Theorem 0reno 28238

Description: Surreal zero is a surreal real. (Contributed by Scott Fenton, 15-Apr-2025.)

Assertion

Ref	Expression
0reno	⊢ 0s ∈ ℝs

Proof of Theorem 0reno

Dummy variables 𝑥 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0sno 27772	. 2 ⊢ 0s ∈ No
2		1nns 28228	. . . 4 ⊢ 1s ∈ ℕs
3		0slt1s 27775	. . . . . . 7 ⊢ 0s <s 1s
4		1sno 27773	. . . . . . . 8 ⊢ 1s ∈ No
5		sltneg 27970	. . . . . . . 8 ⊢ (( 0s ∈ No ∧ 1s ∈ No ) → ( 0s <s 1s ↔ ( -us ‘ 1s ) <s ( -us ‘ 0s )))
6	1, 4, 5	mp2an 691	. . . . . . 7 ⊢ ( 0s <s 1s ↔ ( -us ‘ 1s ) <s ( -us ‘ 0s ))
7	3, 6	mpbi 229	. . . . . 6 ⊢ ( -us ‘ 1s ) <s ( -us ‘ 0s )
8		negs0s 27952	. . . . . 6 ⊢ ( -us ‘ 0s ) = 0s
9	7, 8	breqtri 5173	. . . . 5 ⊢ ( -us ‘ 1s ) <s 0s
10	9, 3	pm3.2i 470	. . . 4 ⊢ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )
11		fveq2 6897	. . . . . . 7 ⊢ (𝑛 = 1s → ( -us ‘𝑛) = ( -us ‘ 1s ))
12	11	breq1d 5158	. . . . . 6 ⊢ (𝑛 = 1s → (( -us ‘𝑛) <s 0s ↔ ( -us ‘ 1s ) <s 0s ))
13		breq2 5152	. . . . . 6 ⊢ (𝑛 = 1s → ( 0s <s 𝑛 ↔ 0s <s 1s ))
14	12, 13	anbi12d 631	. . . . 5 ⊢ (𝑛 = 1s → ((( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛) ↔ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )))
15	14	rspcev 3609	. . . 4 ⊢ (( 1s ∈ ℕs ∧ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛))
16	2, 10, 15	mp2an 691	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛)
17	4	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
18		nnsno 28209	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
19		nnne0s 28218	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
20	17, 18, 19	divscld 28135	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
21	20	negsval2d 27988	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) = ( 0s -s ( 1s /su 𝑛)))
22	21	eqeq2d 2739	. . . . . . . 8 ⊢ (𝑛 ∈ ℕs → (𝑥 = ( -us ‘( 1s /su 𝑛)) ↔ 𝑥 = ( 0s -s ( 1s /su 𝑛))))
23	22	bicomd 222	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → (𝑥 = ( 0s -s ( 1s /su 𝑛)) ↔ 𝑥 = ( -us ‘( 1s /su 𝑛))))
24	23	rexbiia 3089	. . . . . 6 ⊢ (∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)))
25	24	abbii 2798	. . . . 5 ⊢ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))}
26		addslid 27898	. . . . . . . . 9 ⊢ (( 1s /su 𝑛) ∈ No → ( 0s +s ( 1s /su 𝑛)) = ( 1s /su 𝑛))
27	20, 26	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕs → ( 0s +s ( 1s /su 𝑛)) = ( 1s /su 𝑛))
28	27	eqeq2d 2739	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → (𝑥 = ( 0s +s ( 1s /su 𝑛)) ↔ 𝑥 = ( 1s /su 𝑛)))
29	28	rexbiia 3089	. . . . . 6 ⊢ (∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛))
30	29	abbii 2798	. . . . 5 ⊢ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}
31	25, 30	oveq12i 7432	. . . 4 ⊢ ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛))}) = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)})
32		nnsex 28203	. . . . . . . . 9 ⊢ ℕs ∈ V
33	32	abrexex 7966	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∈ V
34	33	a1i 11	. . . . . . 7 ⊢ (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∈ V)
35		snex 5433	. . . . . . . 8 ⊢ { 0s } ∈ V
36	35	a1i 11	. . . . . . 7 ⊢ (⊤ → { 0s } ∈ V)
37	20	negscld 27962	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) ∈ No )
38		eleq1 2817	. . . . . . . . . . 11 ⊢ (𝑥 = ( -us ‘( 1s /su 𝑛)) → (𝑥 ∈ No ↔ ( -us ‘( 1s /su 𝑛)) ∈ No ))
39	37, 38	syl5ibrcom 246	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕs → (𝑥 = ( -us ‘( 1s /su 𝑛)) → 𝑥 ∈ No ))
40	39	rexlimiv 3145	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)) → 𝑥 ∈ No )
41	40	a1i 11	. . . . . . . 8 ⊢ (⊤ → (∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)) → 𝑥 ∈ No ))
42	41	abssdv 4063	. . . . . . 7 ⊢ (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ⊆ No )
43	1	a1i 11	. . . . . . . 8 ⊢ (⊤ → 0s ∈ No )
44	43	snssd 4813	. . . . . . 7 ⊢ (⊤ → { 0s } ⊆ No )
45		vex 3475	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
46		eqeq1 2732	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 = ( -us ‘( 1s /su 𝑛)) ↔ 𝑧 = ( -us ‘( 1s /su 𝑛))))
47	46	rexbidv 3175	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛))))
48	45, 47	elab 3667	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛)))
49		velsn 4645	. . . . . . . . . . 11 ⊢ (𝑦 ∈ { 0s } ↔ 𝑦 = 0s )
50	48, 49	anbi12i 627	. . . . . . . . . 10 ⊢ ((𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) ↔ (∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ))
51		r19.41v 3185	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ ℕs (𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ) ↔ (∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ))
52	50, 51	bitr4i 278	. . . . . . . . 9 ⊢ ((𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) ↔ ∃𝑛 ∈ ℕs (𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ))
53		muls02 28054	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ No → ( 0s ·s 𝑛) = 0s )
54	18, 53	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → ( 0s ·s 𝑛) = 0s )
55	54, 3	eqbrtrdi 5187	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → ( 0s ·s 𝑛) <s 1s )
56	1	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 0s ∈ No )
57		nnsgt0 28220	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 0s <s 𝑛)
58	56, 17, 18, 57	sltmuldivd 28140	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → (( 0s ·s 𝑛) <s 1s ↔ 0s <s ( 1s /su 𝑛)))
59	55, 58	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
60	20	slt0neg2d 27976	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → ( 0s <s ( 1s /su 𝑛) ↔ ( -us ‘( 1s /su 𝑛)) <s 0s ))
61	59, 60	mpbid 231	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) <s 0s )
62		breq12 5153	. . . . . . . . . . 11 ⊢ ((𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ) → (𝑧 <s 𝑦 ↔ ( -us ‘( 1s /su 𝑛)) <s 0s ))
63	61, 62	syl5ibrcom 246	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕs → ((𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ) → 𝑧 <s 𝑦))
64	63	rexlimiv 3145	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕs (𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ) → 𝑧 <s 𝑦)
65	52, 64	sylbi 216	. . . . . . . 8 ⊢ ((𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) → 𝑧 <s 𝑦)
66	65	3adant1 1128	. . . . . . 7 ⊢ ((⊤ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) → 𝑧 <s 𝑦)
67	34, 36, 42, 44, 66	ssltd 27737	. . . . . 6 ⊢ (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} <<s { 0s })
68	32	abrexex 7966	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ∈ V
69	68	a1i 11	. . . . . . 7 ⊢ (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ∈ V)
70		eleq1 2817	. . . . . . . . . . 11 ⊢ (𝑥 = ( 1s /su 𝑛) → (𝑥 ∈ No ↔ ( 1s /su 𝑛) ∈ No ))
71	20, 70	syl5ibrcom 246	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕs → (𝑥 = ( 1s /su 𝑛) → 𝑥 ∈ No ))
72	71	rexlimiv 3145	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛) → 𝑥 ∈ No )
73	72	a1i 11	. . . . . . . 8 ⊢ (⊤ → (∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛) → 𝑥 ∈ No ))
74	73	abssdv 4063	. . . . . . 7 ⊢ (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ⊆ No )
75		eqeq1 2732	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 = ( 1s /su 𝑛) ↔ 𝑧 = ( 1s /su 𝑛)))
76	75	rexbidv 3175	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛) ↔ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛)))
77	45, 76	elab 3667	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ↔ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛))
78	49, 77	anbi12i 627	. . . . . . . . . 10 ⊢ ((𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) ↔ (𝑦 = 0s ∧ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛)))
79		r19.42v 3187	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ ℕs (𝑦 = 0s ∧ 𝑧 = ( 1s /su 𝑛)) ↔ (𝑦 = 0s ∧ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛)))
80	78, 79	bitr4i 278	. . . . . . . . 9 ⊢ ((𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) ↔ ∃𝑛 ∈ ℕs (𝑦 = 0s ∧ 𝑧 = ( 1s /su 𝑛)))
81		breq12 5153	. . . . . . . . . . . 12 ⊢ ((𝑦 = 0s ∧ 𝑧 = ( 1s /su 𝑛)) → (𝑦 <s 𝑧 ↔ 0s <s ( 1s /su 𝑛)))
82	59, 81	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ((𝑦 = 0s ∧ 𝑧 = ( 1s /su 𝑛)) → 𝑦 <s 𝑧))
83	82	rexlimiv 3145	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ ℕs (𝑦 = 0s ∧ 𝑧 = ( 1s /su 𝑛)) → 𝑦 <s 𝑧)
84	83	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (∃𝑛 ∈ ℕs (𝑦 = 0s ∧ 𝑧 = ( 1s /su 𝑛)) → 𝑦 <s 𝑧))
85	80, 84	biimtrid 241	. . . . . . . 8 ⊢ (⊤ → ((𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) → 𝑦 <s 𝑧))
86	85	3impib 1114	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) → 𝑦 <s 𝑧)
87	36, 69, 44, 74, 86	ssltd 27737	. . . . . 6 ⊢ (⊤ → { 0s } <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)})
88	67, 87	cuteq0 27778	. . . . 5 ⊢ (⊤ → ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) = 0s )
89	88	mptru 1541	. . . 4 ⊢ ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) = 0s
90	31, 89	eqtr2i 2757	. . 3 ⊢ 0s = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛))})
91	16, 90	pm3.2i 470	. 2 ⊢ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛) ∧ 0s = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛))}))
92		elreno 28236	. 2 ⊢ ( 0s ∈ ℝs ↔ ( 0s ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛) ∧ 0s = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛))}))))
93	1, 91, 92	mpbir2an 710	1 ⊢ 0s ∈ ℝs

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534  ⊤wtru 1535   ∈ wcel 2099  {cab 2705  ∃wrex 3067  Vcvv 3471  {csn 4629   class class class wbr 5148  ‘cfv 6548  (class class class)co 7420   No csur 27586   <s cslt 27587   |s cscut 27728   0_s c0s 27768   1_s c1s 27769   +_s cadds 27889   -u_s cnegs 27945   -_s csubs 27946   ·_s cmuls 28019   /_su cdivs 28100  ℕ_scnns 28199  ℝ_screno 28234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-dc 10470

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-nadd 8687  df-no 27589  df-slt 27590  df-bday 27591  df-sle 27691  df-sslt 27727  df-scut 27729  df-0s 27770  df-1s 27771  df-made 27787  df-old 27788  df-left 27790  df-right 27791  df-norec 27868  df-norec2 27879  df-adds 27890  df-negs 27947  df-subs 27948  df-muls 28020  df-divs 28101  df-n0s 28200  df-nns 28201  df-reno 28235

This theorem is referenced by: (None)