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Theorem caucvgprlemcl 7674
Description: Lemma for caucvgpr 7680. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-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
caucvgprlemcl (𝜑𝐿P)
Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑙   𝑢,𝐹,𝑗   𝑛,𝐹,𝑘   𝑗,𝑘,𝐿   𝑘,𝑛
Allowed substitution hints:   𝜑(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑙)   𝐿(𝑢,𝑛,𝑙)

Proof of Theorem caucvgprlemcl
Dummy variables 𝑠 𝑎 𝑐 𝑑 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgpr.f . . . 4 (𝜑𝐹:NQ)
2 caucvgpr.cau . . . 4 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
3 caucvgpr.bnd . . . . 5 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
4 fveq2 5515 . . . . . . 7 (𝑗 = 𝑎 → (𝐹𝑗) = (𝐹𝑎))
54breq2d 4015 . . . . . 6 (𝑗 = 𝑎 → (𝐴 <Q (𝐹𝑗) ↔ 𝐴 <Q (𝐹𝑎)))
65cbvralv 2703 . . . . 5 (∀𝑗N 𝐴 <Q (𝐹𝑗) ↔ ∀𝑎N 𝐴 <Q (𝐹𝑎))
73, 6sylib 122 . . . 4 (𝜑 → ∀𝑎N 𝐴 <Q (𝐹𝑎))
8 caucvgpr.lim . . . . 5 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
9 opeq1 3778 . . . . . . . . . . . . 13 (𝑗 = 𝑎 → ⟨𝑗, 1o⟩ = ⟨𝑎, 1o⟩)
109eceq1d 6570 . . . . . . . . . . . 12 (𝑗 = 𝑎 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
1110fveq2d 5519 . . . . . . . . . . 11 (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
1211oveq2d 5890 . . . . . . . . . 10 (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
1312, 4breq12d 4016 . . . . . . . . 9 (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹𝑎)))
1413cbvrexv 2704 . . . . . . . 8 (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹𝑎))
1514a1i 9 . . . . . . 7 (𝑙Q → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹𝑎)))
1615rabbiia 2722 . . . . . 6 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} = {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹𝑎)}
174, 11oveq12d 5892 . . . . . . . . . 10 (𝑗 = 𝑎 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
1817breq1d 4013 . . . . . . . . 9 (𝑗 = 𝑎 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢))
1918cbvrexv 2704 . . . . . . . 8 (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢)
2019a1i 9 . . . . . . 7 (𝑢Q → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢))
2120rabbiia 2722 . . . . . 6 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} = {𝑢Q ∣ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}
2216, 21opeq12i 3783 . . . . 5 ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩ = ⟨{𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹𝑎)}, {𝑢Q ∣ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩
238, 22eqtri 2198 . . . 4 𝐿 = ⟨{𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹𝑎)}, {𝑢Q ∣ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩
241, 2, 7, 23caucvgprlemm 7666 . . 3 (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))
25 ssrab2 3240 . . . . . 6 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ⊆ Q
26 nqex 7361 . . . . . . 7 Q ∈ V
2726elpw2 4157 . . . . . 6 ({𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ 𝒫 Q ↔ {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ⊆ Q)
2825, 27mpbir 146 . . . . 5 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ 𝒫 Q
29 ssrab2 3240 . . . . . 6 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ⊆ Q
3026elpw2 4157 . . . . . 6 ({𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q ↔ {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ⊆ Q)
3129, 30mpbir 146 . . . . 5 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q
32 opelxpi 4658 . . . . 5 (({𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ 𝒫 Q ∧ {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q) → ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ (𝒫 Q × 𝒫 Q))
3328, 31, 32mp2an 426 . . . 4 ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ (𝒫 Q × 𝒫 Q)
348, 33eqeltri 2250 . . 3 𝐿 ∈ (𝒫 Q × 𝒫 Q)
3524, 34jctil 312 . 2 (𝜑 → (𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿))))
361, 2, 7, 23caucvgprlemrnd 7671 . . 3 (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))))
37 breq1 4006 . . . . . . 7 (𝑛 = 𝑐 → (𝑛 <N 𝑘𝑐 <N 𝑘))
38 fveq2 5515 . . . . . . . . 9 (𝑛 = 𝑐 → (𝐹𝑛) = (𝐹𝑐))
39 opeq1 3778 . . . . . . . . . . . 12 (𝑛 = 𝑐 → ⟨𝑛, 1o⟩ = ⟨𝑐, 1o⟩)
4039eceq1d 6570 . . . . . . . . . . 11 (𝑛 = 𝑐 → [⟨𝑛, 1o⟩] ~Q = [⟨𝑐, 1o⟩] ~Q )
4140fveq2d 5519 . . . . . . . . . 10 (𝑛 = 𝑐 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝑐, 1o⟩] ~Q ))
4241oveq2d 5890 . . . . . . . . 9 (𝑛 = 𝑐 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
4338, 42breq12d 4016 . . . . . . . 8 (𝑛 = 𝑐 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))
4438, 41oveq12d 5892 . . . . . . . . 9 (𝑛 = 𝑐 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
4544breq2d 4015 . . . . . . . 8 (𝑛 = 𝑐 → ((𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))
4643, 45anbi12d 473 . . . . . . 7 (𝑛 = 𝑐 → (((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q ))) ↔ ((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))))
4737, 46imbi12d 234 . . . . . 6 (𝑛 = 𝑐 → ((𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))) ↔ (𝑐 <N 𝑘 → ((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))))
48 breq2 4007 . . . . . . 7 (𝑘 = 𝑑 → (𝑐 <N 𝑘𝑐 <N 𝑑))
49 fveq2 5515 . . . . . . . . . 10 (𝑘 = 𝑑 → (𝐹𝑘) = (𝐹𝑑))
5049oveq1d 5889 . . . . . . . . 9 (𝑘 = 𝑑 → ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) = ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
5150breq2d 4015 . . . . . . . 8 (𝑘 = 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ↔ (𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))
5249breq1d 4013 . . . . . . . 8 (𝑘 = 𝑑 → ((𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ↔ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))
5351, 52anbi12d 473 . . . . . . 7 (𝑘 = 𝑑 → (((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))) ↔ ((𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))))
5448, 53imbi12d 234 . . . . . 6 (𝑘 = 𝑑 → ((𝑐 <N 𝑘 → ((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))) ↔ (𝑐 <N 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))))
5547, 54cbvral2v 2716 . . . . 5 (∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))) ↔ ∀𝑐N𝑑N (𝑐 <N 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))))
562, 55sylib 122 . . . 4 (𝜑 → ∀𝑐N𝑑N (𝑐 <N 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))))
571, 56, 7, 23caucvgprlemdisj 7672 . . 3 (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
581, 2, 7, 23caucvgprlemloc 7673 . . 3 (𝜑 → ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))
5936, 57, 583jca 1177 . 2 (𝜑 → ((∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))) ∧ ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) ∧ ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿)))))
60 elnp1st2nd 7474 . 2 (𝐿P ↔ ((𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿))) ∧ ((∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))) ∧ ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) ∧ ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))))
6135, 59, 60sylanbrc 417 1 (𝜑𝐿P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  w3a 978   = wceq 1353  wcel 2148  wral 2455  wrex 2456  {crab 2459  wss 3129  𝒫 cpw 3575  cop 3595   class class class wbr 4003   × cxp 4624  wf 5212  cfv 5216  (class class class)co 5874  1st c1st 6138  2nd c2nd 6139  1oc1o 6409  [cec 6532  Ncnpi 7270   <N clti 7273   ~Q ceq 7277  Qcnq 7278   +Q cplq 7280  *Qcrq 7282   <Q cltq 7283  Pcnp 7289
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-inp 7464
This theorem is referenced by:  caucvgprlemladdfu  7675  caucvgprlemladdrl  7676  caucvgprlem1  7677  caucvgprlem2  7678  caucvgpr  7680
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