Theorem caucvgprprlemopu 8067

Description: Lemma for caucvgprpr 8080. 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 8054	. . . 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 537	. . . . 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 5843	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ N) → (𝐹‘𝑏) ∈ P)
8		recnnpr 7916	. . . . . . . . 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 7905	. . . . . . . 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 415	. . . . . . 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 513	. . . . . 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 493	. . . . . . 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 7915	. . . . . . 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 7874	. . . . . 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 415	. . . . 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 7920	. . . . . . . 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 415	. . . . . . 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 2824	. . . . . . . . 9 ⊢ 𝑠 ∈ V
25		breq1 4133	. . . . . . . . 9 ⊢ (𝑝 = 𝑠 → (𝑝 <Q 𝑡 ↔ 𝑠 <Q 𝑡))
26		ltnqex 7917	. . . . . . . . . 10 ⊢ {𝑝 ∣ 𝑝 <Q 𝑡} ∈ V
27		gtnqex 7918	. . . . . . . . . 10 ⊢ {𝑞 ∣ 𝑡 <Q 𝑞} ∈ V
28	26, 27	op1st 6380	. . . . . . . . 9 ⊢ (1st ‘⟨{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}⟩) = {𝑝 ∣ 𝑝 <Q 𝑡}
29	24, 25, 28	elab2 2974	. . . . . . . 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 477	. . . . . 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 2652	. . . 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 537	. . . . . 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 533	. . . . . . 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 541	. . . . . . . . 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 535	. . . . . . . 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 5695	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑏 → (𝐹‘𝑟) = (𝐹‘𝑏))
41		opeq1 3904	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑏 → ⟨𝑟, 1o⟩ = ⟨𝑏, 1o⟩)
42	41	eceq1d 6843	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑏 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
43	42	fveq2d 5699	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
44	43	breq2d 4142	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
45	44	abbidv 2358	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )})
46	43	breq1d 4140	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞))
47	46	abbidv 2358	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞})
48	45, 47	opeq12d 3912	. . . . . . . . . . 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 6103	. . . . . . . . . 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 4140	. . . . . . . . 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 2929	. . . . . . . 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 415	. . . . . . 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 8054	. . . . . . 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 421	. . . . . 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 2652	. . 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 2673	1 ⊢ ((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))

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:  caucvgprprlemrnd  8069