Theorem caucvgprprlem2 8078

Description: Lemma for caucvgprpr 8080. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)

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}⟩}⟩
caucvgprprlemlim.q	⊢ (𝜑 → 𝑄 ∈ P)
caucvgprprlemlim.jk	⊢ (𝜑 → 𝐽 <N 𝐾)
caucvgprprlemlim.jkq	⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)

Assertion

Ref	Expression
caucvgprprlem2	⊢ (𝜑 → 𝐿<P ((𝐹‘𝐾) +P 𝑄))

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

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

Proof of Theorem caucvgprprlem2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprprlemlim.jk	. . . . 5 ⊢ (𝜑 → 𝐽 <N 𝐾)
2		caucvgprprlemlim.jkq	. . . . 5 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
3	1, 2	caucvgprprlemk 8051	. . . 4 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
4		ltrelpi 7692	. . . . . . . . . 10 ⊢ <N ⊆ (N × N)
5	4	brel 4827	. . . . . . . . 9 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
6	1, 5	syl 14	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
7	6	simprd 114	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ N)
8		nnnq 7790	. . . . . . . 8 ⊢ (𝐾 ∈ N → [⟨𝐾, 1o⟩] ~Q ∈ Q)
9		recclnq 7760	. . . . . . . 8 ⊢ ([⟨𝐾, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q)
10	8, 9	syl 14	. . . . . . 7 ⊢ (𝐾 ∈ N → (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q)
11	7, 10	syl 14	. . . . . 6 ⊢ (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q)
12		nqprlu 7915	. . . . . 6 ⊢ ((*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
13	11, 12	syl 14	. . . . 5 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
14		caucvgprprlemlim.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ P)
15		caucvgprpr.f	. . . . . 6 ⊢ (𝜑 → 𝐹:N⟶P)
16	15, 7	ffvelcdmd 5844	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐾) ∈ P)
17		ltaprg 7987	. . . . 5 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P ∧ 𝑄 ∈ P ∧ (𝐹‘𝐾) ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄 ↔ ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝐾) +P 𝑄)))
18	13, 14, 16, 17	syl3anc 1278	. . . 4 ⊢ (𝜑 → (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄 ↔ ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝐾) +P 𝑄)))
19	3, 18	mpbid 147	. . 3 ⊢ (𝜑 → ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝐾) +P 𝑄))
20		addclpr 7905	. . . . 5 ⊢ (((𝐹‘𝐾) ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P) → ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
21	16, 13, 20	syl2anc 415	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
22		addclpr 7905	. . . . 5 ⊢ (((𝐹‘𝐾) ∈ P ∧ 𝑄 ∈ P) → ((𝐹‘𝐾) +P 𝑄) ∈ P)
23	16, 14, 22	syl2anc 415	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐾) +P 𝑄) ∈ P)
24		ltdfpr 7874	. . . 4 ⊢ ((((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P ∧ ((𝐹‘𝐾) +P 𝑄) ∈ P) → (((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝐾) +P 𝑄) ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄)))))
25	21, 23, 24	syl2anc 415	. . 3 ⊢ (𝜑 → (((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝐾) +P 𝑄) ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄)))))
26	19, 25	mpbid 147	. 2 ⊢ (𝜑 → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))
27		simprl 535	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝑥 ∈ Q)
28	7	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝐾 ∈ N)
29		simprrl 545	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)))
30		breq1 4133	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
31	30	cbvabv 2365	. . . . . . . . . . 11 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}
32		breq2 4134	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑞 → ((*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞))
33	32	cbvabv 2365	. . . . . . . . . . 11 ⊢ {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}
34	31, 33	opeq12i 3909	. . . . . . . . . 10 ⊢ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩
35	34	oveq2i 6096	. . . . . . . . 9 ⊢ ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)
36	35	fveq2i 5698	. . . . . . . 8 ⊢ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) = (2nd ‘((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩))
37	29, 36	eleqtrdi 2331	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)))
38		nqprlu 7915	. . . . . . . . . . 11 ⊢ ((*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
39	11, 38	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
40		addclpr 7905	. . . . . . . . . 10 ⊢ (((𝐹‘𝐾) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
41	16, 39, 40	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
42	41	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
43		nqpru 7920	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P) → (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)) ↔ ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
44	27, 42, 43	syl2anc 415	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)) ↔ ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
45	37, 44	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩)
46		fveq2 5695	. . . . . . . . 9 ⊢ (𝑟 = 𝐾 → (𝐹‘𝑟) = (𝐹‘𝐾))
47		opeq1 3904	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝐾 → ⟨𝑟, 1o⟩ = ⟨𝐾, 1o⟩)
48	47	eceq1d 6843	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝐾 → [⟨𝑟, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
49	48	fveq2d 5699	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝐾 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
50	49	breq2d 4142	. . . . . . . . . . 11 ⊢ (𝑟 = 𝐾 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
51	50	abbidv 2358	. . . . . . . . . 10 ⊢ (𝑟 = 𝐾 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )})
52	49	breq1d 4140	. . . . . . . . . . 11 ⊢ (𝑟 = 𝐾 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞))
53	52	abbidv 2358	. . . . . . . . . 10 ⊢ (𝑟 = 𝐾 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞})
54	51, 53	opeq12d 3912	. . . . . . . . 9 ⊢ (𝑟 = 𝐾 → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)
55	46, 54	oveq12d 6103	. . . . . . . 8 ⊢ (𝑟 = 𝐾 → ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩))
56	55	breq1d 4140	. . . . . . 7 ⊢ (𝑟 = 𝐾 → (((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩ ↔ ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
57	56	rspcev 2929	. . . . . 6 ⊢ ((𝐾 ∈ N ∧ ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <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 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩)
58	28, 45, 57	syl2anc 415	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩)
59		breq2 4134	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑝 <Q 𝑢 ↔ 𝑝 <Q 𝑥))
60	59	abbidv 2358	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → {𝑝 ∣ 𝑝 <Q 𝑢} = {𝑝 ∣ 𝑝 <Q 𝑥})
61		breq1 4133	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 <Q 𝑞 ↔ 𝑥 <Q 𝑞))
62	61	abbidv 2358	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → {𝑞 ∣ 𝑢 <Q 𝑞} = {𝑞 ∣ 𝑥 <Q 𝑞})
63	60, 62	opeq12d 3912	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩)
64	63	breq2d 4142	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (((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 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
65	64	rexbidv 2551	. . . . . 6 ⊢ (𝑢 = 𝑥 → (∃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 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
66		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}⟩}⟩
67	66	fveq2i 5698	. . . . . . 7 ⊢ (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}⟩}⟩)
68		nqex 7731	. . . . . . . . 9 ⊢ Q ∈ V
69	68	rabex 4280	. . . . . . . 8 ⊢ {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
70	68	rabex 4280	. . . . . . . 8 ⊢ {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
71	69, 70	op2nd 6381	. . . . . . 7 ⊢ (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}⟩}
72	67, 71	eqtri 2259	. . . . . 6 ⊢ (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}⟩}
73	65, 72	elrab2 2985	. . . . 5 ⊢ (𝑥 ∈ (2nd ‘𝐿) ↔ (𝑥 ∈ Q ∧ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
74	27, 58, 73	sylanbrc 421	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝑥 ∈ (2nd ‘𝐿))
75		simprrr 546	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄)))
76		rspe 2599	. . . 4 ⊢ ((𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄)))) → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))
77	27, 74, 75, 76	syl12anc 1276	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))
78		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}⟩))))
79		caucvgprpr.bnd	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))
80	15, 78, 79, 66	caucvgprprlemcl 8072	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ P)
81	80	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝐿 ∈ P)
82	23	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → ((𝐹‘𝐾) +P 𝑄) ∈ P)
83		ltdfpr 7874	. . . 4 ⊢ ((𝐿 ∈ P ∧ ((𝐹‘𝐾) +P 𝑄) ∈ P) → (𝐿<P ((𝐹‘𝐾) +P 𝑄) ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄)))))
84	81, 82, 83	syl2anc 415	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → (𝐿<P ((𝐹‘𝐾) +P 𝑄) ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄)))))
85	77, 84	mpbird 167	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝐿<P ((𝐹‘𝐾) +P 𝑄))
86	26, 85	rexlimddv 2673	1 ⊢ (𝜑 → 𝐿<P ((𝐹‘𝐾) +P 𝑄))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  {crab 2532  ⟨cop 3712   class class class wbr 4130  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  1^st c1st 6372  2^nd c2nd 6373  1_oc1o 6680  [cec 6805  Ncnpi 7640   <_N clti 7643   ~_Q ceq 7647  Qcnq 7648   +_Q cplq 7650  *_Qcrq 7652   <_Q cltq 7653  Pcnp 7659   +_P cpp 7661  <_P cltp 7663

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7672  df-pli 7673  df-mi 7674  df-lti 7675  df-plpq 7712  df-mpq 7713  df-enq 7715  df-nqqs 7716  df-plqqs 7717  df-mqqs 7718  df-1nqqs 7719  df-rq 7720  df-ltnqqs 7721  df-enq0 7792  df-nq0 7793  df-0nq0 7794  df-plq0 7795  df-mq0 7796  df-inp 7834  df-iplp 7836  df-iltp 7838

This theorem is used by:  caucvgprprlemlim  8079