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Theorem caucvgprlemopl 7682
Description: Lemma for caucvgpr 7695. 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 5895 . . . . . . 7 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
21breq1d 4025 . . . . . 6 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
32rexbidv 2488 . . . . 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 5530 . . . . . 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 7376 . . . . . . . 8 Q ∈ V
76rabex 4159 . . . . . . 7 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
86rabex 4159 . . . . . . 7 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
97, 8op1st 6161 . . . . . 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 2208 . . . . 5 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
113, 10elrab2 2908 . . . 4 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
1211simprbi 275 . . 3 (𝑠 ∈ (1st𝐿) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
1312adantl 277 . 2 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
14 simprr 531 . . . 4 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
15 ltbtwnnqq 7428 . . . 4 ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑡Q ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡𝑡 <Q (𝐹𝑗)))
1614, 15sylib 122 . . 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 535 . . . . . . . . 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 7435 . . . . . . . . 9 (𝑗N → [⟨𝑗, 1o⟩] ~QQ)
19 recclnq 7405 . . . . . . . . 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 274 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) → 𝑠Q)
2221ad3antlr 493 . . . . . . . 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 7420 . . . . . . . 8 (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q𝑠Q) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠))
2420, 22, 23syl2anc 411 . . . . . . 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 7394 . . . . . . . 8 (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q𝑠Q) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
2620, 22, 25syl2anc 411 . . . . . . 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 4041 . . . . . 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 539 . . . . . 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 7411 . . . . . . 7 <Q Or Q
30 ltrelnq 7378 . . . . . . 7 <Q ⊆ (Q × Q)
3129, 30sotri 5036 . . . . . 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 411 . . . . 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 529 . . . . . 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 7421 . . . . . 6 (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q𝑡Q) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑡 ↔ ∃𝑟Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡))
3520, 33, 34syl2anc 411 . . . . 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 147 . . . 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 488 . . . . . . . . . . 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 488 . . . . . . . . . . 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 7394 . . . . . . . . . . 11 ((𝑠Q ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠))
4037, 38, 39syl2anc 411 . . . . . . . . . 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 488 . . . . . . . . . 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 4039 . . . . . . . . 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 110 . . . . . . . . 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 4043 . . . . . . . 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 528 . . . . . . . . 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 7413 . . . . . . . . 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 1248 . . . . . . . 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 167 . . . . . . 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 488 . . . . . . . . 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 540 . . . . . . . . . . 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 488 . . . . . . . . . 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 7394 . . . . . . . . . . . . 13 (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q𝑟Q) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
5338, 45, 52syl2anc 411 . . . . . . . . . . . 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 2222 . . . . . . . . . . 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 4025 . . . . . . . . . 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 167 . . . . . . . . 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 2536 . . . . . . . . 9 ((𝑗N ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
5849, 56, 57syl2anc 411 . . . . . . . 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 5895 . . . . . . . . . . 11 (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
6059breq1d 4025 . . . . . . . . . 10 (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
6160rexbidv 2488 . . . . . . . . 9 (𝑙 = 𝑟 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
6261, 10elrab2 2908 . . . . . . . 8 (𝑟 ∈ (1st𝐿) ↔ (𝑟Q ∧ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
6345, 58, 62sylanbrc 417 . . . . . . 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 306 . . . . . 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 115 . . . . 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 2589 . . . 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 2609 . 2 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑗N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
6913, 68rexlimddv 2609 1 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158  wral 2465  wrex 2466  {crab 2469  cop 3607   class class class wbr 4015  wf 5224  cfv 5228  (class class class)co 5888  1st c1st 6153  1oc1o 6424  [cec 6547  Ncnpi 7285   <N clti 7288   ~Q ceq 7292  Qcnq 7293   +Q cplq 7295  *Qcrq 7297   <Q cltq 7298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-1o 6431  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-pli 7318  df-mi 7319  df-lti 7320  df-plpq 7357  df-mpq 7358  df-enq 7360  df-nqqs 7361  df-plqqs 7362  df-mqqs 7363  df-1nqqs 7364  df-rq 7365  df-ltnqqs 7366
This theorem is referenced by:  caucvgprlemrnd  7686
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