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Theorem caucvgprlemm 6766
Description: Lemma for caucvgpr 6780. 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 1pi 6413 . . . . 5 1𝑜N
2 caucvgpr.bnd . . . . 5 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
3 fveq2 5178 . . . . . . 7 (𝑗 = 1𝑜 → (𝐹𝑗) = (𝐹‘1𝑜))
43breq2d 3776 . . . . . 6 (𝑗 = 1𝑜 → (𝐴 <Q (𝐹𝑗) ↔ 𝐴 <Q (𝐹‘1𝑜)))
54rspcv 2652 . . . . 5 (1𝑜N → (∀𝑗N 𝐴 <Q (𝐹𝑗) → 𝐴 <Q (𝐹‘1𝑜)))
61, 2, 5mpsyl 59 . . . 4 (𝜑𝐴 <Q (𝐹‘1𝑜))
7 ltrelnq 6463 . . . . . 6 <Q ⊆ (Q × Q)
87brel 4392 . . . . 5 (𝐴 <Q (𝐹‘1𝑜) → (𝐴Q ∧ (𝐹‘1𝑜) ∈ Q))
98simpld 105 . . . 4 (𝐴 <Q (𝐹‘1𝑜) → 𝐴Q)
10 halfnqq 6508 . . . 4 (𝐴Q → ∃𝑠Q (𝑠 +Q 𝑠) = 𝐴)
116, 9, 103syl 17 . . 3 (𝜑 → ∃𝑠Q (𝑠 +Q 𝑠) = 𝐴)
12 simplr 482 . . . . . 6 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠Q)
13 archrecnq 6761 . . . . . . . 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 103 . . . . . . . . . . . 12 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠)
16 simplr 482 . . . . . . . . . . . . . 14 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑗N)
17 nnnq 6520 . . . . . . . . . . . . . 14 (𝑗N → [⟨𝑗, 1𝑜⟩] ~QQ)
18 recclnq 6490 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . 13 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑠Q)
21 ltanqg 6498 . . . . . . . . . . . . 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 1135 . . . . . . . . . . . 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 135 . . . . . . . . . . 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 486 . . . . . . . . . . 11 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) = 𝐴)
2523, 24breqtrd 3788 . . . . . . . . . 10 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝐴)
26 rsp 2369 . . . . . . . . . . . . 13 (∀𝑗N 𝐴 <Q (𝐹𝑗) → (𝑗N𝐴 <Q (𝐹𝑗)))
272, 26syl 14 . . . . . . . . . . . 12 (𝜑 → (𝑗N𝐴 <Q (𝐹𝑗)))
2827ad4antr 463 . . . . . . . . . . 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 6496 . . . . . . . . . . 11 <Q Or Q
3130, 7sotri 4720 . . . . . . . . . 10 (((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝐴𝐴 <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
3225, 29, 31syl2anc 391 . . . . . . . . 9 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
3332ex 108 . . . . . . . 8 ((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
3433reximdva 2421 . . . . . . 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 5519 . . . . . . . . 9 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
3736breq1d 3774 . . . . . . . 8 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
3837rexbidv 2327 . . . . . . 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 5181 . . . . . . . 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 6461 . . . . . . . . . 10 Q ∈ V
4241rabex 3901 . . . . . . . . 9 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
4341rabex 3901 . . . . . . . . 9 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
4442, 43op1st 5773 . . . . . . . 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 2060 . . . . . . 7 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
4638, 45elrab2 2700 . . . . . 6 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
4712, 35, 46sylanbrc 394 . . . . 5 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ (1st𝐿))
4847ex 108 . . . 4 ((𝜑𝑠Q) → ((𝑠 +Q 𝑠) = 𝐴𝑠 ∈ (1st𝐿)))
4948reximdva 2421 . . 3 (𝜑 → (∃𝑠Q (𝑠 +Q 𝑠) = 𝐴 → ∃𝑠Q 𝑠 ∈ (1st𝐿)))
5011, 49mpd 13 . 2 (𝜑 → ∃𝑠Q 𝑠 ∈ (1st𝐿))
51 caucvgpr.f . . . . . 6 (𝜑𝐹:NQ)
521a1i 9 . . . . . 6 (𝜑 → 1𝑜N)
5351, 52ffvelrnd 5303 . . . . 5 (𝜑 → (𝐹‘1𝑜) ∈ Q)
54 1nq 6464 . . . . 5 1QQ
55 addclnq 6473 . . . . 5 (((𝐹‘1𝑜) ∈ Q ∧ 1QQ) → ((𝐹‘1𝑜) +Q 1Q) ∈ Q)
5653, 54, 55sylancl 392 . . . 4 (𝜑 → ((𝐹‘1𝑜) +Q 1Q) ∈ Q)
57 addclnq 6473 . . . 4 ((((𝐹‘1𝑜) +Q 1Q) ∈ Q ∧ 1QQ) → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ Q)
5856, 54, 57sylancl 392 . . 3 (𝜑 → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ Q)
59 df-1nqqs 6449 . . . . . . . . 9 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
6059fveq2i 5181 . . . . . . . 8 (*Q‘1Q) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )
61 rec1nq 6493 . . . . . . . 8 (*Q‘1Q) = 1Q
6260, 61eqtr3i 2062 . . . . . . 7 (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) = 1Q
6362oveq2i 5523 . . . . . 6 ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) = ((𝐹‘1𝑜) +Q 1Q)
64 ltaddnq 6505 . . . . . . 7 ((((𝐹‘1𝑜) +Q 1Q) ∈ Q ∧ 1QQ) → ((𝐹‘1𝑜) +Q 1Q) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
6556, 54, 64sylancl 392 . . . . . 6 (𝜑 → ((𝐹‘1𝑜) +Q 1Q) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
6663, 65syl5eqbr 3797 . . . . 5 (𝜑 → ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
67 opeq1 3549 . . . . . . . . . 10 (𝑗 = 1𝑜 → ⟨𝑗, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
6867eceq1d 6142 . . . . . . . . 9 (𝑗 = 1𝑜 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
6968fveq2d 5182 . . . . . . . 8 (𝑗 = 1𝑜 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ))
703, 69oveq12d 5530 . . . . . . 7 (𝑗 = 1𝑜 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )))
7170breq1d 3774 . . . . . 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)))
7271rspcev 2656 . . . . 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))
7352, 66, 72syl2anc 391 . . . 4 (𝜑 → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
74 breq2 3768 . . . . . 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)))
7574rexbidv 2327 . . . . 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)))
7639fveq2i 5181 . . . . . 6 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)
7742, 43op2nd 5774 . . . . . 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}
7876, 77eqtri 2060 . . . . 5 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
7975, 78elrab2 2700 . . . 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)))
8058, 73, 79sylanbrc 394 . . 3 (𝜑 → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd𝐿))
81 eleq1 2100 . . . 4 (𝑟 = (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd𝐿) ↔ (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd𝐿)))
8281rspcev 2656 . . 3 (((((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ Q ∧ (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd𝐿)) → ∃𝑟Q 𝑟 ∈ (2nd𝐿))
8358, 80, 82syl2anc 391 . 2 (𝜑 → ∃𝑟Q 𝑟 ∈ (2nd𝐿))
8450, 83jca 290 1 (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = 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  2nd c2nd 5766  1𝑜c1o 5994  [cec 6104  Ncnpi 6370   <N clti 6373   ~Q ceq 6377  Qcnq 6378  1Qc1q 6379   +Q cplq 6380  *Qcrq 6382   <Q cltq 6383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451
This theorem is referenced by:  caucvgprlemcl  6774
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