Theorem caucvgprprlemexb 7210

Description: Lemma for caucvgprpr 7215. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-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}⟩}⟩
caucvgprprlemexb.q	⊢ (𝜑 → 𝑄 ∈ P)
caucvgprprlemexb.r	⊢ (𝜑 → 𝑅 ∈ N)

Assertion

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

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

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,𝑞,𝑝,𝑏,𝑙)   𝑅(𝑢,𝑘,𝑚,𝑛,𝑟,𝑙)   𝐹(𝑞,𝑝)   𝐿(𝑢,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemexb

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

Step	Hyp	Ref	Expression
1		caucvgprpr.f	. . . . . 6 ⊢ (𝜑 → 𝐹:N⟶P)
2		caucvgprpr.cau	. . . . . 6 ⊢ (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}⟩))))
3		caucvgprpr.bnd	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))
4		caucvgprpr.lim	. . . . . 6 ⊢ 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}⟩}⟩
5	1, 2, 3, 4	caucvgprprlemclphr 7208	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ P)
6		caucvgprprlemexb.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ N)
7		recnnpr 7051	. . . . . 6 ⊢ (𝑅 ∈ N → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
8	6, 7	syl 14	. . . . 5 ⊢ (𝜑 → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
9		addclpr 7040	. . . . 5 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
10	5, 8, 9	syl2anc 403	. . . 4 ⊢ (𝜑 → (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
11	1, 6	ffvelrnd 5398	. . . 4 ⊢ (𝜑 → (𝐹‘𝑅) ∈ P)
12		caucvgprprlemexb.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ P)
13		ltaprg 7122	. . . 4 ⊢ (((𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ (𝐹‘𝑅) ∈ P ∧ 𝑄 ∈ P) → ((𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅) ↔ (𝑄 +P (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P (𝑄 +P (𝐹‘𝑅))))
14	10, 11, 12, 13	syl3anc 1172	. . 3 ⊢ (𝜑 → ((𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅) ↔ (𝑄 +P (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P (𝑄 +P (𝐹‘𝑅))))
15		addassprg 7082	. . . . . 6 ⊢ ((𝑄 ∈ P ∧ 𝐿 ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝑄 +P 𝐿) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = (𝑄 +P (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
16	12, 5, 8, 15	syl3anc 1172	. . . . 5 ⊢ (𝜑 → ((𝑄 +P 𝐿) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = (𝑄 +P (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
17		addcomprg 7081	. . . . . . 7 ⊢ ((𝑄 ∈ P ∧ 𝐿 ∈ P) → (𝑄 +P 𝐿) = (𝐿 +P 𝑄))
18	12, 5, 17	syl2anc 403	. . . . . 6 ⊢ (𝜑 → (𝑄 +P 𝐿) = (𝐿 +P 𝑄))
19	18	oveq1d 5628	. . . . 5 ⊢ (𝜑 → ((𝑄 +P 𝐿) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = ((𝐿 +P 𝑄) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
20	16, 19	eqtr3d 2119	. . . 4 ⊢ (𝜑 → (𝑄 +P (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) = ((𝐿 +P 𝑄) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
21		addcomprg 7081	. . . . 5 ⊢ ((𝑄 ∈ P ∧ (𝐹‘𝑅) ∈ P) → (𝑄 +P (𝐹‘𝑅)) = ((𝐹‘𝑅) +P 𝑄))
22	12, 11, 21	syl2anc 403	. . . 4 ⊢ (𝜑 → (𝑄 +P (𝐹‘𝑅)) = ((𝐹‘𝑅) +P 𝑄))
23	20, 22	breq12d 3833	. . 3 ⊢ (𝜑 → ((𝑄 +P (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P (𝑄 +P (𝐹‘𝑅)) ↔ ((𝐿 +P 𝑄) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹‘𝑅) +P 𝑄)))
24	14, 23	bitrd 186	. 2 ⊢ (𝜑 → ((𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅) ↔ ((𝐿 +P 𝑄) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹‘𝑅) +P 𝑄)))
25	1	adantr 270	. . . . 5 ⊢ ((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) → 𝐹:N⟶P)
26	2	adantr 270	. . . . 5 ⊢ ((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) → ∀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}⟩))))
27	3	adantr 270	. . . . 5 ⊢ ((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))
28		nnnq 6925	. . . . . . 7 ⊢ (𝑅 ∈ N → [⟨𝑅, 1𝑜⟩] ~Q ∈ Q)
29		recclnq 6895	. . . . . . 7 ⊢ ([⟨𝑅, 1𝑜⟩] ~Q ∈ Q → (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) ∈ Q)
30	6, 28, 29	3syl 17	. . . . . 6 ⊢ (𝜑 → (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) ∈ Q)
31	30	adantr 270	. . . . 5 ⊢ ((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) → (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) ∈ Q)
32	11	adantr 270	. . . . 5 ⊢ ((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) → (𝐹‘𝑅) ∈ P)
33		simpr 108	. . . . 5 ⊢ ((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) → (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅))
34	25, 26, 27, 4, 31, 32, 33	caucvgprprlemexbt 7209	. . . 4 ⊢ ((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) → ∃𝑏 ∈ N (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅))
35		ltaprg 7122	. . . . . . . 8 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
36	35	adantl 271	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
37	25	ffvelrnda 5397	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → (𝐹‘𝑏) ∈ P)
38		recnnpr 7051	. . . . . . . . . 10 ⊢ (𝑏 ∈ N → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
39	38	adantl 271	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
40		addclpr 7040	. . . . . . . . 9 ⊢ (((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)
41	37, 39, 40	syl2anc 403	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
42	6	ad2antrr 472	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → 𝑅 ∈ N)
43	42, 7	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
44		addclpr 7040	. . . . . . . 8 ⊢ ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~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‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
45	41, 43, 44	syl2anc 403	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
46	11	ad2antrr 472	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → (𝐹‘𝑅) ∈ P)
47	12	ad2antrr 472	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → 𝑄 ∈ P)
48		addcomprg 7081	. . . . . . . 8 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
49	48	adantl 271	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
50	36, 45, 46, 47, 49	caovord2d 5771	. . . . . 6 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅) ↔ ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P 𝑄)<P ((𝐹‘𝑅) +P 𝑄)))
51		addassprg 7082	. . . . . . . 8 ⊢ ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P ∧ 𝑄 ∈ P) → ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P 𝑄) = (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄)))
52	41, 43, 47, 51	syl3anc 1172	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P 𝑄) = (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄)))
53	52	breq1d 3830	. . . . . 6 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → (((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P 𝑄)<P ((𝐹‘𝑅) +P 𝑄) ↔ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄))<P ((𝐹‘𝑅) +P 𝑄)))
54		addcomprg 7081	. . . . . . . . 9 ⊢ ((⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P ∧ 𝑄 ∈ P) → (⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄) = (𝑄 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
55	43, 47, 54	syl2anc 403	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → (⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄) = (𝑄 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
56	55	oveq2d 5629	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄)) = (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
57	56	breq1d 3830	. . . . . 6 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄))<P ((𝐹‘𝑅) +P 𝑄) ↔ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹‘𝑅) +P 𝑄)))
58	50, 53, 57	3bitrd 212	. . . . 5 ⊢ (((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅) ↔ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹‘𝑅) +P 𝑄)))
59	58	rexbidva 2373	. . . 4 ⊢ ((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) → (∃𝑏 ∈ N (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅) ↔ ∃𝑏 ∈ N (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹‘𝑅) +P 𝑄)))
60	34, 59	mpbid 145	. . 3 ⊢ ((𝜑 ∧ (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅)) → ∃𝑏 ∈ N (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹‘𝑅) +P 𝑄))
61	60	ex 113	. 2 ⊢ (𝜑 → ((𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝑅) → ∃𝑏 ∈ N (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹‘𝑅) +P 𝑄)))
62	24, 61	sylbird 168	1 ⊢ (𝜑 → (((𝐿 +P 𝑄) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹‘𝑅) +P 𝑄) → ∃𝑏 ∈ N (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹‘𝑅) +P 𝑄)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 922   = wceq 1287   ∈ wcel 1436  {cab 2071  ∀wral 2355  ∃wrex 2356  {crab 2359  ⟨cop 3434   class class class wbr 3820  ⟶wf 4977  ‘cfv 4981  (class class class)co 5613  1_𝑜c1o 6128  [cec 6242  Ncnpi 6775   <_N clti 6778   ~_Q ceq 6782  Qcnq 6783   +_Q cplq 6785  *_Qcrq 6787   <_Q cltq 6788  Pcnp 6794   +_P cpp 6796  <_P cltp 6798

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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376

This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-eprel 4090  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-1o 6135  df-2o 6136  df-oadd 6139  df-omul 6140  df-er 6244  df-ec 6246  df-qs 6250  df-ni 6807  df-pli 6808  df-mi 6809  df-lti 6810  df-plpq 6847  df-mpq 6848  df-enq 6850  df-nqqs 6851  df-plqqs 6852  df-mqqs 6853  df-1nqqs 6854  df-rq 6855  df-ltnqqs 6856  df-enq0 6927  df-nq0 6928  df-0nq0 6929  df-plq0 6930  df-mq0 6931  df-inp 6969  df-iplp 6971  df-iltp 6973

This theorem is referenced by:  caucvgprprlemaddq  7211