Theorem 0reno 28324

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 27714	. 2 ⊢ 0s ∈ No
2		1nns 28217	. . . 4 ⊢ 1s ∈ ℕs
3		0slt1s 27717	. . . . . . 7 ⊢ 0s <s 1s
4		1sno 27715	. . . . . . . 8 ⊢ 1s ∈ No
5		sltneg 27927	. . . . . . . 8 ⊢ (( 0s ∈ No ∧ 1s ∈ No ) → ( 0s <s 1s ↔ ( -us ‘ 1s ) <s ( -us ‘ 0s )))
6	1, 4, 5	mp2an 692	. . . . . . 7 ⊢ ( 0s <s 1s ↔ ( -us ‘ 1s ) <s ( -us ‘ 0s ))
7	3, 6	mpbi 230	. . . . . 6 ⊢ ( -us ‘ 1s ) <s ( -us ‘ 0s )
8		negs0s 27908	. . . . . 6 ⊢ ( -us ‘ 0s ) = 0s
9	7, 8	breqtri 5127	. . . . 5 ⊢ ( -us ‘ 1s ) <s 0s
10	9, 3	pm3.2i 470	. . . 4 ⊢ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )
11		fveq2 6840	. . . . . . 7 ⊢ (𝑛 = 1s → ( -us ‘𝑛) = ( -us ‘ 1s ))
12	11	breq1d 5112	. . . . . 6 ⊢ (𝑛 = 1s → (( -us ‘𝑛) <s 0s ↔ ( -us ‘ 1s ) <s 0s ))
13		breq2 5106	. . . . . 6 ⊢ (𝑛 = 1s → ( 0s <s 𝑛 ↔ 0s <s 1s ))
14	12, 13	anbi12d 632	. . . . 5 ⊢ (𝑛 = 1s → ((( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛) ↔ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )))
15	14	rspcev 3585	. . . 4 ⊢ (( 1s ∈ ℕs ∧ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛))
16	2, 10, 15	mp2an 692	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛)
17	4	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
18		nnsno 28193	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
19		nnne0s 28205	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
20	17, 18, 19	divscld 28102	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
21	20	negsval2d 27947	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) = ( 0s -s ( 1s /su 𝑛)))
22	21	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑛 ∈ ℕs → (𝑥 = ( -us ‘( 1s /su 𝑛)) ↔ 𝑥 = ( 0s -s ( 1s /su 𝑛))))
23	22	bicomd 223	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → (𝑥 = ( 0s -s ( 1s /su 𝑛)) ↔ 𝑥 = ( -us ‘( 1s /su 𝑛))))
24	23	rexbiia 3074	. . . . . 6 ⊢ (∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)))
25	24	abbii 2796	. . . . 5 ⊢ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))}
26		addslid 27851	. . . . . . . . 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 2740	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → (𝑥 = ( 0s +s ( 1s /su 𝑛)) ↔ 𝑥 = ( 1s /su 𝑛)))
29	28	rexbiia 3074	. . . . . 6 ⊢ (∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛))
30	29	abbii 2796	. . . . 5 ⊢ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}
31	25, 30	oveq12i 7381	. . . 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 28187	. . . . . . . . 9 ⊢ ℕs ∈ V
33	32	abrexex 7920	. . . . . . . 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 5386	. . . . . . . 8 ⊢ { 0s } ∈ V
36	35	a1i 11	. . . . . . 7 ⊢ (⊤ → { 0s } ∈ V)
37	20	negscld 27919	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) ∈ No )
38		eleq1 2816	. . . . . . . . . . 11 ⊢ (𝑥 = ( -us ‘( 1s /su 𝑛)) → (𝑥 ∈ No ↔ ( -us ‘( 1s /su 𝑛)) ∈ No ))
39	37, 38	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕs → (𝑥 = ( -us ‘( 1s /su 𝑛)) → 𝑥 ∈ No ))
40	39	rexlimiv 3127	. . . . . . . . 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 4028	. . . . . . 7 ⊢ (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ⊆ No )
43	1	a1i 11	. . . . . . . 8 ⊢ (⊤ → 0s ∈ No )
44	43	snssd 4769	. . . . . . 7 ⊢ (⊤ → { 0s } ⊆ No )
45		vex 3448	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
46		eqeq1 2733	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 = ( -us ‘( 1s /su 𝑛)) ↔ 𝑧 = ( -us ‘( 1s /su 𝑛))))
47	46	rexbidv 3157	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛))))
48	45, 47	elab 3643	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛)))
49		velsn 4601	. . . . . . . . . . 11 ⊢ (𝑦 ∈ { 0s } ↔ 𝑦 = 0s )
50	48, 49	anbi12i 628	. . . . . . . . . 10 ⊢ ((𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) ↔ (∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ))
51		r19.41v 3165	. . . . . . . . . 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 28020	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ No → ( 0s ·s 𝑛) = 0s )
54	18, 53	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → ( 0s ·s 𝑛) = 0s )
55	54, 3	eqbrtrdi 5141	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → ( 0s ·s 𝑛) <s 1s )
56	1	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 0s ∈ No )
57		nnsgt0 28207	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 0s <s 𝑛)
58	56, 17, 18, 57	sltmuldivd 28107	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → (( 0s ·s 𝑛) <s 1s ↔ 0s <s ( 1s /su 𝑛)))
59	55, 58	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
60	20	slt0neg2d 27933	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → ( 0s <s ( 1s /su 𝑛) ↔ ( -us ‘( 1s /su 𝑛)) <s 0s ))
61	59, 60	mpbid 232	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) <s 0s )
62		breq12 5107	. . . . . . . . . . 11 ⊢ ((𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ) → (𝑧 <s 𝑦 ↔ ( -us ‘( 1s /su 𝑛)) <s 0s ))
63	61, 62	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕs → ((𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ) → 𝑧 <s 𝑦))
64	63	rexlimiv 3127	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕs (𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ) → 𝑧 <s 𝑦)
65	52, 64	sylbi 217	. . . . . . . 8 ⊢ ((𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) → 𝑧 <s 𝑦)
66	65	3adant1 1130	. . . . . . 7 ⊢ ((⊤ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) → 𝑧 <s 𝑦)
67	34, 36, 42, 44, 66	ssltd 27679	. . . . . 6 ⊢ (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} <<s { 0s })
68	32	abrexex 7920	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ∈ V
69	68	a1i 11	. . . . . . 7 ⊢ (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ∈ V)
70		eleq1 2816	. . . . . . . . . . 11 ⊢ (𝑥 = ( 1s /su 𝑛) → (𝑥 ∈ No ↔ ( 1s /su 𝑛) ∈ No ))
71	20, 70	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕs → (𝑥 = ( 1s /su 𝑛) → 𝑥 ∈ No ))
72	71	rexlimiv 3127	. . . . . . . . 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 4028	. . . . . . 7 ⊢ (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ⊆ No )
75		eqeq1 2733	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 = ( 1s /su 𝑛) ↔ 𝑧 = ( 1s /su 𝑛)))
76	75	rexbidv 3157	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛) ↔ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛)))
77	45, 76	elab 3643	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ↔ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛))
78	49, 77	anbi12i 628	. . . . . . . . . 10 ⊢ ((𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) ↔ (𝑦 = 0s ∧ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛)))
79		r19.42v 3167	. . . . . . . . . 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 5107	. . . . . . . . . . . 12 ⊢ ((𝑦 = 0s ∧ 𝑧 = ( 1s /su 𝑛)) → (𝑦 <s 𝑧 ↔ 0s <s ( 1s /su 𝑛)))
82	59, 81	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ((𝑦 = 0s ∧ 𝑧 = ( 1s /su 𝑛)) → 𝑦 <s 𝑧))
83	82	rexlimiv 3127	. . . . . . . . . 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 242	. . . . . . . 8 ⊢ (⊤ → ((𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) → 𝑦 <s 𝑧))
86	85	3impib 1116	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) → 𝑦 <s 𝑧)
87	36, 69, 44, 74, 86	ssltd 27679	. . . . . 6 ⊢ (⊤ → { 0s } <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)})
88	67, 87	cuteq0 27720	. . . . 5 ⊢ (⊤ → ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) = 0s )
89	88	mptru 1547	. . . 4 ⊢ ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) = 0s
90	31, 89	eqtr2i 2753	. . 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 28322	. 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 711	1 ⊢ 0s ∈ ℝs

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3444  {csn 4585   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369   No csur 27527   <s cslt 27528   |s cscut 27670   0_s c0s 27710   1_s c1s 27711   +_s cadds 27842   -u_s cnegs 27901   -_s csubs 27902   ·_s cmuls 27985   /_su cdivs 28066  ℕ_scnns 28183  ℝ_screno 28320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-dc 10375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-nadd 8607  df-no 27530  df-slt 27531  df-bday 27532  df-sle 27633  df-sslt 27669  df-scut 27671  df-0s 27712  df-1s 27713  df-made 27731  df-old 27732  df-left 27734  df-right 27735  df-norec 27821  df-norec2 27832  df-adds 27843  df-negs 27903  df-subs 27904  df-muls 27986  df-divs 28067  df-n0s 28184  df-nns 28185  df-reno 28321

This theorem is referenced by: (None)