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Theorem caucvgprprlemopl 6852
Description: Lemma for caucvgprpr 6867. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprpr.bnd (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
caucvgprpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
Assertion
Ref Expression
caucvgprprlemopl ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐹,𝑙,𝑡,𝑟   𝑢,𝐹,𝑡   𝑡,𝐿   𝑝,𝑙,𝑞,𝑟,𝑠,𝑡   𝑢,𝑝,𝑞,𝑟,𝑠   𝜑,𝑟,𝑡
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑠,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑡,𝑘,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑘,𝑛,𝑠,𝑞,𝑝)   𝐿(𝑢,𝑘,𝑚,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemopl
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.lim . . . . 5 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
21caucvgprprlemell 6840 . . . 4 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑏N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
32simprbi 264 . . 3 (𝑠 ∈ (1st𝐿) → ∃𝑏N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
43adantl 266 . 2 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑏N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
5 caucvgprpr.f . . . . . . 7 (𝜑𝐹:NP)
65ad2antrr 465 . . . . . 6 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → 𝐹:NP)
7 simprl 491 . . . . . 6 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → 𝑏N)
86, 7ffvelrnd 5330 . . . . 5 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → (𝐹𝑏) ∈ P)
9 prop 6630 . . . . 5 ((𝐹𝑏) ∈ P → ⟨(1st ‘(𝐹𝑏)), (2nd ‘(𝐹𝑏))⟩ ∈ P)
108, 9syl 14 . . . 4 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → ⟨(1st ‘(𝐹𝑏)), (2nd ‘(𝐹𝑏))⟩ ∈ P)
11 simprr 492 . . . . 5 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
121caucvgprprlemell 6840 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
1312simplbi 263 . . . . . . . 8 (𝑠 ∈ (1st𝐿) → 𝑠Q)
1413ad2antlr 466 . . . . . . 7 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → 𝑠Q)
15 nnnq 6577 . . . . . . . . 9 (𝑏N → [⟨𝑏, 1𝑜⟩] ~QQ)
16 recclnq 6547 . . . . . . . . 9 ([⟨𝑏, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q)
1715, 16syl 14 . . . . . . . 8 (𝑏N → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q)
1817ad2antrl 467 . . . . . . 7 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q)
19 addclnq 6530 . . . . . . 7 ((𝑠Q ∧ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q)
2014, 18, 19syl2anc 397 . . . . . 6 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q)
21 nqprl 6706 . . . . . 6 (((𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q ∧ (𝐹𝑏) ∈ P) → ((𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐹𝑏)) ↔ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
2220, 8, 21syl2anc 397 . . . . 5 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → ((𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐹𝑏)) ↔ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
2311, 22mpbird 160 . . . 4 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐹𝑏)))
24 prnmaxl 6643 . . . 4 ((⟨(1st ‘(𝐹𝑏)), (2nd ‘(𝐹𝑏))⟩ ∈ P ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐹𝑏))) → ∃𝑎 ∈ (1st ‘(𝐹𝑏))(𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)
2510, 23, 24syl2anc 397 . . 3 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → ∃𝑎 ∈ (1st ‘(𝐹𝑏))(𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)
2618adantr 265 . . . . . . . 8 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q)
2714adantr 265 . . . . . . . 8 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → 𝑠Q)
28 ltaddnq 6562 . . . . . . . 8 (((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q𝑠Q) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠))
2926, 27, 28syl2anc 397 . . . . . . 7 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠))
30 addcomnqg 6536 . . . . . . . 8 (((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q𝑠Q) → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
3126, 27, 30syl2anc 397 . . . . . . 7 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
3229, 31breqtrd 3815 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
33 simprr 492 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)
34 ltsonq 6553 . . . . . . 7 <Q Or Q
35 ltrelnq 6520 . . . . . . 7 <Q ⊆ (Q × Q)
3634, 35sotri 4747 . . . . . 6 (((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑎)
3732, 33, 36syl2anc 397 . . . . 5 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑎)
3810adantr 265 . . . . . . 7 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → ⟨(1st ‘(𝐹𝑏)), (2nd ‘(𝐹𝑏))⟩ ∈ P)
39 simprl 491 . . . . . . 7 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → 𝑎 ∈ (1st ‘(𝐹𝑏)))
40 elprnql 6636 . . . . . . 7 ((⟨(1st ‘(𝐹𝑏)), (2nd ‘(𝐹𝑏))⟩ ∈ P𝑎 ∈ (1st ‘(𝐹𝑏))) → 𝑎Q)
4138, 39, 40syl2anc 397 . . . . . 6 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → 𝑎Q)
42 ltexnqq 6563 . . . . . 6 (((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q𝑎Q) → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑎 ↔ ∃𝑡Q ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎))
4326, 41, 42syl2anc 397 . . . . 5 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑎 ↔ ∃𝑡Q ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎))
4437, 43mpbid 139 . . . 4 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → ∃𝑡Q ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎)
4527ad2antrr 465 . . . . . . . . . . 11 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑠Q)
4626ad2antrr 465 . . . . . . . . . . 11 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q)
47 addcomnqg 6536 . . . . . . . . . . 11 ((𝑠Q ∧ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) = ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠))
4845, 46, 47syl2anc 397 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) = ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠))
4933ad2antrr 465 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)
5048, 49eqbrtrrd 3813 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠) <Q 𝑎)
51 simpr 107 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎)
5250, 51breqtrrd 3817 . . . . . . . 8 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡))
53 simplr 490 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑡Q)
54 ltanqg 6555 . . . . . . . . 9 ((𝑠Q𝑡Q ∧ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 <Q 𝑡 ↔ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡)))
5545, 53, 46, 54syl3anc 1146 . . . . . . . 8 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑠 <Q 𝑡 ↔ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡)))
5652, 55mpbird 160 . . . . . . 7 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑠 <Q 𝑡)
577ad3antrrr 469 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑏N)
58 addcomnqg 6536 . . . . . . . . . . . . 13 (((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q𝑡Q) → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
5946, 53, 58syl2anc 397 . . . . . . . . . . . 12 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
6059, 51eqtr3d 2090 . . . . . . . . . . 11 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) = 𝑎)
6139ad2antrr 465 . . . . . . . . . . 11 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑎 ∈ (1st ‘(𝐹𝑏)))
6260, 61eqeltrd 2130 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐹𝑏)))
63 addclnq 6530 . . . . . . . . . . . 12 ((𝑡Q ∧ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q) → (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q)
6453, 46, 63syl2anc 397 . . . . . . . . . . 11 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q)
658ad3antrrr 469 . . . . . . . . . . 11 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝐹𝑏) ∈ P)
66 nqprl 6706 . . . . . . . . . . 11 (((𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q ∧ (𝐹𝑏) ∈ P) → ((𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐹𝑏)) ↔ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
6764, 65, 66syl2anc 397 . . . . . . . . . 10 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → ((𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐹𝑏)) ↔ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
6862, 67mpbid 139 . . . . . . . . 9 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
69 opeq1 3576 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑏 → ⟨𝑟, 1𝑜⟩ = ⟨𝑏, 1𝑜⟩)
7069eceq1d 6172 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑏 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝑏, 1𝑜⟩] ~Q )
7170fveq2d 5209 . . . . . . . . . . . . . . 15 (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))
7271oveq2d 5555 . . . . . . . . . . . . . 14 (𝑟 = 𝑏 → (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) = (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
7372breq2d 3803 . . . . . . . . . . . . 13 (𝑟 = 𝑏 → (𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))))
7473abbidv 2171 . . . . . . . . . . . 12 (𝑟 = 𝑏 → {𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))})
7572breq1d 3801 . . . . . . . . . . . . 13 (𝑟 = 𝑏 → ((𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞))
7675abbidv 2171 . . . . . . . . . . . 12 (𝑟 = 𝑏 → {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞})
7774, 76opeq12d 3584 . . . . . . . . . . 11 (𝑟 = 𝑏 → ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩)
78 fveq2 5205 . . . . . . . . . . 11 (𝑟 = 𝑏 → (𝐹𝑟) = (𝐹𝑏))
7977, 78breq12d 3804 . . . . . . . . . 10 (𝑟 = 𝑏 → (⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟) ↔ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
8079rspcev 2673 . . . . . . . . 9 ((𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)) → ∃𝑟N ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟))
8157, 68, 80syl2anc 397 . . . . . . . 8 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → ∃𝑟N ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟))
821caucvgprprlemell 6840 . . . . . . . 8 (𝑡 ∈ (1st𝐿) ↔ (𝑡Q ∧ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
8353, 81, 82sylanbrc 402 . . . . . . 7 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑡 ∈ (1st𝐿))
8456, 83jca 294 . . . . . 6 ((((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) ∧ ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)))
8584ex 112 . . . . 5 (((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) ∧ 𝑡Q) → (((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎 → (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿))))
8685reximdva 2438 . . . 4 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → (∃𝑡Q ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑡) = 𝑎 → ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿))))
8744, 86mpd 13 . . 3 ((((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) ∧ (𝑎 ∈ (1st ‘(𝐹𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑎)) → ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)))
8825, 87rexlimddv 2454 . 2 (((𝜑𝑠 ∈ (1st𝐿)) ∧ (𝑏N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))) → ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)))
894, 88rexlimddv 2454 1 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  {crab 2327  cop 3405   class class class wbr 3791  wf 4925  cfv 4929  (class class class)co 5539  1st c1st 5792  2nd c2nd 5793  1𝑜c1o 6024  [cec 6134  Ncnpi 6427   <N clti 6430   ~Q ceq 6434  Qcnq 6435   +Q cplq 6437  *Qcrq 6439   <Q cltq 6440  Pcnp 6446   +P cpp 6448  <P cltp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-inp 6621  df-iltp 6625
This theorem is referenced by:  caucvgprprlemrnd  6856
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