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Theorem caucvgprlemcl 7235
Description: Lemma for caucvgpr 7241. 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 5305 . . . . . . 7 (𝑗 = 𝑎 → (𝐹𝑗) = (𝐹𝑎))
54breq2d 3857 . . . . . 6 (𝑗 = 𝑎 → (𝐴 <Q (𝐹𝑗) ↔ 𝐴 <Q (𝐹𝑎)))
65cbvralv 2590 . . . . 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 3622 . . . . . . . . . . . . 13 (𝑗 = 𝑎 → ⟨𝑗, 1𝑜⟩ = ⟨𝑎, 1𝑜⟩)
109eceq1d 6328 . . . . . . . . . . . 12 (𝑗 = 𝑎 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑎, 1𝑜⟩] ~Q )
1110fveq2d 5309 . . . . . . . . . . 11 (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))
1211oveq2d 5668 . . . . . . . . . 10 (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
1312, 4breq12d 3858 . . . . . . . . 9 (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)))
1413cbvrexv 2591 . . . . . . . 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 2604 . . . . . 6 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} = {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)}
174, 11oveq12d 5670 . . . . . . . . . 10 (𝑗 = 𝑎 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
1817breq1d 3855 . . . . . . . . 9 (𝑗 = 𝑎 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢))
1918cbvrexv 2591 . . . . . . . 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 2604 . . . . . 6 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} = {𝑢Q ∣ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}
2216, 21opeq12i 3627 . . . . 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 2108 . . . 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 7227 . . 3 (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))
25 ssrab2 3106 . . . . . 6 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ⊆ Q
26 nqex 6922 . . . . . . 7 Q ∈ V
2726elpw2 3993 . . . . . 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 3106 . . . . . 6 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ⊆ Q
3026elpw2 3993 . . . . . 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 4469 . . . . 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 2160 . . 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 7232 . . 3 (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))))
37 breq1 3848 . . . . . . 7 (𝑛 = 𝑐 → (𝑛 <N 𝑘𝑐 <N 𝑘))
38 fveq2 5305 . . . . . . . . 9 (𝑛 = 𝑐 → (𝐹𝑛) = (𝐹𝑐))
39 opeq1 3622 . . . . . . . . . . . 12 (𝑛 = 𝑐 → ⟨𝑛, 1𝑜⟩ = ⟨𝑐, 1𝑜⟩)
4039eceq1d 6328 . . . . . . . . . . 11 (𝑛 = 𝑐 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝑐, 1𝑜⟩] ~Q )
4140fveq2d 5309 . . . . . . . . . 10 (𝑛 = 𝑐 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))
4241oveq2d 5668 . . . . . . . . 9 (𝑛 = 𝑐 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
4338, 42breq12d 3858 . . . . . . . 8 (𝑛 = 𝑐 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
4438, 41oveq12d 5670 . . . . . . . . 9 (𝑛 = 𝑐 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
4544breq2d 3857 . . . . . . . 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 3849 . . . . . . 7 (𝑘 = 𝑑 → (𝑐 <N 𝑘𝑐 <N 𝑑))
49 fveq2 5305 . . . . . . . . . 10 (𝑘 = 𝑑 → (𝐹𝑘) = (𝐹𝑑))
5049oveq1d 5667 . . . . . . . . 9 (𝑘 = 𝑑 → ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) = ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
5150breq2d 3857 . . . . . . . 8 (𝑘 = 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ↔ (𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
5249breq1d 3855 . . . . . . . 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 2598 . . . . 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 7233 . . 3 (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
581, 2, 7, 23caucvgprlemloc 7234 . . 3 (𝜑 → ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))
5936, 57, 583jca 1123 . 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 7035 . 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 664  w3a 924   = wceq 1289  wcel 1438  wral 2359  wrex 2360  {crab 2363  wss 2999  𝒫 cpw 3429  cop 3449   class class class wbr 3845   × cxp 4436  wf 5011  cfv 5015  (class class class)co 5652  1st c1st 5909  2nd c2nd 5910  1𝑜c1o 6174  [cec 6290  Ncnpi 6831   <N clti 6834   ~Q ceq 6838  Qcnq 6839   +Q cplq 6841  *Qcrq 6843   <Q cltq 6844  Pcnp 6850
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 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
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 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-inp 7025
This theorem is referenced by:  caucvgprlemladdfu  7236  caucvgprlemladdrl  7237  caucvgprlem1  7238  caucvgprlem2  7239  caucvgpr  7241
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