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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
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