Theorem caucvgprprlemopu 7783

Description: Lemma for caucvgprpr 7796. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-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}⟩}⟩

Assertion

Ref	Expression
caucvgprprlemopu	⊢ ((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))

Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐹,𝑙,𝑟,𝑠   𝑢,𝐹,𝑟,𝑠   𝐿,𝑠   𝑝,𝑙,𝑞,𝑡,𝑟,𝑠   𝑢,𝑝,𝑞,𝑡   𝜑,𝑟,𝑠

Allowed substitution hints:   𝜑(𝑢,𝑡,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑡,𝑘,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑡,𝑘,𝑛,𝑞,𝑝)   𝐿(𝑢,𝑡,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemopu

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprpr.lim	. . . . 5 ⊢ 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}⟩}⟩
2	1	caucvgprprlemelu 7770	. . . 4 ⊢ (𝑡 ∈ (2nd ‘𝐿) ↔ (𝑡 ∈ Q ∧ ∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩))
3	2	simprbi 275	. . 3 ⊢ (𝑡 ∈ (2nd ‘𝐿) → ∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)
4	3	adantl 277	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) → ∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)
5		simprr 531	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)
6		caucvgprpr.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:N⟶P)
7	6	ffvelcdmda 5700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → (𝐹‘𝑏) ∈ P)
8		recnnpr 7632	. . . . . . . . 9 ⊢ (𝑏 ∈ N → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
9	8	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
10		addclpr 7621	. . . . . . . 8 ⊢ (((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)
11	7, 9, 10	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
12	11	ad2ant2r 509	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
13	2	simplbi 274	. . . . . . . 8 ⊢ (𝑡 ∈ (2nd ‘𝐿) → 𝑡 ∈ Q)
14	13	ad2antlr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → 𝑡 ∈ Q)
15		nqprlu 7631	. . . . . . 7 ⊢ (𝑡 ∈ Q → ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩ ∈ P)
16	14, 15	syl 14	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩ ∈ P)
17		ltdfpr 7590	. . . . . 6 ⊢ ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩ ∈ P) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩ ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩))))
18	12, 16, 17	syl2anc 411	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩ ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩))))
19	5, 18	mpbid 147	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)))
20		simpr 110	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) → 𝑠 ∈ Q)
21	12	adantr 276	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) → ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
22		nqpru 7636	. . . . . . . 8 ⊢ ((𝑠 ∈ Q ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P) → (𝑠 ∈ (2nd ‘((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)) ↔ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩))
23	20, 21, 22	syl2anc 411	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) → (𝑠 ∈ (2nd ‘((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)) ↔ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩))
24		vex 2766	. . . . . . . . 9 ⊢ 𝑠 ∈ V
25		breq1 4037	. . . . . . . . 9 ⊢ (𝑝 = 𝑠 → (𝑝 <Q 𝑡 ↔ 𝑠 <Q 𝑡))
26		ltnqex 7633	. . . . . . . . . 10 ⊢ {𝑝 ∣ 𝑝 <Q 𝑡} ∈ V
27		gtnqex 7634	. . . . . . . . . 10 ⊢ {𝑞 ∣ 𝑡 <Q 𝑞} ∈ V
28	26, 27	op1st 6213	. . . . . . . . 9 ⊢ (1st ‘⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩) = {𝑝 ∣ 𝑝 <Q 𝑡}
29	24, 25, 28	elab2 2912	. . . . . . . 8 ⊢ (𝑠 ∈ (1st ‘⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩) ↔ 𝑠 <Q 𝑡)
30	29	a1i 9	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) → (𝑠 ∈ (1st ‘⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩) ↔ 𝑠 <Q 𝑡))
31	23, 30	anbi12d 473	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) → ((𝑠 ∈ (2nd ‘((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ↔ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)))
32	31	biimpd 144	. . . . 5 ⊢ ((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) → ((𝑠 ∈ (2nd ‘((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)))
33	32	reximdva 2599	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → (∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → ∃𝑠 ∈ Q (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)))
34	19, 33	mpd 13	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → ∃𝑠 ∈ Q (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡))
35		simprr 531	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → 𝑠 <Q 𝑡)
36		simplr 528	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → 𝑠 ∈ Q)
37		simplrl 535	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) → 𝑏 ∈ N)
38	37	adantr 276	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → 𝑏 ∈ N)
39		simprl 529	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩)
40		fveq2 5561	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑏 → (𝐹‘𝑟) = (𝐹‘𝑏))
41		opeq1 3809	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑏 → ⟨𝑟, 1o⟩ = ⟨𝑏, 1o⟩)
42	41	eceq1d 6637	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑏 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
43	42	fveq2d 5565	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
44	43	breq2d 4046	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
45	44	abbidv 2314	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )})
46	43	breq1d 4044	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞))
47	46	abbidv 2314	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞})
48	45, 47	opeq12d 3817	. . . . . . . . . . 11 ⊢ (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}⟩)
49	40, 48	oveq12d 5943	. . . . . . . . . 10 ⊢ (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}⟩))
50	49	breq1d 4044	. . . . . . . . 9 ⊢ (𝑟 = 𝑏 → (((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ↔ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩))
51	50	rspcev 2868	. . . . . . . 8 ⊢ ((𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <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 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩)
52	38, 39, 51	syl2anc 411	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩)
53	1	caucvgprprlemelu 7770	. . . . . . 7 ⊢ (𝑠 ∈ (2nd ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩))
54	36, 52, 53	sylanbrc 417	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → 𝑠 ∈ (2nd ‘𝐿))
55	35, 54	jca 306	. . . . 5 ⊢ (((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))
56	55	ex 115	. . . 4 ⊢ ((((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) ∧ 𝑠 ∈ Q) → ((((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡) → (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿))))
57	56	reximdva 2599	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → (∃𝑠 ∈ Q (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡) → ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿))))
58	34, 57	mpd 13	. 2 ⊢ (((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) ∧ (𝑏 ∈ N ∧ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))
59	4, 58	rexlimddv 2619	1 ⊢ ((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∃wrex 2476  {crab 2479  ⟨cop 3626   class class class wbr 4034  ⟶wf 5255  ‘cfv 5259  (class class class)co 5925  1^st c1st 6205  2^nd c2nd 6206  1_oc1o 6476  [cec 6599  Ncnpi 7356   <_N clti 7359   ~_Q ceq 7363  Qcnq 7364   +_Q cplq 7366  *_Qcrq 7368   <_Q cltq 7369  Pcnp 7375   +_P cpp 7377  <_P cltp 7379

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iplp 7552  df-iltp 7554

This theorem is referenced by:  caucvgprprlemrnd  7785