Theorem archsr 8155

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 8100	. 2 ⊢ R = ((P × P) / ~R )
2		breq1 4134	. . 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 2551	. 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 7927	. . . . . . 7 ⊢ 1P ∈ P
5		addclpr 7910	. . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 1P ∈ P) → (𝑧 +P 1P) ∈ P)
6	4, 5	mpan2 429	. . . . . 6 ⊢ (𝑧 ∈ P → (𝑧 +P 1P) ∈ P)
7		archpr 8016	. . . . . 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 7926	. . . . . . . . . 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 7910	. . . . . . . . 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 417	. . . . . . . 8 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
14		simplr 533	. . . . . . . 8 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑥 ∈ N) → 𝑤 ∈ P)
15		ltaddpr 7970	. . . . . . . 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 415	. . . . . . 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 7951	. . . . . . . 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 415	. . . . . . 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 4159	. . . . . 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 7970	. . . . . . . 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 417	. . . . . . 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 7969	. . . . . . . . 9 ⊢ <P Or P
23		ltrelpr 7878	. . . . . . . . 9 ⊢ <P ⊆ (P × P)
24	22, 23	sotri 5186	. . . . . . . 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 5186	. . . . . . 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 2652	. . . 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 8120	. . . . 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 1276	. . . 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 2547	. . 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 6900	1 ⊢ (𝐴 ∈ R → ∃𝑥 ∈ N 𝐴 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529  ⟨cop 3712   class class class wbr 4131  (class class class)co 6088  1_oc1o 6684  [cec 6809  Ncnpi 7645   ~_Q ceq 7652   <_Q cltq 7658  Pcnp 7664  1_Pc1p 7665   +_P cpp 7666  <_P cltp 7668   ~_R cer 7669  Rcnr 7670   <_R cltr 7676

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-i1p 7840  df-iplp 7841  df-iltp 7843  df-enr 8099  df-nr 8100  df-ltr 8103

This theorem is used by:  axarch  8264