Theorem caucvgprprlemexbt 8063

Description: Lemma for caucvgprpr 8069. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)

Hypotheses

Ref	Expression
caucvgprpr.f	⊢ (𝜑 → 𝐹:N⟶P)
caucvgprpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprpr.bnd	⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))
caucvgprpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩
caucvgprprlemexbt.q	⊢ (𝜑 → 𝑄 ∈ Q)
caucvgprprlemexbt.t	⊢ (𝜑 → 𝑇 ∈ P)
caucvgprprlemexbt.lt	⊢ (𝜑 → (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P 𝑇)

Assertion

Ref	Expression
caucvgprprlemexbt	⊢ (𝜑 → ∃𝑏 ∈ N (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P 𝑇)

Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟,𝑚   𝐹,𝑏   𝑘,𝐹,𝑙,𝑛,𝑢   𝐹,𝑟   𝐿,𝑏   𝑘,𝐿   𝑄,𝑏,𝑝,𝑞   𝑇,𝑏   𝜑,𝑏   𝑟,𝑏,𝑝,𝑞   𝑘,𝑝,𝑞,𝑟,𝑙,𝑢

Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑞,𝑝,𝑏,𝑙)   𝑄(𝑢,𝑘,𝑚,𝑛,𝑟,𝑙)   𝑇(𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑞,𝑝)   𝐿(𝑢,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemexbt

Dummy variables 𝑓 𝑔 ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprprlemexbt.lt	. . . . 5 ⊢ (𝜑 → (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P 𝑇)
2		caucvgprpr.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:N⟶P)
3		caucvgprpr.cau	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
4		caucvgprpr.bnd	. . . . . . . 8 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))
5		caucvgprpr.lim	. . . . . . . 8 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩
6	2, 3, 4, 5	caucvgprprlemclphr 8062	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ P)
7		caucvgprprlemexbt.q	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ Q)
8		nqprlu 7904	. . . . . . . 8 ⊢ (𝑄 ∈ Q → ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩ ∈ P)
9	7, 8	syl 14	. . . . . . 7 ⊢ (𝜑 → ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩ ∈ P)
10		addclpr 7894	. . . . . . 7 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩ ∈ P) → (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩) ∈ P)
11	6, 9, 10	syl2anc 415	. . . . . 6 ⊢ (𝜑 → (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩) ∈ P)
12		caucvgprprlemexbt.t	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ P)
13		ltdfpr 7863	. . . . . 6 ⊢ (((𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩) ∈ P ∧ 𝑇 ∈ P) → ((𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P 𝑇 ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇))))
14	11, 12, 13	syl2anc 415	. . . . 5 ⊢ (𝜑 → ((𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P 𝑇 ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇))))
15	1, 14	mpbid 147	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))
16	6	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) → 𝐿 ∈ P)
17	7	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) → 𝑄 ∈ Q)
18		simprrl 545	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) → 𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)))
19	16, 17, 18	prplnqu 7977	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) → ∃𝑦 ∈ (2nd ‘𝐿)(𝑦 +Q 𝑄) = 𝑥)
20		simprl 535	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) → 𝑦 ∈ (2nd ‘𝐿))
21		breq2 4129	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑦 → (𝑝 <Q 𝑢 ↔ 𝑝 <Q 𝑦))
22	21	abbidv 2358	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → {𝑝 ∣ 𝑝 <Q 𝑢} = {𝑝 ∣ 𝑝 <Q 𝑦})
23		breq1 4128	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑦 → (𝑢 <Q 𝑞 ↔ 𝑦 <Q 𝑞))
24	23	abbidv 2358	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → {𝑞 ∣ 𝑢 <Q 𝑞} = {𝑞 ∣ 𝑦 <Q 𝑞})
25	22, 24	opeq12d 3907	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑦 → ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩)
26	25	breq2d 4137	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑦 → (((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩ ↔ ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩))
27	26	rexbidv 2551	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑦 → (∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩ ↔ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩))
28	5	fveq2i 5693	. . . . . . . . . . . . . 14 ⊢ (2nd ‘𝐿) = (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩)
29		nqex 7720	. . . . . . . . . . . . . . . 16 ⊢ Q ∈ V
30	29	rabex 4275	. . . . . . . . . . . . . . 15 ⊢ {𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)} ∈ V
31	29	rabex 4275	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩} ∈ V
32	30, 31	op2nd 6371	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩) = {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}
33	28, 32	eqtri 2259	. . . . . . . . . . . . 13 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}
34	27, 33	elrab2 2985	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (2nd ‘𝐿) ↔ (𝑦 ∈ Q ∧ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩))
35	34	biimpi 120	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (2nd ‘𝐿) → (𝑦 ∈ Q ∧ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩))
36	35	simprd 114	. . . . . . . . . 10 ⊢ (𝑦 ∈ (2nd ‘𝐿) → ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩)
37	20, 36	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) → ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩)
38		fveq2 5690	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → (𝐹‘𝑟) = (𝐹‘𝑏))
39		opeq1 3899	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑏 → ⟨𝑟, 1o⟩ = ⟨𝑏, 1o⟩)
40	39	eceq1d 6833	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑏 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
41	40	fveq2d 5694	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
42	41	breq2d 4137	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑏 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
43	42	abbidv 2358	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )})
44	41	breq1d 4135	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑏 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞))
45	44	abbidv 2358	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞})
46	43, 45	opeq12d 3907	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)
47	38, 46	oveq12d 6093	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑏 → ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩))
48	47	breq1d 4135	. . . . . . . . . 10 ⊢ (𝑟 = 𝑏 → (((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩ ↔ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩))
49	48	cbvrexv 2787	. . . . . . . . 9 ⊢ (∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩ ↔ ∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩)
50	37, 49	sylib 122	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) → ∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩)
51		simpr 110	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩)
52		ltaprg 7976	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
53	52	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
54	2	ad4antr 498	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → 𝐹:N⟶P)
55		simplr 533	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → 𝑏 ∈ N)
56	54, 55	ffvelcdmd 5835	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → (𝐹‘𝑏) ∈ P)
57		recnnpr 7905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ N → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
58	55, 57	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
59		addclpr 7894	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑏) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
60	56, 58, 59	syl2anc 415	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
61	20	ad2antrr 492	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → 𝑦 ∈ (2nd ‘𝐿))
62	35	simpld 112	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (2nd ‘𝐿) → 𝑦 ∈ Q)
63	61, 62	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → 𝑦 ∈ Q)
64		nqprlu 7904	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ Q → ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩ ∈ P)
65	63, 64	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩ ∈ P)
66	9	ad4antr 498	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩ ∈ P)
67		addcomprg 7935	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
68	67	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
69	53, 60, 65, 66, 68	caovord2d 6249	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩ ↔ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P (⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)))
70	51, 69	mpbid 147	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P (⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩))
71	7	ad4antr 498	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → 𝑄 ∈ Q)
72		addnqpr 7918	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Q ∧ 𝑄 ∈ Q) → ⟨{𝑝 ∣ 𝑝 <Q (𝑦 +Q 𝑄)}, {𝑞 ∣ (𝑦 +Q 𝑄) <Q 𝑞}⟩ = (⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩))
73	63, 71, 72	syl2anc 415	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → ⟨{𝑝 ∣ 𝑝 <Q (𝑦 +Q 𝑄)}, {𝑞 ∣ (𝑦 +Q 𝑄) <Q 𝑞}⟩ = (⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩))
74	70, 73	breqtrrd 4153	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑦 +Q 𝑄)}, {𝑞 ∣ (𝑦 +Q 𝑄) <Q 𝑞}⟩)
75		simplrr 542	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) → (𝑦 +Q 𝑄) = 𝑥)
76	75	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → (𝑦 +Q 𝑄) = 𝑥)
77		breq2 4129	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 +Q 𝑄) = 𝑥 → (𝑝 <Q (𝑦 +Q 𝑄) ↔ 𝑝 <Q 𝑥))
78	77	abbidv 2358	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 +Q 𝑄) = 𝑥 → {𝑝 ∣ 𝑝 <Q (𝑦 +Q 𝑄)} = {𝑝 ∣ 𝑝 <Q 𝑥})
79		breq1 4128	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 +Q 𝑄) = 𝑥 → ((𝑦 +Q 𝑄) <Q 𝑞 ↔ 𝑥 <Q 𝑞))
80	79	abbidv 2358	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 +Q 𝑄) = 𝑥 → {𝑞 ∣ (𝑦 +Q 𝑄) <Q 𝑞} = {𝑞 ∣ 𝑥 <Q 𝑞})
81	78, 80	opeq12d 3907	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 +Q 𝑄) = 𝑥 → ⟨{𝑝 ∣ 𝑝 <Q (𝑦 +Q 𝑄)}, {𝑞 ∣ (𝑦 +Q 𝑄) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩)
82	81	breq2d 4137	. . . . . . . . . . . . . 14 ⊢ ((𝑦 +Q 𝑄) = 𝑥 → ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑦 +Q 𝑄)}, {𝑞 ∣ (𝑦 +Q 𝑄) <Q 𝑞}⟩ ↔ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
83	76, 82	syl 14	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑦 +Q 𝑄)}, {𝑞 ∣ (𝑦 +Q 𝑄) <Q 𝑞}⟩ ↔ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
84	74, 83	mpbid 147	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩)
85		simplrl 541	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) → 𝑥 ∈ Q)
86	85	ad2antrr 492	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → 𝑥 ∈ Q)
87		addclpr 7894	. . . . . . . . . . . . . 14 ⊢ ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩ ∈ P) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩) ∈ P)
88	60, 66, 87	syl2anc 415	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩) ∈ P)
89		nqpru 7909	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩) ∈ P) → (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ↔ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
90	86, 88, 89	syl2anc 415	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ↔ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
91	84, 90	mpbird 167	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → 𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)))
92		simprrr 546	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) → 𝑥 ∈ (1st ‘𝑇))
93	92	ad3antrrr 496	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → 𝑥 ∈ (1st ‘𝑇))
94	91, 93	jca 306	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩) → (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))
95	94	ex 115	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) ∧ 𝑏 ∈ N) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩ → (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇))))
96	95	reximdva 2652	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) → (∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩ → ∃𝑏 ∈ N (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇))))
97	50, 96	mpd 13	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) ∧ (𝑦 ∈ (2nd ‘𝐿) ∧ (𝑦 +Q 𝑄) = 𝑥)) → ∃𝑏 ∈ N (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))
98	19, 97	rexlimddv 2673	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) → ∃𝑏 ∈ N (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))
99	98	expr 375	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ Q) → ((𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)) → ∃𝑏 ∈ N (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇))))
100	99	reximdva 2652	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)) → ∃𝑥 ∈ Q ∃𝑏 ∈ N (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇))))
101	15, 100	mpd 13	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ Q ∃𝑏 ∈ N (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))
102		rexcom 2715	. . 3 ⊢ (∃𝑥 ∈ Q ∃𝑏 ∈ N (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)) ↔ ∃𝑏 ∈ N ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))
103	101, 102	sylib 122	. 2 ⊢ (𝜑 → ∃𝑏 ∈ N ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))
104	2	ffvelcdmda 5834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → (𝐹‘𝑏) ∈ P)
105	57	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
106	104, 105, 59	syl2anc 415	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
107	9	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩ ∈ P)
108	106, 107, 87	syl2anc 415	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩) ∈ P)
109	12	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → 𝑇 ∈ P)
110		ltdfpr 7863	. . . 4 ⊢ (((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩) ∈ P ∧ 𝑇 ∈ P) → ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P 𝑇 ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇))))
111	108, 109, 110	syl2anc 415	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P 𝑇 ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇))))
112	111	rexbidva 2547	. 2 ⊢ (𝜑 → (∃𝑏 ∈ N (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P 𝑇 ↔ ∃𝑏 ∈ N ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘(((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇))))
113	103, 112	mpbird 167	1 ⊢ (𝜑 → ∃𝑏 ∈ N (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P 𝑇)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  {crab 2532  ⟨cop 3708   class class class wbr 4125  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075  1^st c1st 6362  2^nd c2nd 6363  1_oc1o 6670  [cec 6795  Ncnpi 7629   <_N clti 7632   ~_Q ceq 7636  Qcnq 7637   +_Q cplq 7639  *_Qcrq 7641   <_Q cltq 7642  Pcnp 7648   +_P cpp 7650  <_P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iplp 7825  df-iltp 7827

This theorem is referenced by:  caucvgprprlemexb  8064