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Theorem caucvgprlemm 7499
Description: Lemma for caucvgpr 7513. 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 5428 . . . . . 6 (𝑗 = 1o → (𝐹𝑗) = (𝐹‘1o))
21breq2d 3948 . . . . 5 (𝑗 = 1o → (𝐴 <Q (𝐹𝑗) ↔ 𝐴 <Q (𝐹‘1o)))
3 caucvgpr.bnd . . . . 5 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
4 1pi 7146 . . . . . 6 1oN
54a1i 9 . . . . 5 (𝜑 → 1oN)
62, 3, 5rspcdva 2797 . . . 4 (𝜑𝐴 <Q (𝐹‘1o))
7 ltrelnq 7196 . . . . . 6 <Q ⊆ (Q × Q)
87brel 4598 . . . . 5 (𝐴 <Q (𝐹‘1o) → (𝐴Q ∧ (𝐹‘1o) ∈ Q))
98simpld 111 . . . 4 (𝐴 <Q (𝐹‘1o) → 𝐴Q)
10 halfnqq 7241 . . . 4 (𝐴Q → ∃𝑠Q (𝑠 +Q 𝑠) = 𝐴)
116, 9, 103syl 17 . . 3 (𝜑 → ∃𝑠Q (𝑠 +Q 𝑠) = 𝐴)
12 simplr 520 . . . . . 6 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠Q)
13 archrecnq 7494 . . . . . . . 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 109 . . . . . . . . . . . 12 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠)
16 simplr 520 . . . . . . . . . . . . . 14 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → 𝑗N)
17 nnnq 7253 . . . . . . . . . . . . . 14 (𝑗N → [⟨𝑗, 1o⟩] ~QQ)
18 recclnq 7223 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → 𝑠Q)
21 ltanqg 7231 . . . . . . . . . . . . 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 1217 . . . . . . . . . . . 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 146 . . . . . . . . . . 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 524 . . . . . . . . . . 11 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) = 𝐴)
2523, 24breqtrd 3961 . . . . . . . . . 10 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝐴)
26 rsp 2483 . . . . . . . . . . . . 13 (∀𝑗N 𝐴 <Q (𝐹𝑗) → (𝑗N𝐴 <Q (𝐹𝑗)))
273, 26syl 14 . . . . . . . . . . . 12 (𝜑 → (𝑗N𝐴 <Q (𝐹𝑗)))
2827ad4antr 486 . . . . . . . . . . 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 7229 . . . . . . . . . . 11 <Q Or Q
3130, 7sotri 4941 . . . . . . . . . 10 (((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝐴𝐴 <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
3225, 29, 31syl2anc 409 . . . . . . . . 9 (((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
3332ex 114 . . . . . . . 8 ((((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗N) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠 → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
3433reximdva 2537 . . . . . . 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 5788 . . . . . . . . 9 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
3736breq1d 3946 . . . . . . . 8 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
3837rexbidv 2439 . . . . . . 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 5431 . . . . . . . 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 7194 . . . . . . . . . 10 Q ∈ V
4241rabex 4079 . . . . . . . . 9 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
4341rabex 4079 . . . . . . . . 9 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
4442, 43op1st 6051 . . . . . . . 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 2161 . . . . . . 7 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
4638, 45elrab2 2846 . . . . . 6 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
4712, 35, 46sylanbrc 414 . . . . 5 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ (1st𝐿))
4847ex 114 . . . 4 ((𝜑𝑠Q) → ((𝑠 +Q 𝑠) = 𝐴𝑠 ∈ (1st𝐿)))
4948reximdva 2537 . . 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 5563 . . . . 5 (𝜑 → (𝐹‘1o) ∈ Q)
53 1nq 7197 . . . . 5 1QQ
54 addclnq 7206 . . . . 5 (((𝐹‘1o) ∈ Q ∧ 1QQ) → ((𝐹‘1o) +Q 1Q) ∈ Q)
5552, 53, 54sylancl 410 . . . 4 (𝜑 → ((𝐹‘1o) +Q 1Q) ∈ Q)
56 addclnq 7206 . . . 4 ((((𝐹‘1o) +Q 1Q) ∈ Q ∧ 1QQ) → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ Q)
5755, 53, 56sylancl 410 . . 3 (𝜑 → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ Q)
58 df-1nqqs 7182 . . . . . . . . 9 1Q = [⟨1o, 1o⟩] ~Q
5958fveq2i 5431 . . . . . . . 8 (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
60 rec1nq 7226 . . . . . . . 8 (*Q‘1Q) = 1Q
6159, 60eqtr3i 2163 . . . . . . 7 (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
6261oveq2i 5792 . . . . . 6 ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )) = ((𝐹‘1o) +Q 1Q)
63 ltaddnq 7238 . . . . . . 7 ((((𝐹‘1o) +Q 1Q) ∈ Q ∧ 1QQ) → ((𝐹‘1o) +Q 1Q) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
6455, 53, 63sylancl 410 . . . . . 6 (𝜑 → ((𝐹‘1o) +Q 1Q) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
6562, 64eqbrtrid 3970 . . . . 5 (𝜑 → ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
66 opeq1 3712 . . . . . . . . . 10 (𝑗 = 1o → ⟨𝑗, 1o⟩ = ⟨1o, 1o⟩)
6766eceq1d 6472 . . . . . . . . 9 (𝑗 = 1o → [⟨𝑗, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
6867fveq2d 5432 . . . . . . . 8 (𝑗 = 1o → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
691, 68oveq12d 5799 . . . . . . 7 (𝑗 = 1o → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )))
7069breq1d 3946 . . . . . 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 2792 . . . . 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 409 . . . 4 (𝜑 → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
73 breq2 3940 . . . . . 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 2439 . . . . 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 5431 . . . . . 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 6052 . . . . . 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 2161 . . . . 5 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
7874, 77elrab2 2846 . . . 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 414 . . 3 (𝜑 → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd𝐿))
80 eleq1 2203 . . . 4 (𝑟 = (((𝐹‘1o) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd𝐿) ↔ (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd𝐿)))
8180rspcev 2792 . . 3 (((((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ Q ∧ (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd𝐿)) → ∃𝑟Q 𝑟 ∈ (2nd𝐿))
8257, 79, 81syl2anc 409 . 2 (𝜑 → ∃𝑟Q 𝑟 ∈ (2nd𝐿))
8350, 82jca 304 1 (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  wral 2417  wrex 2418  {crab 2421  cop 3534   class class class wbr 3936  wf 5126  cfv 5130  (class class class)co 5781  1st c1st 6043  2nd c2nd 6044  1oc1o 6313  [cec 6434  Ncnpi 7103   <N clti 7106   ~Q ceq 7110  Qcnq 7111  1Qc1q 7112   +Q cplq 7113  *Qcrq 7115   <Q cltq 7116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184
This theorem is referenced by:  caucvgprlemcl  7507
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