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Theorem caucvgprprlemopl 7549
 Description: Lemma for caucvgprpr 7564. 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 7537 . . . 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 273 . . 3 (𝑠 ∈ (1st𝐿) → ∃𝑏N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
43adantl 275 . 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 480 . . . . . 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 521 . . . . . 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 5565 . . . . 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 7327 . . . . 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 522 . . . . 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 7537 . . . . . . . . 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 272 . . . . . . . 8 (𝑠 ∈ (1st𝐿) → 𝑠Q)
1413ad2antlr 481 . . . . . . 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 7274 . . . . . . . . 9 (𝑏N → [⟨𝑏, 1o⟩] ~QQ)
16 recclnq 7244 . . . . . . . . 9 ([⟨𝑏, 1o⟩] ~QQ → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
1715, 16syl 14 . . . . . . . 8 (𝑏N → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
1817ad2antrl 482 . . . . . . 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 7227 . . . . . . 7 ((𝑠Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
2014, 18, 19syl2anc 409 . . . . . 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 7403 . . . . . 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 409 . . . . 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 166 . . . 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 7340 . . . 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 409 . . 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 274 . . . . . . . 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 274 . . . . . . . 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 7259 . . . . . . . 8 (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q𝑠Q) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠))
2926, 27, 28syl2anc 409 . . . . . . 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 7233 . . . . . . . 8 (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q𝑠Q) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
3126, 27, 30syl2anc 409 . . . . . . 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 3963 . . . . . 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 522 . . . . . 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 7250 . . . . . . 7 <Q Or Q
35 ltrelnq 7217 . . . . . . 7 <Q ⊆ (Q × Q)
3634, 35sotri 4943 . . . . . 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 409 . . . . 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 274 . . . . . . 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 521 . . . . . . 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 7333 . . . . . . 7 ((⟨(1st ‘(𝐹𝑏)), (2nd ‘(𝐹𝑏))⟩ ∈ P𝑎 ∈ (1st ‘(𝐹𝑏))) → 𝑎Q)
4138, 39, 40syl2anc 409 . . . . . 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 7260 . . . . . 6 (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q𝑎Q) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑎 ↔ ∃𝑡Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎))
4326, 41, 42syl2anc 409 . . . . 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 146 . . . 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 480 . . . . . . . . . . 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 480 . . . . . . . . . . 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 7233 . . . . . . . . . . 11 ((𝑠Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) = ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠))
4845, 46, 47syl2anc 409 . . . . . . . . . 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 480 . . . . . . . . . 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 3961 . . . . . . . . 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 109 . . . . . . . . 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 3965 . . . . . . . 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 520 . . . . . . . . 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 7252 . . . . . . . . 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 1217 . . . . . . . 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 166 . . . . . . 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 484 . . . . . . . . 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 7233 . . . . . . . . . . . . 13 (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q𝑡Q) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
5946, 53, 58syl2anc 409 . . . . . . . . . . . 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 2175 . . . . . . . . . . 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 480 . . . . . . . . . . 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 2217 . . . . . . . . . 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 7227 . . . . . . . . . . . 12 ((𝑡Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
6453, 46, 63syl2anc 409 . . . . . . . . . . 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 484 . . . . . . . . . . 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 7403 . . . . . . . . . . 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 409 . . . . . . . . . 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 146 . . . . . . . . 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 3714 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑏 → ⟨𝑟, 1o⟩ = ⟨𝑏, 1o⟩)
7069eceq1d 6474 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑏 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
7170fveq2d 5434 . . . . . . . . . . . . . . 15 (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
7271oveq2d 5799 . . . . . . . . . . . . . 14 (𝑟 = 𝑏 → (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
7372breq2d 3950 . . . . . . . . . . . . 13 (𝑟 = 𝑏 → (𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))))
7473abbidv 2258 . . . . . . . . . . . 12 (𝑟 = 𝑏 → {𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))})
7572breq1d 3948 . . . . . . . . . . . . 13 (𝑟 = 𝑏 → ((𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞))
7675abbidv 2258 . . . . . . . . . . . 12 (𝑟 = 𝑏 → {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞})
7774, 76opeq12d 3722 . . . . . . . . . . 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 5430 . . . . . . . . . . 11 (𝑟 = 𝑏 → (𝐹𝑟) = (𝐹𝑏))
7977, 78breq12d 3951 . . . . . . . . . 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 2794 . . . . . . . . 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 409 . . . . . . . 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 7537 . . . . . . . 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 414 . . . . . . 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 304 . . . . . 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 114 . . . . 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 2538 . . . 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 2558 . 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 2558 1 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∃wrex 2418  {crab 2421  ⟨cop 3536   class class class wbr 3938  ⟶wf 5128  ‘cfv 5132  (class class class)co 5783  1st c1st 6045  2nd c2nd 6046  1oc1o 6315  [cec 6436  Ncnpi 7124
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