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Theorem caucvgprlemm 7667
Description: Lemma for caucvgpr 7681. 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 5516 . . . . . 6 (𝑗 = 1o → (𝐹𝑗) = (𝐹‘1o))
21breq2d 4016 . . . . 5 (𝑗 = 1o → (𝐴 <Q (𝐹𝑗) ↔ 𝐴 <Q (𝐹‘1o)))
3 caucvgpr.bnd . . . . 5 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
4 1pi 7314 . . . . . 6 1oN
54a1i 9 . . . . 5 (𝜑 → 1oN)
62, 3, 5rspcdva 2847 . . . 4 (𝜑𝐴 <Q (𝐹‘1o))
7 ltrelnq 7364 . . . . . 6 <Q ⊆ (Q × Q)
87brel 4679 . . . . 5 (𝐴 <Q (𝐹‘1o) → (𝐴Q ∧ (𝐹‘1o) ∈ Q))
98simpld 112 . . . 4 (𝐴 <Q (𝐹‘1o) → 𝐴Q)
10 halfnqq 7409 . . . 4 (𝐴Q → ∃𝑠Q (𝑠 +Q 𝑠) = 𝐴)
116, 9, 103syl 17 . . 3 (𝜑 → ∃𝑠Q (𝑠 +Q 𝑠) = 𝐴)
12 simplr 528 . . . . . 6 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠Q)
13 archrecnq 7662 . . . . . . . 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 110 . . . . . . . . . . . 12 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠)
16 simplr 528 . . . . . . . . . . . . . 14 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → 𝑗N)
17 nnnq 7421 . . . . . . . . . . . . . 14 (𝑗N → [⟨𝑗, 1o⟩] ~QQ)
18 recclnq 7391 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 13 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → 𝑠Q)
21 ltanqg 7399 . . . . . . . . . . . . 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 1238 . . . . . . . . . . . 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 147 . . . . . . . . . . 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 534 . . . . . . . . . . 11 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) = 𝐴)
2523, 24breqtrd 4030 . . . . . . . . . 10 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝐴)
26 rsp 2524 . . . . . . . . . . . . 13 (∀𝑗N 𝐴 <Q (𝐹𝑗) → (𝑗N𝐴 <Q (𝐹𝑗)))
273, 26syl 14 . . . . . . . . . . . 12 (𝜑 → (𝑗N𝐴 <Q (𝐹𝑗)))
2827ad4antr 494 . . . . . . . . . . 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 7397 . . . . . . . . . . 11 <Q Or Q
3130, 7sotri 5025 . . . . . . . . . 10 (((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝐴𝐴 <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
3225, 29, 31syl2anc 411 . . . . . . . . 9 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
3332ex 115 . . . . . . . 8 ((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠 → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
3433reximdva 2579 . . . . . . 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 5882 . . . . . . . . 9 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
3736breq1d 4014 . . . . . . . 8 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
3837rexbidv 2478 . . . . . . 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 5519 . . . . . . . 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 7362 . . . . . . . . . 10 Q ∈ V
4241rabex 4148 . . . . . . . . 9 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
4341rabex 4148 . . . . . . . . 9 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
4442, 43op1st 6147 . . . . . . . 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 2198 . . . . . . 7 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
4638, 45elrab2 2897 . . . . . 6 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
4712, 35, 46sylanbrc 417 . . . . 5 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ (1st𝐿))
4847ex 115 . . . 4 ((𝜑𝑠Q) → ((𝑠 +Q 𝑠) = 𝐴𝑠 ∈ (1st𝐿)))
4948reximdva 2579 . . 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, 5ffvelcdmd 5653 . . . . 5 (𝜑 → (𝐹‘1o) ∈ Q)
53 1nq 7365 . . . . 5 1QQ
54 addclnq 7374 . . . . 5 (((𝐹‘1o) ∈ Q ∧ 1QQ) → ((𝐹‘1o) +Q 1Q) ∈ Q)
5552, 53, 54sylancl 413 . . . 4 (𝜑 → ((𝐹‘1o) +Q 1Q) ∈ Q)
56 addclnq 7374 . . . 4 ((((𝐹‘1o) +Q 1Q) ∈ Q ∧ 1QQ) → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ Q)
5755, 53, 56sylancl 413 . . 3 (𝜑 → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ Q)
58 df-1nqqs 7350 . . . . . . . . 9 1Q = [⟨1o, 1o⟩] ~Q
5958fveq2i 5519 . . . . . . . 8 (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
60 rec1nq 7394 . . . . . . . 8 (*Q‘1Q) = 1Q
6159, 60eqtr3i 2200 . . . . . . 7 (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
6261oveq2i 5886 . . . . . 6 ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )) = ((𝐹‘1o) +Q 1Q)
63 ltaddnq 7406 . . . . . . 7 ((((𝐹‘1o) +Q 1Q) ∈ Q ∧ 1QQ) → ((𝐹‘1o) +Q 1Q) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
6455, 53, 63sylancl 413 . . . . . 6 (𝜑 → ((𝐹‘1o) +Q 1Q) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
6562, 64eqbrtrid 4039 . . . . 5 (𝜑 → ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
66 opeq1 3779 . . . . . . . . . 10 (𝑗 = 1o → ⟨𝑗, 1o⟩ = ⟨1o, 1o⟩)
6766eceq1d 6571 . . . . . . . . 9 (𝑗 = 1o → [⟨𝑗, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
6867fveq2d 5520 . . . . . . . 8 (𝑗 = 1o → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
691, 68oveq12d 5893 . . . . . . 7 (𝑗 = 1o → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )))
7069breq1d 4014 . . . . . 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 2842 . . . . 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 411 . . . 4 (𝜑 → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
73 breq2 4008 . . . . . 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 2478 . . . . 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 5519 . . . . . 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 6148 . . . . . 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 2198 . . . . 5 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
7874, 77elrab2 2897 . . . 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 417 . . 3 (𝜑 → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd𝐿))
80 eleq1 2240 . . . 4 (𝑟 = (((𝐹‘1o) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd𝐿) ↔ (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd𝐿)))
8180rspcev 2842 . . 3 (((((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ Q ∧ (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd𝐿)) → ∃𝑟Q 𝑟 ∈ (2nd𝐿))
8257, 79, 81syl2anc 411 . 2 (𝜑 → ∃𝑟Q 𝑟 ∈ (2nd𝐿))
8350, 82jca 306 1 (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  wrex 2456  {crab 2459  cop 3596   class class class wbr 4004  wf 5213  cfv 5217  (class class class)co 5875  1st c1st 6139  2nd c2nd 6140  1oc1o 6410  [cec 6533  Ncnpi 7271   <N clti 7274   ~Q ceq 7278  Qcnq 7279  1Qc1q 7280   +Q cplq 7281  *Qcrq 7283   <Q cltq 7284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352
This theorem is referenced by:  caucvgprlemcl  7675
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