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Theorem caucvgprlemcl 6980
Description: Lemma for caucvgpr 6986. 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 5229 . . . . . . 7 (𝑗 = 𝑎 → (𝐹𝑗) = (𝐹𝑎))
54breq2d 3817 . . . . . 6 (𝑗 = 𝑎 → (𝐴 <Q (𝐹𝑗) ↔ 𝐴 <Q (𝐹𝑎)))
65cbvralv 2582 . . . . 5 (∀𝑗N 𝐴 <Q (𝐹𝑗) ↔ ∀𝑎N 𝐴 <Q (𝐹𝑎))
73, 6sylib 120 . . . 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 3590 . . . . . . . . . . . . 13 (𝑗 = 𝑎 → ⟨𝑗, 1𝑜⟩ = ⟨𝑎, 1𝑜⟩)
109eceq1d 6229 . . . . . . . . . . . 12 (𝑗 = 𝑎 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑎, 1𝑜⟩] ~Q )
1110fveq2d 5233 . . . . . . . . . . 11 (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))
1211oveq2d 5579 . . . . . . . . . 10 (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
1312, 4breq12d 3818 . . . . . . . . 9 (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)))
1413cbvrexv 2583 . . . . . . . 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 2596 . . . . . 6 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} = {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)}
174, 11oveq12d 5581 . . . . . . . . . 10 (𝑗 = 𝑎 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
1817breq1d 3815 . . . . . . . . 9 (𝑗 = 𝑎 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢))
1918cbvrexv 2583 . . . . . . . 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 2596 . . . . . 6 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} = {𝑢Q ∣ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}
2216, 21opeq12i 3595 . . . . 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 2103 . . . 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 6972 . . 3 (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))
25 ssrab2 3088 . . . . . 6 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ⊆ Q
26 nqex 6667 . . . . . . 7 Q ∈ V
2726elpw2 3952 . . . . . 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 144 . . . . 5 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ 𝒫 Q
29 ssrab2 3088 . . . . . 6 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ⊆ Q
3026elpw2 3952 . . . . . 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 144 . . . . 5 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q
32 opelxpi 4422 . . . . 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 417 . . . 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 2155 . . 3 𝐿 ∈ (𝒫 Q × 𝒫 Q)
3524, 34jctil 305 . 2 (𝜑 → (𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿))))
361, 2, 7, 23caucvgprlemrnd 6977 . . 3 (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))))
37 breq1 3808 . . . . . . 7 (𝑛 = 𝑐 → (𝑛 <N 𝑘𝑐 <N 𝑘))
38 fveq2 5229 . . . . . . . . 9 (𝑛 = 𝑐 → (𝐹𝑛) = (𝐹𝑐))
39 opeq1 3590 . . . . . . . . . . . 12 (𝑛 = 𝑐 → ⟨𝑛, 1𝑜⟩ = ⟨𝑐, 1𝑜⟩)
4039eceq1d 6229 . . . . . . . . . . 11 (𝑛 = 𝑐 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝑐, 1𝑜⟩] ~Q )
4140fveq2d 5233 . . . . . . . . . 10 (𝑛 = 𝑐 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))
4241oveq2d 5579 . . . . . . . . 9 (𝑛 = 𝑐 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
4338, 42breq12d 3818 . . . . . . . 8 (𝑛 = 𝑐 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
4438, 41oveq12d 5581 . . . . . . . . 9 (𝑛 = 𝑐 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
4544breq2d 3817 . . . . . . . 8 (𝑛 = 𝑐 → ((𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
4643, 45anbi12d 457 . . . . . . 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 232 . . . . . 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 3809 . . . . . . 7 (𝑘 = 𝑑 → (𝑐 <N 𝑘𝑐 <N 𝑑))
49 fveq2 5229 . . . . . . . . . 10 (𝑘 = 𝑑 → (𝐹𝑘) = (𝐹𝑑))
5049oveq1d 5578 . . . . . . . . 9 (𝑘 = 𝑑 → ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) = ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
5150breq2d 3817 . . . . . . . 8 (𝑘 = 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ↔ (𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
5249breq1d 3815 . . . . . . . 8 (𝑘 = 𝑑 → ((𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ↔ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
5351, 52anbi12d 457 . . . . . . 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 232 . . . . . 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 2590 . . . . 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 120 . . . 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 6978 . . 3 (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
581, 2, 7, 23caucvgprlemloc 6979 . . 3 (𝜑 → ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))
5936, 57, 583jca 1119 . 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 6780 . 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 408 1 (𝜑𝐿P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3a 920   = wceq 1285  wcel 1434  wral 2353  wrex 2354  {crab 2357  wss 2982  𝒫 cpw 3400  cop 3419   class class class wbr 3805   × cxp 4389  wf 4948  cfv 4952  (class class class)co 5563  1st c1st 5816  2nd c2nd 5817  1𝑜c1o 6078  [cec 6191  Ncnpi 6576   <N clti 6579   ~Q ceq 6583  Qcnq 6584   +Q cplq 6586  *Qcrq 6588   <Q cltq 6589  Pcnp 6595
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-oadd 6089  df-omul 6090  df-er 6193  df-ec 6195  df-qs 6199  df-ni 6608  df-pli 6609  df-mi 6610  df-lti 6611  df-plpq 6648  df-mpq 6649  df-enq 6651  df-nqqs 6652  df-plqqs 6653  df-mqqs 6654  df-1nqqs 6655  df-rq 6656  df-ltnqqs 6657  df-inp 6770
This theorem is referenced by:  caucvgprlemladdfu  6981  caucvgprlemladdrl  6982  caucvgprlem1  6983  caucvgprlem2  6984  caucvgpr  6986
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