ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprlemm GIF version

Theorem caucvgprlemm 7206
Description: Lemma for caucvgpr 7220. 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 5289 . . . . . 6 (𝑗 = 1𝑜 → (𝐹𝑗) = (𝐹‘1𝑜))
21breq2d 3849 . . . . 5 (𝑗 = 1𝑜 → (𝐴 <Q (𝐹𝑗) ↔ 𝐴 <Q (𝐹‘1𝑜)))
3 caucvgpr.bnd . . . . 5 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
4 1pi 6853 . . . . . 6 1𝑜N
54a1i 9 . . . . 5 (𝜑 → 1𝑜N)
62, 3, 5rspcdva 2727 . . . 4 (𝜑𝐴 <Q (𝐹‘1𝑜))
7 ltrelnq 6903 . . . . . 6 <Q ⊆ (Q × Q)
87brel 4478 . . . . 5 (𝐴 <Q (𝐹‘1𝑜) → (𝐴Q ∧ (𝐹‘1𝑜) ∈ Q))
98simpld 110 . . . 4 (𝐴 <Q (𝐹‘1𝑜) → 𝐴Q)
10 halfnqq 6948 . . . 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 7201 . . . . . . . 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 6960 . . . . . . . . . . . . . 14 (𝑗N → [⟨𝑗, 1𝑜⟩] ~QQ)
18 recclnq 6930 . . . . . . . . . . . . . 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 6938 . . . . . . . . . . . . 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 1174 . . . . . . . . . . . 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 3861 . . . . . . . . . 10 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝐴)
26 rsp 2423 . . . . . . . . . . . . 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 6936 . . . . . . . . . . 11 <Q Or Q
3130, 7sotri 4814 . . . . . . . . . 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 2475 . . . . . . 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 5641 . . . . . . . . 9 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
3736breq1d 3847 . . . . . . . 8 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
3837rexbidv 2381 . . . . . . 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 5292 . . . . . . . 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 6901 . . . . . . . . . 10 Q ∈ V
4241rabex 3975 . . . . . . . . 9 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
4341rabex 3975 . . . . . . . . 9 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
4442, 43op1st 5899 . . . . . . . 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 2108 . . . . . . 7 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
4638, 45elrab2 2772 . . . . . 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 2475 . . 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 5419 . . . . 5 (𝜑 → (𝐹‘1𝑜) ∈ Q)
53 1nq 6904 . . . . 5 1QQ
54 addclnq 6913 . . . . 5 (((𝐹‘1𝑜) ∈ Q ∧ 1QQ) → ((𝐹‘1𝑜) +Q 1Q) ∈ Q)
5552, 53, 54sylancl 404 . . . 4 (𝜑 → ((𝐹‘1𝑜) +Q 1Q) ∈ Q)
56 addclnq 6913 . . . 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 6889 . . . . . . . . 9 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
5958fveq2i 5292 . . . . . . . 8 (*Q‘1Q) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )
60 rec1nq 6933 . . . . . . . 8 (*Q‘1Q) = 1Q
6159, 60eqtr3i 2110 . . . . . . 7 (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) = 1Q
6261oveq2i 5645 . . . . . 6 ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) = ((𝐹‘1𝑜) +Q 1Q)
63 ltaddnq 6945 . . . . . . 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 3870 . . . . 5 (𝜑 → ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
66 opeq1 3617 . . . . . . . . . 10 (𝑗 = 1𝑜 → ⟨𝑗, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
6766eceq1d 6308 . . . . . . . . 9 (𝑗 = 1𝑜 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
6867fveq2d 5293 . . . . . . . 8 (𝑗 = 1𝑜 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ))
691, 68oveq12d 5652 . . . . . . 7 (𝑗 = 1𝑜 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )))
7069breq1d 3847 . . . . . 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 2722 . . . . 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 3841 . . . . . 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 2381 . . . . 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 5292 . . . . . 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 5900 . . . . . 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 2108 . . . . 5 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
7874, 77elrab2 2772 . . . 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 2150 . . . 4 (𝑟 = (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd𝐿) ↔ (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd𝐿)))
8180rspcev 2722 . . 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 1289  wcel 1438  wral 2359  wrex 2360  {crab 2363  cop 3444   class class class wbr 3837  wf 4998  cfv 5002  (class class class)co 5634  1st c1st 5891  2nd c2nd 5892  1𝑜c1o 6156  [cec 6270  Ncnpi 6810   <N clti 6813   ~Q ceq 6817  Qcnq 6818  1Qc1q 6819   +Q cplq 6820  *Qcrq 6822   <Q cltq 6823
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891
This theorem is referenced by:  caucvgprlemcl  7214
  Copyright terms: Public domain W3C validator