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Theorem caucvgprlemopl 7131
Description: Lemma for caucvgpr 7144. 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 5598 . . . . . . 7 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
21breq1d 3821 . . . . . 6 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
32rexbidv 2375 . . . . 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 5256 . . . . . 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 6825 . . . . . . . 8 Q ∈ V
76rabex 3948 . . . . . . 7 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
86rabex 3948 . . . . . . 7 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
97, 8op1st 5852 . . . . . 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 2103 . . . . 5 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
113, 10elrab2 2762 . . . 4 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
1211simprbi 269 . . 3 (𝑠 ∈ (1st𝐿) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
1312adantl 271 . 2 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
14 simprr 499 . . . 4 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
15 ltbtwnnqq 6877 . . . 4 ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑡Q ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))
1614, 15sylib 120 . . 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 502 . . . . . . . . 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 6884 . . . . . . . . 9 (𝑗N → [⟨𝑗, 1𝑜⟩] ~QQ)
19 recclnq 6854 . . . . . . . . 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 268 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) → 𝑠Q)
2221ad3antlr 477 . . . . . . . 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 6869 . . . . . . . 8 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q𝑠Q) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠))
2420, 22, 23syl2anc 403 . . . . . . 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 6843 . . . . . . . 8 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q𝑠Q) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
2620, 22, 25syl2anc 403 . . . . . . 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 3835 . . . . . 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 506 . . . . . 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 6860 . . . . . . 7 <Q Or Q
30 ltrelnq 6827 . . . . . . 7 <Q ⊆ (Q × Q)
3129, 30sotri 4782 . . . . . 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 403 . . . . 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 498 . . . . . 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 6870 . . . . . 6 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q𝑡Q) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑡 ↔ ∃𝑟Q ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = 𝑡))
3520, 33, 34syl2anc 403 . . . . 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 145 . . . 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 472 . . . . . . . . . . 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 472 . . . . . . . . . . 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 6843 . . . . . . . . . . 11 ((𝑠Q ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑠))
4037, 38, 39syl2anc 403 . . . . . . . . . 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 472 . . . . . . . . . 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 3833 . . . . . . . . 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 108 . . . . . . . . 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 3837 . . . . . . . 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 497 . . . . . . . . 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 6862 . . . . . . . . 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 1170 . . . . . . . 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 165 . . . . . . 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 472 . . . . . . . . 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 507 . . . . . . . . . . 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 472 . . . . . . . . . 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 6843 . . . . . . . . . . . . 13 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q𝑟Q) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) +Q 𝑟) = (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
5338, 45, 52syl2anc 403 . . . . . . . . . . . 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 2117 . . . . . . . . . . 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 3821 . . . . . . . . . 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 165 . . . . . . . . 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 2418 . . . . . . . . 9 ((𝑗N ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
5849, 56, 57syl2anc 403 . . . . . . . 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 5598 . . . . . . . . . . 11 (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
6059breq1d 3821 . . . . . . . . . 10 (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
6160rexbidv 2375 . . . . . . . . 9 (𝑙 = 𝑟 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
6261, 10elrab2 2762 . . . . . . . 8 (𝑟 ∈ (1st𝐿) ↔ (𝑟Q ∧ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
6345, 58, 62sylanbrc 408 . . . . . . 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 300 . . . . . 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 113 . . . . 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 2469 . . . 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 2487 . 2 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
6913, 68rexlimddv 2487 1 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  wral 2353  wrex 2354  {crab 2357  cop 3425   class class class wbr 3811  wf 4965  cfv 4969  (class class class)co 5591  1st c1st 5844  1𝑜c1o 6106  [cec 6220  Ncnpi 6734   <N clti 6737   ~Q ceq 6741  Qcnq 6742   +Q cplq 6744  *Qcrq 6746   <Q cltq 6747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815
This theorem is referenced by:  caucvgprlemrnd  7135
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