Theorem archsr 7781

Description: For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑥, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝑥, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R is the embedding of the positive integer 𝑥 into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.)

Assertion

Ref	Expression
archsr	⊢ (𝐴 ∈ R → ∃𝑥 ∈ N 𝐴 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )

Distinct variable group:   𝐴,𝑙,𝑢,𝑥

Proof of Theorem archsr

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 7726	. 2 ⊢ R = ((P × P) / ~R )
2		breq1 4007	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐴 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ 𝐴 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
3	2	rexbidv 2478	. 2 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐴 → (∃𝑥 ∈ N [⟨𝑧, 𝑤⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ ∃𝑥 ∈ N 𝐴 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
4		1pr 7553	. . . . . . 7 ⊢ 1P ∈ P
5		addclpr 7536	. . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 1P ∈ P) → (𝑧 +P 1P) ∈ P)
6	4, 5	mpan2 425	. . . . . 6 ⊢ (𝑧 ∈ P → (𝑧 +P 1P) ∈ P)
7		archpr 7642	. . . . . 6 ⊢ ((𝑧 +P 1P) ∈ P → ∃𝑥 ∈ N (𝑧 +P 1P)<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
8	6, 7	syl 14	. . . . 5 ⊢ (𝑧 ∈ P → ∃𝑥 ∈ N (𝑧 +P 1P)<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
9	8	adantr 276	. . . 4 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P) → ∃𝑥 ∈ N (𝑧 +P 1P)<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
10		nnprlu 7552	. . . . . . . . . 10 ⊢ (𝑥 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
11	10	adantl 277	. . . . . . . . 9 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
12		addclpr 7536	. . . . . . . . 9 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ 1P ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
13	11, 4, 12	sylancl 413	. . . . . . . 8 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
14		simplr 528	. . . . . . . 8 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → 𝑤 ∈ P)
15		ltaddpr 7596	. . . . . . . 8 ⊢ (((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 𝑤 ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)<P ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 𝑤))
16	13, 14, 15	syl2anc 411	. . . . . . 7 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)<P ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 𝑤))
17		addcomprg 7577	. . . . . . . 8 ⊢ ((𝑤 ∈ P ∧ (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P) → (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)) = ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 𝑤))
18	14, 13, 17	syl2anc 411	. . . . . . 7 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)) = ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 𝑤))
19	16, 18	breqtrrd 4032	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)<P (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
20		ltaddpr 7596	. . . . . . . 8 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ 1P ∈ P) → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩<P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
21	11, 4, 20	sylancl 413	. . . . . . 7 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩<P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
22		ltsopr 7595	. . . . . . . . 9 ⊢ <P Or P
23		ltrelpr 7504	. . . . . . . . 9 ⊢ <P ⊆ (P × P)
24	22, 23	sotri 5025	. . . . . . . 8 ⊢ (((𝑧 +P 1P)<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∧ ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩<P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)) → (𝑧 +P 1P)<P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
25	24	expcom 116	. . . . . . 7 ⊢ (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩<P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) → ((𝑧 +P 1P)<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ → (𝑧 +P 1P)<P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
26	21, 25	syl 14	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → ((𝑧 +P 1P)<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ → (𝑧 +P 1P)<P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
27	22, 23	sotri 5025	. . . . . . 7 ⊢ (((𝑧 +P 1P)<P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∧ (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)<P (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))) → (𝑧 +P 1P)<P (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
28	27	expcom 116	. . . . . 6 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)<P (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)) → ((𝑧 +P 1P)<P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) → (𝑧 +P 1P)<P (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
29	19, 26, 28	sylsyld 58	. . . . 5 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → ((𝑧 +P 1P)<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ → (𝑧 +P 1P)<P (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
30	29	reximdva 2579	. . . 4 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P) → (∃𝑥 ∈ N (𝑧 +P 1P)<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ → ∃𝑥 ∈ N (𝑧 +P 1P)<P (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
31	9, 30	mpd 13	. . 3 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P) → ∃𝑥 ∈ N (𝑧 +P 1P)<P (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
32		simpl 109	. . . . 5 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → (𝑧 ∈ P ∧ 𝑤 ∈ P))
33	4	a1i 9	. . . . 5 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → 1P ∈ P)
34		ltsrprg 7746	. . . . 5 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ (𝑧 +P 1P)<P (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
35	32, 13, 33, 34	syl12anc 1236	. . . 4 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ (𝑧 +P 1P)<P (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
36	35	rexbidva 2474	. . 3 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P) → (∃𝑥 ∈ N [⟨𝑧, 𝑤⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ ∃𝑥 ∈ N (𝑧 +P 1P)<P (𝑤 +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
37	31, 36	mpbird 167	. 2 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P) → ∃𝑥 ∈ N [⟨𝑧, 𝑤⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
38	1, 3, 37	ecoptocl 6622	1 ⊢ (𝐴 ∈ R → ∃𝑥 ∈ N 𝐴 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163  ∃wrex 2456  ⟨cop 3596   class class class wbr 4004  (class class class)co 5875  1_oc1o 6410  [cec 6533  Ncnpi 7271   ~_Q ceq 7278   <_Q cltq 7284  Pcnp 7290  1_Pc1p 7291   +_P cpp 7292  <_P cltp 7294   ~_R cer 7295  Rcnr 7296   <_R cltr 7302

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-i1p 7466  df-iplp 7467  df-iltp 7469  df-enr 7725  df-nr 7726  df-ltr 7729

This theorem is referenced by:  axarch  7890