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Theorem caucvgprlemm 6639
Description: Lemma for caucvgpr 6653. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-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
caucvgprlemm (φ → (𝑠 Q 𝑠 (1st𝐿) 𝑟 Q 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 1pi 6299 . . . . 5 1𝑜 N
2 caucvgpr.bnd . . . . 5 (φ𝑗 N A <Q (𝐹𝑗))
3 fveq2 5121 . . . . . . 7 (𝑗 = 1𝑜 → (𝐹𝑗) = (𝐹‘1𝑜))
43breq2d 3767 . . . . . 6 (𝑗 = 1𝑜 → (A <Q (𝐹𝑗) ↔ A <Q (𝐹‘1𝑜)))
54rspcv 2646 . . . . 5 (1𝑜 N → (𝑗 N A <Q (𝐹𝑗) → A <Q (𝐹‘1𝑜)))
61, 2, 5mpsyl 59 . . . 4 (φA <Q (𝐹‘1𝑜))
7 ltrelnq 6349 . . . . . 6 <Q ⊆ (Q × Q)
87brel 4335 . . . . 5 (A <Q (𝐹‘1𝑜) → (A Q (𝐹‘1𝑜) Q))
98simpld 105 . . . 4 (A <Q (𝐹‘1𝑜) → A Q)
10 halfnqq 6393 . . . 4 (A Q𝑠 Q (𝑠 +Q 𝑠) = A)
116, 9, 103syl 17 . . 3 (φ𝑠 Q (𝑠 +Q 𝑠) = A)
12 simplr 482 . . . . . 6 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → 𝑠 Q)
13 archrecnq 6635 . . . . . . . 8 (𝑠 Q𝑗 N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠)
1412, 13syl 14 . . . . . . 7 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → 𝑗 N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠)
15 simpr 103 . . . . . . . . . . . 12 (((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠)
16 simplr 482 . . . . . . . . . . . . . 14 (((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑗 N)
17 nnnq 6405 . . . . . . . . . . . . . 14 (𝑗 N → [⟨𝑗, 1𝑜⟩] ~Q Q)
18 recclnq 6376 . . . . . . . . . . . . . 14 ([⟨𝑗, 1𝑜⟩] ~Q Q → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) Q)
1916, 17, 183syl 17 . . . . . . . . . . . . 13 (((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) Q)
2012ad2antrr 457 . . . . . . . . . . . . 13 (((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑠 Q)
21 ltanqg 6384 . . . . . . . . . . . . 13 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) Q 𝑠 Q 𝑠 Q) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝑠 +Q 𝑠)))
2219, 20, 20, 21syl3anc 1134 . . . . . . . . . . . 12 (((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝑠 +Q 𝑠)))
2315, 22mpbid 135 . . . . . . . . . . 11 (((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝑠 +Q 𝑠))
24 simpllr 486 . . . . . . . . . . 11 (((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) = A)
2523, 24breqtrd 3779 . . . . . . . . . 10 (((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q A)
26 rsp 2363 . . . . . . . . . . . . 13 (𝑗 N A <Q (𝐹𝑗) → (𝑗 NA <Q (𝐹𝑗)))
272, 26syl 14 . . . . . . . . . . . 12 (φ → (𝑗 NA <Q (𝐹𝑗)))
2827ad4antr 463 . . . . . . . . . . 11 (((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑗 NA <Q (𝐹𝑗)))
2916, 28mpd 13 . . . . . . . . . 10 (((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → A <Q (𝐹𝑗))
30 ltsonq 6382 . . . . . . . . . . 11 <Q Or Q
3130, 7sotri 4663 . . . . . . . . . 10 (((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q A A <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
3225, 29, 31syl2anc 391 . . . . . . . . 9 (((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
3332ex 108 . . . . . . . 8 ((((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) 𝑗 N) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
3433reximdva 2415 . . . . . . 7 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → (𝑗 N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
3514, 34mpd 13 . . . . . 6 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
36 oveq1 5462 . . . . . . . . 9 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
3736breq1d 3765 . . . . . . . 8 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
3837rexbidv 2321 . . . . . . 7 (𝑙 = 𝑠 → (𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
39 caucvgpr.lim . . . . . . . . 9 𝐿 = ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
4039fveq2i 5124 . . . . . . . 8 (1st𝐿) = (1st ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩)
41 nqex 6347 . . . . . . . . . 10 Q V
4241rabex 3892 . . . . . . . . 9 {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} V
4341rabex 3892 . . . . . . . . 9 {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} V
4442, 43op1st 5715 . . . . . . . 8 (1st ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩) = {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
4540, 44eqtri 2057 . . . . . . 7 (1st𝐿) = {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
4638, 45elrab2 2694 . . . . . 6 (𝑠 (1st𝐿) ↔ (𝑠 Q 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
4712, 35, 46sylanbrc 394 . . . . 5 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → 𝑠 (1st𝐿))
4847ex 108 . . . 4 ((φ 𝑠 Q) → ((𝑠 +Q 𝑠) = A𝑠 (1st𝐿)))
4948reximdva 2415 . . 3 (φ → (𝑠 Q (𝑠 +Q 𝑠) = A𝑠 Q 𝑠 (1st𝐿)))
5011, 49mpd 13 . 2 (φ𝑠 Q 𝑠 (1st𝐿))
51 caucvgpr.f . . . . . 6 (φ𝐹:NQ)
521a1i 9 . . . . . 6 (φ → 1𝑜 N)
5351, 52ffvelrnd 5246 . . . . 5 (φ → (𝐹‘1𝑜) Q)
54 1nq 6350 . . . . 5 1Q Q
55 addclnq 6359 . . . . 5 (((𝐹‘1𝑜) Q 1Q Q) → ((𝐹‘1𝑜) +Q 1Q) Q)
5653, 54, 55sylancl 392 . . . 4 (φ → ((𝐹‘1𝑜) +Q 1Q) Q)
57 addclnq 6359 . . . 4 ((((𝐹‘1𝑜) +Q 1Q) Q 1Q Q) → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) Q)
5856, 54, 57sylancl 392 . . 3 (φ → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) Q)
59 df-1nqqs 6335 . . . . . . . . 9 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
6059fveq2i 5124 . . . . . . . 8 (*Q‘1Q) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )
61 rec1nq 6379 . . . . . . . 8 (*Q‘1Q) = 1Q
6260, 61eqtr3i 2059 . . . . . . 7 (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) = 1Q
6362oveq2i 5466 . . . . . 6 ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) = ((𝐹‘1𝑜) +Q 1Q)
64 ltaddnq 6390 . . . . . . 7 ((((𝐹‘1𝑜) +Q 1Q) Q 1Q Q) → ((𝐹‘1𝑜) +Q 1Q) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
6556, 54, 64sylancl 392 . . . . . 6 (φ → ((𝐹‘1𝑜) +Q 1Q) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
6663, 65syl5eqbr 3788 . . . . 5 (φ → ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
67 opeq1 3540 . . . . . . . . . 10 (𝑗 = 1𝑜 → ⟨𝑗, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
6867eceq1d 6078 . . . . . . . . 9 (𝑗 = 1𝑜 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
6968fveq2d 5125 . . . . . . . 8 (𝑗 = 1𝑜 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ))
703, 69oveq12d 5473 . . . . . . 7 (𝑗 = 1𝑜 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )))
7170breq1d 3765 . . . . . 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)))
7271rspcev 2650 . . . . 5 ((1𝑜 N ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)) → 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
7352, 66, 72syl2anc 391 . . . 4 (φ𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
74 breq2 3759 . . . . . 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)))
7574rexbidv 2321 . . . . 5 (u = (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) → (𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)))
7639fveq2i 5124 . . . . . 6 (2nd𝐿) = (2nd ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩)
7742, 43op2nd 5716 . . . . . 6 (2nd ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩) = {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
7876, 77eqtri 2057 . . . . 5 (2nd𝐿) = {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
7975, 78elrab2 2694 . . . 4 ((((𝐹‘1𝑜) +Q 1Q) +Q 1Q) (2nd𝐿) ↔ ((((𝐹‘1𝑜) +Q 1Q) +Q 1Q) Q 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)))
8058, 73, 79sylanbrc 394 . . 3 (φ → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) (2nd𝐿))
81 eleq1 2097 . . . 4 (𝑟 = (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) → (𝑟 (2nd𝐿) ↔ (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) (2nd𝐿)))
8281rspcev 2650 . . 3 (((((𝐹‘1𝑜) +Q 1Q) +Q 1Q) Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) (2nd𝐿)) → 𝑟 Q 𝑟 (2nd𝐿))
8358, 80, 82syl2anc 391 . 2 (φ𝑟 Q 𝑟 (2nd𝐿))
8450, 83jca 290 1 (φ → (𝑠 Q 𝑠 (1st𝐿) 𝑟 Q 𝑟 (2nd𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  wrex 2301  {crab 2304  cop 3370   class class class wbr 3755  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  1Qc1q 6265   +Q cplq 6266  *Qcrq 6268   <Q cltq 6269
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
This theorem is referenced by:  caucvgprlemcl  6647
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