Theorem caucvgprprlemopu 8014

Description: Lemma for caucvgprpr 8027. 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 8001	. . . 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 533	. . . . 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 5812	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → (𝐹‘𝑏) ∈ P)
8		recnnpr 7863	. . . . . . . . 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 7852	. . . . . . . 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 7862	. . . . . . 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 7821	. . . . . 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 7867	. . . . . . . 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 2816	. . . . . . . . 9 ⊢ 𝑠 ∈ V
25		breq1 4112	. . . . . . . . 9 ⊢ (𝑝 = 𝑠 → (𝑝 <Q 𝑡 ↔ 𝑠 <Q 𝑡))
26		ltnqex 7864	. . . . . . . . . 10 ⊢ {𝑝 ∣ 𝑝 <Q 𝑡} ∈ V
27		gtnqex 7865	. . . . . . . . . 10 ⊢ {𝑞 ∣ 𝑡 <Q 𝑞} ∈ V
28	26, 27	op1st 6340	. . . . . . . . 9 ⊢ (1st ‘⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩) = {𝑝 ∣ 𝑝 <Q 𝑡}
29	24, 25, 28	elab2 2965	. . . . . . . 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 2644	. . . 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 533	. . . . . 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 529	. . . . . . 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 537	. . . . . . . . 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 531	. . . . . . . 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 5670	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑏 → (𝐹‘𝑟) = (𝐹‘𝑏))
41		opeq1 3883	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑏 → ⟨𝑟, 1o⟩ = ⟨𝑏, 1o⟩)
42	41	eceq1d 6803	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑏 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
43	42	fveq2d 5674	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
44	43	breq2d 4121	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
45	44	abbidv 2352	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )})
46	43	breq1d 4119	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞))
47	46	abbidv 2352	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞})
48	45, 47	opeq12d 3891	. . . . . . . . . . 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 6068	. . . . . . . . . 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 4119	. . . . . . . . 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 2921	. . . . . . . 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 8001	. . . . . . 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 2644	. . 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 2665	1 ⊢ ((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  {cab 2218  ∀wral 2520  ∃wrex 2521  {crab 2524  ⟨cop 3692   class class class wbr 4109  ⟶wf 5348  ‘cfv 5352  (class class class)co 6050  1^st c1st 6332  2^nd c2nd 6333  1_oc1o 6640  [cec 6765  Ncnpi 7587   <_N clti 7590   ~_Q ceq 7594  Qcnq 7595   +_Q cplq 7597  *_Qcrq 7599   <_Q cltq 7600  Pcnp 7606   +_P cpp 7608  <_P cltp 7610

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iplp 7783  df-iltp 7785

This theorem is referenced by:  caucvgprprlemrnd  8016