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

Theorem caucvgprlemopl 6767
 Description: Lemma for caucvgpr 6780. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-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
caucvgprlemopl ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
Distinct variable groups:   𝐴,𝑗   𝐹,𝑙,𝑟,𝑠   𝑢,𝐹   𝑗,𝐿,𝑟,𝑠   𝑗,𝑙,𝑠   𝜑,𝑗,𝑟,𝑠   𝑢,𝑗,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑠,𝑟,𝑙)   𝐹(𝑗,𝑘,𝑛)   𝐿(𝑢,𝑘,𝑛,𝑙)

Proof of Theorem caucvgprlemopl
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 oveq1 5519 . . . . . . 7 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
21breq1d 3774 . . . . . 6 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
32rexbidv 2327 . . . . 5 (𝑙 = 𝑠 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
4 caucvgpr.lim . . . . . . 7 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
54fveq2i 5181 . . . . . 6 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)
6 nqex 6461 . . . . . . . 8 Q ∈ V
76rabex 3901 . . . . . . 7 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
86rabex 3901 . . . . . . 7 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
97, 8op1st 5773 . . . . . 6 (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 2060 . . . . 5 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
113, 10elrab2 2700 . . . 4 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
1211simprbi 260 . . 3 (𝑠 ∈ (1st𝐿) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
1312adantl 262 . 2 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
14 simprr 484 . . . 4 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
15 ltbtwnnqq 6513 . . . 4 ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑡Q ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))
1614, 15sylib 127 . . 3 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) → ∃𝑡Q ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))
17 simplrl 487 . . . . . . . . 9 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → 𝑗N)
18 nnnq 6520 . . . . . . . . 9 (𝑗N → [⟨𝑗, 1𝑜⟩] ~QQ)
19 recclnq 6490 . . . . . . . . 9 ([⟨𝑗, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
2017, 18, 193syl 17 . . . . . . . 8 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
2111simplbi 259 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) → 𝑠Q)
2221ad3antlr 462 . . . . . . . 8 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → 𝑠Q)
23 ltaddnq 6505 . . . . . . . 8 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q𝑠Q) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠))
2420, 22, 23syl2anc 391 . . . . . . 7 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠))
25 addcomnqg 6479 . . . . . . . 8 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q𝑠Q) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
2620, 22, 25syl2anc 391 . . . . . . 7 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
2724, 26breqtrd 3788 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
28 simprrl 491 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡)
29 ltsonq 6496 . . . . . . 7 <Q Or Q
30 ltrelnq 6463 . . . . . . 7 <Q ⊆ (Q × Q)
3129, 30sotri 4720 . . . . . 6 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑡)
3227, 28, 31syl2anc 391 . . . . 5 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑡)
33 simprl 483 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → 𝑡Q)
34 ltexnqq 6506 . . . . . 6 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q𝑡Q) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑡 ↔ ∃𝑟Q ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡))
3520, 33, 34syl2anc 391 . . . . 5 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑡 ↔ ∃𝑟Q ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡))
3632, 35mpbid 135 . . . 4 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → ∃𝑟Q ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡)
3722ad2antrr 457 . . . . . . . . . . 11 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑠Q)
3820ad2antrr 457 . . . . . . . . . . 11 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
39 addcomnqg 6479 . . . . . . . . . . 11 ((𝑠Q ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠))
4037, 38, 39syl2anc 391 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠))
4128ad2antrr 457 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡)
4240, 41eqbrtrrd 3786 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠) <Q 𝑡)
43 simpr 103 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡)
4442, 43breqtrrd 3790 . . . . . . . 8 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟))
45 simplr 482 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑟Q)
46 ltanqg 6498 . . . . . . . . 9 ((𝑠Q𝑟Q ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 <Q 𝑟 ↔ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟)))
4737, 45, 38, 46syl3anc 1135 . . . . . . . 8 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑠 <Q 𝑟 ↔ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟)))
4844, 47mpbird 156 . . . . . . 7 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑠 <Q 𝑟)
4917ad2antrr 457 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑗N)
50 simprrr 492 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → 𝑡 <Q (𝐹𝑗))
5150ad2antrr 457 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑡 <Q (𝐹𝑗))
52 addcomnqg 6479 . . . . . . . . . . . . 13 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q𝑟Q) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
5338, 45, 52syl2anc 391 . . . . . . . . . . . 12 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
5453, 43eqtr3d 2074 . . . . . . . . . . 11 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = 𝑡)
5554breq1d 3774 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → ((𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ 𝑡 <Q (𝐹𝑗)))
5651, 55mpbird 156 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
57 rspe 2370 . . . . . . . . 9 ((𝑗N ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
5849, 56, 57syl2anc 391 . . . . . . . 8 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
59 oveq1 5519 . . . . . . . . . . 11 (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
6059breq1d 3774 . . . . . . . . . 10 (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
6160rexbidv 2327 . . . . . . . . 9 (𝑙 = 𝑟 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
6261, 10elrab2 2700 . . . . . . . 8 (𝑟 ∈ (1st𝐿) ↔ (𝑟Q ∧ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
6345, 58, 62sylanbrc 394 . . . . . . 7 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑟 ∈ (1st𝐿))
6448, 63jca 290 . . . . . 6 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) ∧ ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
6564ex 108 . . . . 5 (((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) ∧ 𝑟Q) → (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡 → (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))))
6665reximdva 2421 . . . 4 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → (∃𝑟Q ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡 → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))))
6736, 66mpd 13 . . 3 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) ∧ (𝑡Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
6816, 67rexlimddv 2437 . 2 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
6913, 68rexlimddv 2437 1 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  {crab 2310  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1st c1st 5765  1𝑜c1o 5994  [cec 6104  Ncnpi 6370
 Copyright terms: Public domain W3C validator