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Theorem caucvgprlemcl 6647
Description: Lemma for caucvgpr 6653. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)
Hypotheses
Ref Expression
caucvgpr.f (φ𝐹:NQ)
caucvgpr.cau (φ𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
caucvgpr.bnd (φ𝑗 N A <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
Assertion
Ref Expression
caucvgprlemcl (φ𝐿 P)
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 (φ𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
3 caucvgpr.bnd . . . . 5 (φ𝑗 N A <Q (𝐹𝑗))
4 fveq2 5121 . . . . . . 7 (𝑗 = 𝑎 → (𝐹𝑗) = (𝐹𝑎))
54breq2d 3767 . . . . . 6 (𝑗 = 𝑎 → (A <Q (𝐹𝑗) ↔ A <Q (𝐹𝑎)))
65cbvralv 2527 . . . . 5 (𝑗 N A <Q (𝐹𝑗) ↔ 𝑎 N A <Q (𝐹𝑎))
73, 6sylib 127 . . . 4 (φ𝑎 N A <Q (𝐹𝑎))
8 caucvgpr.lim . . . . 5 𝐿 = ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
9 opeq1 3540 . . . . . . . . . . . . 13 (𝑗 = 𝑎 → ⟨𝑗, 1𝑜⟩ = ⟨𝑎, 1𝑜⟩)
109eceq1d 6078 . . . . . . . . . . . 12 (𝑗 = 𝑎 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑎, 1𝑜⟩] ~Q )
1110fveq2d 5125 . . . . . . . . . . 11 (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))
1211oveq2d 5471 . . . . . . . . . 10 (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
1312, 4breq12d 3768 . . . . . . . . 9 (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)))
1413cbvrexv 2528 . . . . . . . 8 (𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ 𝑎 N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎))
1514a1i 9 . . . . . . 7 (𝑙 Q → (𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ 𝑎 N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)))
1615rabbiia 2541 . . . . . 6 {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} = {𝑙 Q𝑎 N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)}
174, 11oveq12d 5473 . . . . . . . . . 10 (𝑗 = 𝑎 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
1817breq1d 3765 . . . . . . . . 9 (𝑗 = 𝑎 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u ↔ ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q u))
1918cbvrexv 2528 . . . . . . . 8 (𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u𝑎 N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q u)
2019a1i 9 . . . . . . 7 (u Q → (𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u𝑎 N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q u))
2120rabbiia 2541 . . . . . 6 {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} = {u Q𝑎 N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q u}
2216, 21opeq12i 3545 . . . . 5 ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩ = ⟨{𝑙 Q𝑎 N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)}, {u Q𝑎 N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q u}⟩
238, 22eqtri 2057 . . . 4 𝐿 = ⟨{𝑙 Q𝑎 N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)}, {u Q𝑎 N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q u}⟩
241, 2, 7, 23caucvgprlemm 6639 . . 3 (φ → (𝑠 Q 𝑠 (1st𝐿) 𝑟 Q 𝑟 (2nd𝐿)))
25 ssrab2 3019 . . . . . 6 {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ⊆ Q
26 nqex 6347 . . . . . . 7 Q V
2726elpw2 3902 . . . . . 6 ({𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} 𝒫 Q ↔ {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ⊆ Q)
2825, 27mpbir 134 . . . . 5 {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} 𝒫 Q
29 ssrab2 3019 . . . . . 6 {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} ⊆ Q
3026elpw2 3902 . . . . . 6 ({u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} 𝒫 Q ↔ {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} ⊆ Q)
3129, 30mpbir 134 . . . . 5 {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} 𝒫 Q
32 opelxpi 4319 . . . . 5 (({𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} 𝒫 Q {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} 𝒫 Q) → ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩ (𝒫 Q × 𝒫 Q))
3328, 31, 32mp2an 402 . . . 4 ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩ (𝒫 Q × 𝒫 Q)
348, 33eqeltri 2107 . . 3 𝐿 (𝒫 Q × 𝒫 Q)
3524, 34jctil 295 . 2 (φ → (𝐿 (𝒫 Q × 𝒫 Q) (𝑠 Q 𝑠 (1st𝐿) 𝑟 Q 𝑟 (2nd𝐿))))
361, 2, 7, 23caucvgprlemrnd 6644 . . 3 (φ → (𝑠 Q (𝑠 (1st𝐿) ↔ 𝑟 Q (𝑠 <Q 𝑟 𝑟 (1st𝐿))) 𝑟 Q (𝑟 (2nd𝐿) ↔ 𝑠 Q (𝑠 <Q 𝑟 𝑠 (2nd𝐿)))))
37 breq1 3758 . . . . . . 7 (𝑛 = 𝑐 → (𝑛 <N 𝑘𝑐 <N 𝑘))
38 fveq2 5121 . . . . . . . . 9 (𝑛 = 𝑐 → (𝐹𝑛) = (𝐹𝑐))
39 opeq1 3540 . . . . . . . . . . . 12 (𝑛 = 𝑐 → ⟨𝑛, 1𝑜⟩ = ⟨𝑐, 1𝑜⟩)
4039eceq1d 6078 . . . . . . . . . . 11 (𝑛 = 𝑐 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝑐, 1𝑜⟩] ~Q )
4140fveq2d 5125 . . . . . . . . . 10 (𝑛 = 𝑐 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))
4241oveq2d 5471 . . . . . . . . 9 (𝑛 = 𝑐 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
4338, 42breq12d 3768 . . . . . . . 8 (𝑛 = 𝑐 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
4438, 41oveq12d 5473 . . . . . . . . 9 (𝑛 = 𝑐 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
4544breq2d 3767 . . . . . . . 8 (𝑛 = 𝑐 → ((𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
4643, 45anbi12d 442 . . . . . . 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 223 . . . . . 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 3759 . . . . . . 7 (𝑘 = 𝑑 → (𝑐 <N 𝑘𝑐 <N 𝑑))
49 fveq2 5121 . . . . . . . . . 10 (𝑘 = 𝑑 → (𝐹𝑘) = (𝐹𝑑))
5049oveq1d 5470 . . . . . . . . 9 (𝑘 = 𝑑 → ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) = ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
5150breq2d 3767 . . . . . . . 8 (𝑘 = 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ↔ (𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
5249breq1d 3765 . . . . . . . 8 (𝑘 = 𝑑 → ((𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ↔ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
5351, 52anbi12d 442 . . . . . . 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 223 . . . . . 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 2535 . . . . 5 (𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) ↔ 𝑐 N 𝑑 N (𝑐 <N 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))))
562, 55sylib 127 . . . 4 (φ𝑐 N 𝑑 N (𝑐 <N 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))))
571, 56, 7, 23caucvgprlemdisj 6645 . . 3 (φ𝑠 Q ¬ (𝑠 (1st𝐿) 𝑠 (2nd𝐿)))
581, 2, 7, 23caucvgprlemloc 6646 . . 3 (φ𝑠 Q 𝑟 Q (𝑠 <Q 𝑟 → (𝑠 (1st𝐿) 𝑟 (2nd𝐿))))
5936, 57, 583jca 1083 . 2 (φ → ((𝑠 Q (𝑠 (1st𝐿) ↔ 𝑟 Q (𝑠 <Q 𝑟 𝑟 (1st𝐿))) 𝑟 Q (𝑟 (2nd𝐿) ↔ 𝑠 Q (𝑠 <Q 𝑟 𝑠 (2nd𝐿)))) 𝑠 Q ¬ (𝑠 (1st𝐿) 𝑠 (2nd𝐿)) 𝑠 Q 𝑟 Q (𝑠 <Q 𝑟 → (𝑠 (1st𝐿) 𝑟 (2nd𝐿)))))
60 elnp1st2nd 6459 . 2 (𝐿 P ↔ ((𝐿 (𝒫 Q × 𝒫 Q) (𝑠 Q 𝑠 (1st𝐿) 𝑟 Q 𝑟 (2nd𝐿))) ((𝑠 Q (𝑠 (1st𝐿) ↔ 𝑟 Q (𝑠 <Q 𝑟 𝑟 (1st𝐿))) 𝑟 Q (𝑟 (2nd𝐿) ↔ 𝑠 Q (𝑠 <Q 𝑟 𝑠 (2nd𝐿)))) 𝑠 Q ¬ (𝑠 (1st𝐿) 𝑠 (2nd𝐿)) 𝑠 Q 𝑟 Q (𝑠 <Q 𝑟 → (𝑠 (1st𝐿) 𝑟 (2nd𝐿))))))
6135, 59, 60sylanbrc 394 1 (φ𝐿 P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3a 884   = wceq 1242   wcel 1390  wral 2300  wrex 2301  {crab 2304  wss 2911  𝒫 cpw 3351  cop 3370   class class class wbr 3755   × cxp 4286  wf 4841  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  1𝑜c1o 5933  [cec 6040  Ncnpi 6256   <N clti 6259   ~Q ceq 6263  Qcnq 6264   +Q cplq 6266  *Qcrq 6268   <Q cltq 6269  Pcnp 6275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6449
This theorem is referenced by:  caucvgprlemladdfu  6648  caucvgprlemladdrl  6649  caucvgprlem1  6650  caucvgprlem2  6651  caucvgpr  6653
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