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Theorem caucvgprprlemopl 7674
Description: Lemma for caucvgprpr 7689. 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 7662 . . . 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 275 . . 3 (𝑠 ∈ (1st𝐿) → ∃𝑏N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
43adantl 277 . 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 488 . . . . . 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 529 . . . . . 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, 7ffvelcdmd 5647 . . . . 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 7452 . . . . 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 531 . . . . 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 7662 . . . . . . . . 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 274 . . . . . . . 8 (𝑠 ∈ (1st𝐿) → 𝑠Q)
1413ad2antlr 489 . . . . . . 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 7399 . . . . . . . . 9 (𝑏N → [⟨𝑏, 1o⟩] ~QQ)
16 recclnq 7369 . . . . . . . . 9 ([⟨𝑏, 1o⟩] ~QQ → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
1715, 16syl 14 . . . . . . . 8 (𝑏N → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
1817ad2antrl 490 . . . . . . 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 7352 . . . . . . 7 ((𝑠Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
2014, 18, 19syl2anc 411 . . . . . 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 7528 . . . . . 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 411 . . . . 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 167 . . . 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 7465 . . . 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 411 . . 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 276 . . . . . . . 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 276 . . . . . . . 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 7384 . . . . . . . 8 (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q𝑠Q) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠))
2926, 27, 28syl2anc 411 . . . . . . 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 7358 . . . . . . . 8 (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q𝑠Q) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
3126, 27, 30syl2anc 411 . . . . . . 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 4026 . . . . . 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 531 . . . . . 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 7375 . . . . . . 7 <Q Or Q
35 ltrelnq 7342 . . . . . . 7 <Q ⊆ (Q × Q)
3634, 35sotri 5019 . . . . . 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 411 . . . . 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 276 . . . . . . 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 529 . . . . . . 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 7458 . . . . . . 7 ((⟨(1st ‘(𝐹𝑏)), (2nd ‘(𝐹𝑏))⟩ ∈ P𝑎 ∈ (1st ‘(𝐹𝑏))) → 𝑎Q)
4138, 39, 40syl2anc 411 . . . . . 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 7385 . . . . . 6 (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q𝑎Q) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑎 ↔ ∃𝑡Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎))
4326, 41, 42syl2anc 411 . . . . 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 147 . . . 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 488 . . . . . . . . . . 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 488 . . . . . . . . . . 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 7358 . . . . . . . . . . 11 ((𝑠Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) = ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠))
4845, 46, 47syl2anc 411 . . . . . . . . . 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 488 . . . . . . . . . 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 4024 . . . . . . . . 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 110 . . . . . . . . 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 4028 . . . . . . . 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 528 . . . . . . . . 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 7377 . . . . . . . . 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 1238 . . . . . . . 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 167 . . . . . . 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 492 . . . . . . . . 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 7358 . . . . . . . . . . . . 13 (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q𝑡Q) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
5946, 53, 58syl2anc 411 . . . . . . . . . . . 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 2212 . . . . . . . . . . 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 488 . . . . . . . . . . 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 2254 . . . . . . . . . 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 7352 . . . . . . . . . . . 12 ((𝑡Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
6453, 46, 63syl2anc 411 . . . . . . . . . . 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 492 . . . . . . . . . . 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 7528 . . . . . . . . . . 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 411 . . . . . . . . . 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 147 . . . . . . . . 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 3776 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑏 → ⟨𝑟, 1o⟩ = ⟨𝑏, 1o⟩)
7069eceq1d 6564 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑏 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
7170fveq2d 5514 . . . . . . . . . . . . . . 15 (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
7271oveq2d 5884 . . . . . . . . . . . . . 14 (𝑟 = 𝑏 → (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
7372breq2d 4012 . . . . . . . . . . . . 13 (𝑟 = 𝑏 → (𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))))
7473abbidv 2295 . . . . . . . . . . . 12 (𝑟 = 𝑏 → {𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))})
7572breq1d 4010 . . . . . . . . . . . . 13 (𝑟 = 𝑏 → ((𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞))
7675abbidv 2295 . . . . . . . . . . . 12 (𝑟 = 𝑏 → {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞})
7774, 76opeq12d 3784 . . . . . . . . . . 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 5510 . . . . . . . . . . 11 (𝑟 = 𝑏 → (𝐹𝑟) = (𝐹𝑏))
7977, 78breq12d 4013 . . . . . . . . . 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 2841 . . . . . . . . 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 411 . . . . . . . 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 7662 . . . . . . . 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 417 . . . . . . 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 306 . . . . . 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 115 . . . . 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 2579 . . . 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 2599 . 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 2599 1 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  {crab 2459  cop 3594   class class class wbr 4000  wf 5207  cfv 5211  (class class class)co 5868  1st c1st 6132  2nd c2nd 6133  1oc1o 6403  [cec 6526  Ncnpi 7249   <N clti 7252   ~Q ceq 7256  Qcnq 7257   +Q cplq 7259  *Qcrq 7261   <Q cltq 7262  Pcnp 7268   +P cpp 7270  <P cltp 7272
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4285  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-1o 6410  df-oadd 6414  df-omul 6415  df-er 6528  df-ec 6530  df-qs 6534  df-ni 7281  df-pli 7282  df-mi 7283  df-lti 7284  df-plpq 7321  df-mpq 7322  df-enq 7324  df-nqqs 7325  df-plqqs 7326  df-mqqs 7327  df-1nqqs 7328  df-rq 7329  df-ltnqqs 7330  df-inp 7443  df-iltp 7447
This theorem is referenced by:  caucvgprprlemrnd  7678
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