Theorem caucvgprprlem2 7770

Description: Lemma for caucvgprpr 7772. 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 7743	. . . 4 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
4		ltrelpi 7384	. . . . . . . . . 10 ⊢ <N ⊆ (N × N)
5	4	brel 4711	. . . . . . . . 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 7482	. . . . . . . 8 ⊢ (𝐾 ∈ N → [⟨𝐾, 1o⟩] ~Q ∈ Q)
9		recclnq 7452	. . . . . . . 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 7607	. . . . . 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 5694	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐾) ∈ P)
17		ltaprg 7679	. . . . 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 1249	. . . 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 7597	. . . . 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 411	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
22		addclpr 7597	. . . . 5 ⊢ (((𝐹‘𝐾) ∈ P ∧ 𝑄 ∈ P) → ((𝐹‘𝐾) +P 𝑄) ∈ P)
23	16, 14, 22	syl2anc 411	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐾) +P 𝑄) ∈ P)
24		ltdfpr 7566	. . . 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 411	. . 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 529	. . . 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 539	. . . . . . . 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 4032	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
31	30	cbvabv 2318	. . . . . . . . . . 11 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}
32		breq2 4033	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑞 → ((*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞))
33	32	cbvabv 2318	. . . . . . . . . . 11 ⊢ {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}
34	31, 33	opeq12i 3809	. . . . . . . . . 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 5929	. . . . . . . . 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 5557	. . . . . . . 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 2286	. . . . . . 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 7607	. . . . . . . . . . 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 7597	. . . . . . . . . 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 411	. . . . . . . . 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 7612	. . . . . . . 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 411	. . . . . . 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 5554	. . . . . . . . 9 ⊢ (𝑟 = 𝐾 → (𝐹‘𝑟) = (𝐹‘𝐾))
47		opeq1 3804	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝐾 → ⟨𝑟, 1o⟩ = ⟨𝐾, 1o⟩)
48	47	eceq1d 6623	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝐾 → [⟨𝑟, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
49	48	fveq2d 5558	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝐾 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
50	49	breq2d 4041	. . . . . . . . . . 11 ⊢ (𝑟 = 𝐾 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
51	50	abbidv 2311	. . . . . . . . . 10 ⊢ (𝑟 = 𝐾 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )})
52	49	breq1d 4039	. . . . . . . . . . 11 ⊢ (𝑟 = 𝐾 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞))
53	52	abbidv 2311	. . . . . . . . . 10 ⊢ (𝑟 = 𝐾 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞})
54	51, 53	opeq12d 3812	. . . . . . . . 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 5936	. . . . . . . 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 4039	. . . . . . 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 2864	. . . . . 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 411	. . . . 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 4033	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑝 <Q 𝑢 ↔ 𝑝 <Q 𝑥))
60	59	abbidv 2311	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → {𝑝 ∣ 𝑝 <Q 𝑢} = {𝑝 ∣ 𝑝 <Q 𝑥})
61		breq1 4032	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 <Q 𝑞 ↔ 𝑥 <Q 𝑞))
62	61	abbidv 2311	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → {𝑞 ∣ 𝑢 <Q 𝑞} = {𝑞 ∣ 𝑥 <Q 𝑞})
63	60, 62	opeq12d 3812	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩)
64	63	breq2d 4041	. . . . . . 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 2495	. . . . . 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 5557	. . . . . . 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 7423	. . . . . . . . 9 ⊢ Q ∈ V
69	68	rabex 4173	. . . . . . . 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 4173	. . . . . . . 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 6200	. . . . . . 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 2214	. . . . . 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 2919	. . . . 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 417	. . . 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 540	. . . 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 2543	. . . 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 1247	. . 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 7764	. . . . 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 7566	. . . 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 411	. . 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 2616	1 ⊢ (𝜑 → 𝐿<P ((𝐹‘𝐾) +P 𝑄))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  {crab 2476  ⟨cop 3621   class class class wbr 4029  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918  1^st c1st 6191  2^nd c2nd 6192  1_oc1o 6462  [cec 6585  Ncnpi 7332   <_N clti 7335   ~_Q ceq 7339  Qcnq 7340   +_Q cplq 7342  *_Qcrq 7344   <_Q cltq 7345  Pcnp 7351   +_P cpp 7353  <_P cltp 7355

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530

This theorem is referenced by:  caucvgprprlemlim  7771