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Theorem caucvgprlemopl 7489
Description: Lemma for caucvgpr 7502. 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 5781 . . . . . . 7 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
21breq1d 3939 . . . . . 6 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
32rexbidv 2438 . . . . 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 5424 . . . . . 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 7183 . . . . . . . 8 Q ∈ V
76rabex 4072 . . . . . . 7 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
86rabex 4072 . . . . . . 7 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
97, 8op1st 6044 . . . . . 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 2160 . . . . 5 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
113, 10elrab2 2843 . . . 4 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
1211simprbi 273 . . 3 (𝑠 ∈ (1st𝐿) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
1312adantl 275 . 2 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
14 simprr 521 . . . 4 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
15 ltbtwnnqq 7235 . . . 4 ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑡Q ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))
1614, 15sylib 121 . . 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 524 . . . . . . . . 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 7242 . . . . . . . . 9 (𝑗N → [⟨𝑗, 1o⟩] ~QQ)
19 recclnq 7212 . . . . . . . . 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 272 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) → 𝑠Q)
2221ad3antlr 484 . . . . . . . 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 7227 . . . . . . . 8 (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q𝑠Q) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠))
2420, 22, 23syl2anc 408 . . . . . . 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 7201 . . . . . . . 8 (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q𝑠Q) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
2620, 22, 25syl2anc 408 . . . . . . 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 3954 . . . . . 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 528 . . . . . 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 7218 . . . . . . 7 <Q Or Q
30 ltrelnq 7185 . . . . . . 7 <Q ⊆ (Q × Q)
3129, 30sotri 4934 . . . . . 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 408 . . . . 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 520 . . . . . 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 7228 . . . . . 6 (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q𝑡Q) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑡 ↔ ∃𝑟Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡))
3520, 33, 34syl2anc 408 . . . . 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 146 . . . 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 479 . . . . . . . . . . 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 479 . . . . . . . . . . 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 7201 . . . . . . . . . . 11 ((𝑠Q ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠))
4037, 38, 39syl2anc 408 . . . . . . . . . 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 479 . . . . . . . . . 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 3952 . . . . . . . . 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 109 . . . . . . . . 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 3956 . . . . . . . 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 519 . . . . . . . . 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 7220 . . . . . . . . 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 1216 . . . . . . . 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 166 . . . . . . 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 479 . . . . . . . . 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 529 . . . . . . . . . . 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 479 . . . . . . . . . 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 7201 . . . . . . . . . . . . 13 (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q𝑟Q) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
5338, 45, 52syl2anc 408 . . . . . . . . . . . 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 2174 . . . . . . . . . . 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 3939 . . . . . . . . . 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 166 . . . . . . . . 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 2481 . . . . . . . . 9 ((𝑗N ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
5849, 56, 57syl2anc 408 . . . . . . . 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 5781 . . . . . . . . . . 11 (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
6059breq1d 3939 . . . . . . . . . 10 (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
6160rexbidv 2438 . . . . . . . . 9 (𝑙 = 𝑟 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
6261, 10elrab2 2843 . . . . . . . 8 (𝑟 ∈ (1st𝐿) ↔ (𝑟Q ∧ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
6345, 58, 62sylanbrc 413 . . . . . . 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 304 . . . . . 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 114 . . . . 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 2534 . . . 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 2554 . 2 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
6913, 68rexlimddv 2554 1 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  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  1oc1o 6306  [cec 6427  Ncnpi 7092   <N clti 7095   ~Q ceq 7099  Qcnq 7100   +Q cplq 7102  *Qcrq 7104   <Q cltq 7105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173
This theorem is referenced by:  caucvgprlemrnd  7493
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