Theorem caucvgprprlemexbt 6896

Description: Lemma for caucvgprpr 6902. 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 6895	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ P)
7		caucvgprprlemexbt.q	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ Q)
8		nqprlu 6737	. . . . . . . 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 6727	. . . . . . 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 403	. . . . . 6 ⊢ (𝜑 → (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩) ∈ P)
12		caucvgprprlemexbt.t	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ P)
13		ltdfpr 6696	. . . . . 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 403	. . . . 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 145	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))
16	6	adantr 270	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) → 𝐿 ∈ P)
17	7	adantr 270	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘(𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)) ∧ 𝑥 ∈ (1st ‘𝑇)))) → 𝑄 ∈ Q)
18		simprrl 505	. . . . . . . 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 6810	. . . . . . 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 497	. . . . . . . . . 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 3789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑦 → (𝑝 <Q 𝑢 ↔ 𝑝 <Q 𝑦))
22	21	abbidv 2196	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → {𝑝 ∣ 𝑝 <Q 𝑢} = {𝑝 ∣ 𝑝 <Q 𝑦})
23		breq1 3788	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑦 → (𝑢 <Q 𝑞 ↔ 𝑦 <Q 𝑞))
24	23	abbidv 2196	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → {𝑞 ∣ 𝑢 <Q 𝑞} = {𝑞 ∣ 𝑦 <Q 𝑞})
25	22, 24	opeq12d 3578	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑦 → ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q 𝑦}, {𝑞 ∣ 𝑦 <Q 𝑞}⟩)
26	25	breq2d 3797	. . . . . . . . . . . . . 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 2369	. . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . 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 6553	. . . . . . . . . . . . . . . 16 ⊢ Q ∈ V
30	29	rabex 3922	. . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . 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 5794	. . . . . . . . . . . . . 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 2101	. . . . . . . . . . . . 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 2751	. . . . . . . . . . . 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 118	. . . . . . . . . . 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 112	. . . . . . . . . 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 5198	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → (𝐹‘𝑟) = (𝐹‘𝑏))
39		opeq1 3570	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑏 → ⟨𝑟, 1𝑜⟩ = ⟨𝑏, 1𝑜⟩)
40	39	eceq1d 6165	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑏 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝑏, 1𝑜⟩] ~Q )
41	40	fveq2d 5202	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))
42	41	breq2d 3797	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑏 → (𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
43	42	abbidv 2196	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )})
44	41	breq1d 3795	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑏 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞))
45	44	abbidv 2196	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞})
46	43, 45	opeq12d 3578	. . . . . . . . . . . 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 5550	. . . . . . . . . . 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 3795	. . . . . . . . . 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 2578	. . . . . . . . 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 120	. . . . . . . 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 108	. . . . . . . . . . . . . . 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 6809	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
53	52	adantl 271	. . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . 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	ffvelrnd 5324	. . . . . . . . . . . . . . . . 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 6738	. . . . . . . . . . . . . . . . . 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 6727	. . . . . . . . . . . . . . . . 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 403	. . . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . . . . 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 110	. . . . . . . . . . . . . . . . . 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 6737	. . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . 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 6768	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
68	67	adantl 271	. . . . . . . . . . . . . . . 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 5690	. . . . . . . . . . . . . . 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 145	. . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 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 6751	. . . . . . . . . . . . . . 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 403	. . . . . . . . . . . . . 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 3811	. . . . . . . . . . . . 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 502	. . . . . . . . . . . . . . 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 270	. . . . . . . . . . . . . 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 3789	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 +Q 𝑄) = 𝑥 → (𝑝 <Q (𝑦 +Q 𝑄) ↔ 𝑝 <Q 𝑥))
78	77	abbidv 2196	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 +Q 𝑄) = 𝑥 → {𝑝 ∣ 𝑝 <Q (𝑦 +Q 𝑄)} = {𝑝 ∣ 𝑝 <Q 𝑥})
79		breq1 3788	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 +Q 𝑄) = 𝑥 → ((𝑦 +Q 𝑄) <Q 𝑞 ↔ 𝑥 <Q 𝑞))
80	79	abbidv 2196	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 +Q 𝑄) = 𝑥 → {𝑞 ∣ (𝑦 +Q 𝑄) <Q 𝑞} = {𝑞 ∣ 𝑥 <Q 𝑞})
81	78, 80	opeq12d 3578	. . . . . . . . . . . . . . 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 3797	. . . . . . . . . . . . . 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 145	. . . . . . . . . . . 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 501	. . . . . . . . . . . . . 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 471	. . . . . . . . . . . . 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 6727	. . . . . . . . . . . . . 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 403	. . . . . . . . . . . . 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 6742	. . . . . . . . . . . . 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 403	. . . . . . . . . . . 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 165	. . . . . . . . . . 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 506	. . . . . . . . . . . 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 475	. . . . . . . . . . 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 300	. . . . . . . . . 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 113	. . . . . . . . 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 2463	. . . . . . . 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 2481	. . . . . 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 367	. . . . 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 2463	. . . 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 2518	. . 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 120	. 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	ffvelrnda 5323	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → (𝐹‘𝑏) ∈ P)
105	57	adantl 271	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
106	104, 105, 59	syl2anc 403	. . . . 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 270	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩ ∈ P)
108	106, 107, 87	syl2anc 403	. . . 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 270	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → 𝑇 ∈ P)
110		ltdfpr 6696	. . . 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 403	. . 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 2365	. 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 165	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 102   ↔ wb 103   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  {cab 2067  ∀wral 2348  ∃wrex 2349  {crab 2352  ⟨cop 3401   class class class wbr 3785  ⟶wf 4918  ‘cfv 4922  (class class class)co 5532  1^st c1st 5785  2^nd c2nd 5786  1_𝑜c1o 6017  [cec 6127  Ncnpi 6462   <_N clti 6465   ~_Q ceq 6469  Qcnq 6470   +_Q cplq 6472  *_Qcrq 6474   <_Q cltq 6475  Pcnp 6481   +_P cpp 6483  <_P cltp 6485

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-iplp 6658  df-iltp 6660

This theorem is referenced by:  caucvgprprlemexb  6897