Q)
10 halfnqq 6872 . . . 4 (𝐴Q → ∃𝑠Q (𝑠 +Q 𝑠) = 𝐴)
116, 9, 103syl 17 . . 3 (𝜑 → ∃𝑠Q (𝑠 +Q 𝑠) = 𝐴)
12 simplr 497 . . . . . 6 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠Q)
13 archrecnq 7125 . . . . . . . 8 (𝑠Q → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠)
1412, 13syl 14 . . . . . . 7 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠)
15 simpr 108 . . . . . . . . . . . 12 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠)
16 simplr 497 . . . . . . . . . . . . . 14 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑗N)
17 nnnq 6884 . . . . . . . . . . . . . 14 (𝑗N → [⟨𝑗, 1𝑜⟩] ~QQ)
18 recclnq 6854 . . . . . . . . . . . . . 14 ([⟨𝑗, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
1916, 17, 183syl 17 . . . . . . . . . . . . 13 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
2012ad2antrr 472 . . . . . . . . . . . . 13 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑠Q)
21 ltanqg 6862 . . . . . . . . . . . . 13 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q𝑠Q𝑠Q) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝑠 +Q 𝑠)))
2219, 20, 20, 21syl3anc 1170 . . . . . . . . . . . 12 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝑠 +Q 𝑠)))
2315, 22mpbid 145 . . . . . . . . . . 11 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝑠 +Q 𝑠))
24 simpllr 501 . . . . . . . . . . 11 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) = 𝐴)
2523, 24breqtrd 3835 . . . . . . . . . 10 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝐴)
26 rsp 2417 . . . . . . . . . . . . 13 (∀𝑗N 𝐴 <Q (𝐹𝑗) → (𝑗N𝐴 <Q (𝐹𝑗)))
273, 26syl 14 . . . . . . . . . . . 12 (𝜑 → (𝑗N𝐴 <Q (𝐹𝑗)))
2827ad4antr 478 . . . . . . . . . . 11 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑗N𝐴 <Q (𝐹𝑗)))
2916, 28mpd 13 . . . . . . . . . 10 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝐴 <Q (𝐹𝑗))
30 ltsonq 6860 . . . . . . . . . . 11 <Q Or Q
3130, 7sotri 4782 . . . . . . . . . 10 (((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝐴𝐴 <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
3225, 29, 31syl2anc 403 . . . . . . . . 9 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
3332ex 113 . . . . . . . 8 ((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
3433reximdva 2469 . . . . . . 7 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → (∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
3514, 34mpd 13 . . . . . 6 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
36 oveq1 5598 . . . . . . . . 9 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
3736breq1d 3821 . . . . . . . 8 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
3837rexbidv 2375 . . . . . . 7 (𝑙 = 𝑠 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
39 caucvgpr.lim . . . . . . . . 9 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
4039fveq2i 5256 . . . . . . . 8 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)
41 nqex 6825 . . . . . . . . . 10 Q ∈ V
4241rabex 3948 . . . . . . . . 9 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
4341rabex 3948 . . . . . . . . 9 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
4442, 43op1st 5852 . . . . . . . 8 (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)}
4540, 44eqtri 2103 . . . . . . 7 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
4638, 45elrab2 2762 . . . . . 6 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
4712, 35, 46sylanbrc 408 . . . . 5 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ (1st𝐿))
4847ex 113 . . . 4 ((𝜑𝑠Q) → ((𝑠 +Q 𝑠) = 𝐴𝑠 ∈ (1st𝐿)))
4948reximdva 2469 . . 3 (𝜑 → (∃𝑠Q (𝑠 +Q 𝑠) = 𝐴 → ∃𝑠Q 𝑠 ∈ (1st𝐿)))
5011, 49mpd 13 . 2 (𝜑 → ∃𝑠Q 𝑠 ∈ (1st𝐿))
51 caucvgpr.f . . . . . 6 (𝜑𝐹:NQ)
5251, 5ffvelrnd 5380 . . . . 5 (𝜑 → (𝐹‘1𝑜) ∈ Q)
53 1nq 6828 . . . . 5 1QQ
54 addclnq 6837 . . . . 5 (((𝐹‘1𝑜) ∈ Q ∧ 1QQ) → ((𝐹‘1𝑜) +Q 1Q) ∈ Q)
5552, 53, 54sylancl 404 . . . 4 (𝜑 → ((𝐹‘1𝑜) +Q 1Q) ∈ Q)
56 addclnq 6837 . . . 4 ((((𝐹‘1𝑜) +Q 1Q) ∈ Q ∧ 1QQ) → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ Q)
5755, 53, 56sylancl 404 . . 3 (𝜑 → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ Q)
58 df-1nqqs 6813 . . . . . . . . 9 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
5958fveq2i 5256 . . . . . . . 8 (*Q‘1Q) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )
60 rec1nq 6857 . . . . . . . 8 (*Q‘1Q) = 1Q
6159, 60eqtr3i 2105 . . . . . . 7 (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) = 1Q
6261oveq2i 5602 . . . . . 6 ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) = ((𝐹‘1𝑜) +Q 1Q)
63 ltaddnq 6869 . . . . . . 7 ((((𝐹‘1𝑜) +Q 1Q) ∈ Q ∧ 1QQ) → ((𝐹‘1𝑜) +Q 1Q) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
6455, 53, 63sylancl 404 . . . . . 6 (𝜑 → ((𝐹‘1𝑜) +Q 1Q) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
6562, 64syl5eqbr 3844 . . . . 5 (𝜑 → ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
66 opeq1 3596 . . . . . . . . . 10 (𝑗 = 1𝑜 → ⟨𝑗, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
6766eceq1d 6258 . . . . . . . . 9 (𝑗 = 1𝑜 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
6867fveq2d 5257 . . . . . . . 8 (𝑗 = 1𝑜 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ))
691, 68oveq12d 5609 . . . . . . 7 (𝑗 = 1𝑜 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )))
7069breq1d 3821 . . . . . 6 (𝑗 = 1𝑜 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ↔ ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)))
7170rspcev 2712 . . . . 5 ((1𝑜N ∧ ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
725, 65, 71syl2anc 403 . . . 4 (𝜑 → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
73 breq2 3815 . . . . . 6 (𝑢 = (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)))
7473rexbidv 2375 . . . . 5 (𝑢 = (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)))
7539fveq2i 5256 . . . . . 6 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)
7642, 43op2nd 5853 . . . . . 6 (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
7775, 76eqtri 2103 . . . . 5 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
7874, 77elrab2 2762 . . . 4 ((((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd𝐿) ↔ ((((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)))
7957, 72, 78sylanbrc 408 . . 3 (𝜑 → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd𝐿))
80 eleq1 2145 . . . 4 (𝑟 = (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd𝐿) ↔ (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd𝐿)))
8180rspcev 2712 . . 3 (((((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ Q ∧ (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd𝐿)) → ∃𝑟Q 𝑟 ∈ (2nd𝐿))
8257, 79, 81syl2anc 403 . 2 (𝜑 → ∃𝑟Q 𝑟 ∈ (2nd𝐿))
8350, 82jca 300 1 (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  wral 2353  wrex 2354  {crab 2357  cop 3425   class class class wbr 3811  wf 4965  cfv 4969  (class class class)co 5591  1st c1st 5844  2nd c2nd 5845  1𝑜c1o 6106  [cec 6220  Ncnpi 6734   <N clti 6737   ~Q ceq 6741  Qcnq 6742  1Qc1q 6743   +Q cplq 6744  *Qcrq 6746   <Q cltq 6747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815
This theorem is referenced by:  caucvgprlemcl  7138
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