Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprlemdisj GIF version

Theorem caucvgprlemdisj 7482
 Description: Lemma for caucvgpr 7490. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-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
caucvgprlemdisj (𝜑 → ∀𝑠Q ¬ (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 5781 . . . . . . . . . . . 12 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
21breq1d 3939 . . . . . . . . . . 11 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
32rexbidv 2438 . . . . . . . . . 10 (𝑙 = 𝑠 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
4 caucvgpr.lim . . . . . . . . . . . 12 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
54fveq2i 5424 . . . . . . . . . . 11 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
6 nqex 7171 . . . . . . . . . . . . 13 Q ∈ V
76rabex 4072 . . . . . . . . . . . 12 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
86rabex 4072 . . . . . . . . . . . 12 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
97, 8op1st 6044 . . . . . . . . . . 11 (1st ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
105, 9eqtri 2160 . . . . . . . . . 10 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
113, 10elrab2 2843 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
1211simprbi 273 . . . . . . . 8 (𝑠 ∈ (1st𝐿) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
13 opeq1 3705 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → ⟨𝑗, 1o⟩ = ⟨𝑘, 1o⟩)
1413eceq1d 6465 . . . . . . . . . . . 12 (𝑗 = 𝑘 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑘, 1o⟩] ~Q )
1514fveq2d 5425 . . . . . . . . . . 11 (𝑗 = 𝑘 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑘, 1o⟩] ~Q ))
1615oveq2d 5790 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )))
17 fveq2 5421 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
1816, 17breq12d 3942 . . . . . . . . 9 (𝑗 = 𝑘 → ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘)))
1918cbvrexv 2655 . . . . . . . 8 (∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑘N (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘))
2012, 19sylib 121 . . . . . . 7 (𝑠 ∈ (1st𝐿) → ∃𝑘N (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘))
21 breq2 3933 . . . . . . . . . 10 (𝑢 = 𝑠 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
2221rexbidv 2438 . . . . . . . . 9 (𝑢 = 𝑠 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
234fveq2i 5424 . . . . . . . . . 10 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
247, 8op2nd 6045 . . . . . . . . . 10 (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
2523, 24eqtri 2160 . . . . . . . . 9 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
2622, 25elrab2 2843 . . . . . . . 8 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
2726simprbi 273 . . . . . . 7 (𝑠 ∈ (2nd𝐿) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠)
2820, 27anim12i 336 . . . . . 6 ((𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) → (∃𝑘N (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘) ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
29 reeanv 2600 . . . . . 6 (∃𝑘N𝑗N ((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠) ↔ (∃𝑘N (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘) ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
3028, 29sylibr 133 . . . . 5 ((𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) → ∃𝑘N𝑗N ((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
3130adantl 275 . . . 4 ((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) → ∃𝑘N𝑗N ((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
32 caucvgpr.f . . . . . . . 8 (𝜑𝐹:NQ)
3332ad2antrr 479 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → 𝐹:NQ)
34 caucvgpr.cau . . . . . . . 8 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
3534ad2antrr 479 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
36 simprl 520 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → 𝑘N)
37 simprr 521 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → 𝑗N)
3811simplbi 272 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) → 𝑠Q)
3938ad2antrl 481 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) → 𝑠Q)
4039adantr 274 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → 𝑠Q)
4133, 35, 36, 37, 40caucvgprlemnkj 7474 . . . . . 6 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → ¬ ((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
4241pm2.21d 608 . . . . 5 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → (((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠) → ⊥))
4342rexlimdvva 2557 . . . 4 ((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) → (∃𝑘N𝑗N ((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 1350 . 2 (𝜑 → ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
4645ralrimivw 2506 1 (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331  ⊥wfal 1336   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420  ⟨cop 3530   class class class wbr 3929  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  1oc1o 6306  [cec 6427  Ncnpi 7080
 Copyright terms: Public domain W3C validator