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Theorem caucvgprlemdisj 6645
 Description: Lemma for caucvgpr 6653. The putative limit is disjoint. (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
caucvgprlemdisj (φ𝑠 Q ¬ (𝑠 (1st𝐿) 𝑠 (2nd𝐿)))
Distinct variable groups:   A,𝑗   𝑗,𝐹,𝑘   𝐹,𝑙,𝑗   u,𝐹,𝑗   𝑛,𝐹   𝑗,𝐿,𝑘   φ,𝑗,𝑠,𝑘   𝑠,𝑙   u,𝑠   𝑘,𝑛
Allowed substitution hints:   φ(u,𝑛,𝑙)   A(u,𝑘,𝑛,𝑠,𝑙)   𝐹(𝑠)   𝐿(u,𝑛,𝑠,𝑙)

Proof of Theorem caucvgprlemdisj
StepHypRef Expression
1 oveq1 5462 . . . . . . . . . . . 12 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
21breq1d 3765 . . . . . . . . . . 11 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
32rexbidv 2321 . . . . . . . . . 10 (𝑙 = 𝑠 → (𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
4 caucvgpr.lim . . . . . . . . . . . 12 𝐿 = ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
54fveq2i 5124 . . . . . . . . . . 11 (1st𝐿) = (1st ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩)
6 nqex 6347 . . . . . . . . . . . . 13 Q V
76rabex 3892 . . . . . . . . . . . 12 {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} V
86rabex 3892 . . . . . . . . . . . 12 {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} V
97, 8op1st 5715 . . . . . . . . . . 11 (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)}
105, 9eqtri 2057 . . . . . . . . . 10 (1st𝐿) = {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
113, 10elrab2 2694 . . . . . . . . 9 (𝑠 (1st𝐿) ↔ (𝑠 Q 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
1211simprbi 260 . . . . . . . 8 (𝑠 (1st𝐿) → 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
13 opeq1 3540 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → ⟨𝑗, 1𝑜⟩ = ⟨𝑘, 1𝑜⟩)
1413eceq1d 6078 . . . . . . . . . . . 12 (𝑗 = 𝑘 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑘, 1𝑜⟩] ~Q )
1514fveq2d 5125 . . . . . . . . . . 11 (𝑗 = 𝑘 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑘, 1𝑜⟩] ~Q ))
1615oveq2d 5471 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )))
17 fveq2 5121 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
1816, 17breq12d 3768 . . . . . . . . 9 (𝑗 = 𝑘 → ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘)))
1918cbvrexv 2528 . . . . . . . 8 (𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ 𝑘 N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘))
2012, 19sylib 127 . . . . . . 7 (𝑠 (1st𝐿) → 𝑘 N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘))
21 breq2 3759 . . . . . . . . . 10 (u = 𝑠 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2221rexbidv 2321 . . . . . . . . 9 (u = 𝑠 → (𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
234fveq2i 5124 . . . . . . . . . 10 (2nd𝐿) = (2nd ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩)
247, 8op2nd 5716 . . . . . . . . . 10 (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}
2523, 24eqtri 2057 . . . . . . . . 9 (2nd𝐿) = {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
2622, 25elrab2 2694 . . . . . . . 8 (𝑠 (2nd𝐿) ↔ (𝑠 Q 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2726simprbi 260 . . . . . . 7 (𝑠 (2nd𝐿) → 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠)
2820, 27anim12i 321 . . . . . 6 ((𝑠 (1st𝐿) 𝑠 (2nd𝐿)) → (𝑘 N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
29 reeanv 2473 . . . . . 6 (𝑘 N 𝑗 N ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠) ↔ (𝑘 N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
3028, 29sylibr 137 . . . . 5 ((𝑠 (1st𝐿) 𝑠 (2nd𝐿)) → 𝑘 N 𝑗 N ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
3130adantl 262 . . . 4 ((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) → 𝑘 N 𝑗 N ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
32 caucvgpr.f . . . . . . . 8 (φ𝐹:NQ)
3332ad2antrr 457 . . . . . . 7 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → 𝐹:NQ)
34 caucvgpr.cau . . . . . . . 8 (φ𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
3534ad2antrr 457 . . . . . . 7 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → 𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
36 simprl 483 . . . . . . 7 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → 𝑘 N)
37 simprr 484 . . . . . . 7 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → 𝑗 N)
3811simplbi 259 . . . . . . . . 9 (𝑠 (1st𝐿) → 𝑠 Q)
3938ad2antrl 459 . . . . . . . 8 ((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) → 𝑠 Q)
4039adantr 261 . . . . . . 7 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → 𝑠 Q)
4133, 35, 36, 37, 40caucvgprlemnkj 6637 . . . . . 6 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → ¬ ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
4241pm2.21d 549 . . . . 5 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → (((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠) → ⊥ ))
4342rexlimdvva 2434 . . . 4 ((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) → (𝑘 N 𝑗 N ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠) → ⊥ ))
4431, 43mpd 13 . . 3 ((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) → ⊥ )
4544inegd 1262 . 2 (φ → ¬ (𝑠 (1st𝐿) 𝑠 (2nd𝐿)))
4645ralrimivw 2387 1 (φ𝑠 Q ¬ (𝑠 (1st𝐿) 𝑠 (2nd𝐿)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1242   ⊥ wfal 1247   ∈ 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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