Theorem n0subs 28258

Description: Subtraction of non-negative surreal integers. (Contributed by Scott Fenton, 26-May-2025.)

Assertion

Ref	Expression
n0subs	⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕ0s))

Proof of Theorem n0subs

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

Step	Hyp	Ref	Expression
1		breq2 5096	. . . . . 6 ⊢ (𝑥 = 0s → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 0s ))
2		oveq1 7356	. . . . . . 7 ⊢ (𝑥 = 0s → (𝑥 -s 𝑧) = ( 0s -s 𝑧))
3	2	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = 0s → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ ( 0s -s 𝑧) ∈ ℕ0s))
4	1, 3	imbi12d 344	. . . . 5 ⊢ (𝑥 = 0s → ((𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s)))
5	4	ralbidv 3152	. . . 4 ⊢ (𝑥 = 0s → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s)))
6		breq2 5096	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 𝑦))
7		oveq1 7356	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 -s 𝑧) = (𝑦 -s 𝑧))
8	7	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ (𝑦 -s 𝑧) ∈ ℕ0s))
9	6, 8	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s)))
10	9	ralbidv 3152	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s)))
11		breq2 5096	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s (𝑦 +s 1s )))
12		oveq1 7356	. . . . . . 7 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 𝑧) = ((𝑦 +s 1s ) -s 𝑧))
13	12	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))
14	11, 13	imbi12d 344	. . . . 5 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s)))
15	14	ralbidv 3152	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s)))
16		breq2 5096	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 𝑁))
17		oveq1 7356	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 -s 𝑧) = (𝑁 -s 𝑧))
18	17	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ (𝑁 -s 𝑧) ∈ ℕ0s))
19	16, 18	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s)))
20	19	ralbidv 3152	. . . 4 ⊢ (𝑥 = 𝑁 → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s)))
21		n0sge0 28235	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → 0s ≤s 𝑧)
22	21	biantrud 531	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
23		n0sno 28221	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → 𝑧 ∈ No )
24		0sno 27740	. . . . . . . 8 ⊢ 0s ∈ No
25		sletri3 27665	. . . . . . . 8 ⊢ ((𝑧 ∈ No ∧ 0s ∈ No ) → (𝑧 = 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
26	23, 24, 25	sylancl 586	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → (𝑧 = 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
27	22, 26	bitr4d 282	. . . . . 6 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s ↔ 𝑧 = 0s ))
28		oveq2 7357	. . . . . . 7 ⊢ (𝑧 = 0s → ( 0s -s 𝑧) = ( 0s -s 0s ))
29		subsid 27978	. . . . . . . . 9 ⊢ ( 0s ∈ No → ( 0s -s 0s ) = 0s )
30	24, 29	ax-mp 5	. . . . . . . 8 ⊢ ( 0s -s 0s ) = 0s
31		0n0s 28227	. . . . . . . 8 ⊢ 0s ∈ ℕ0s
32	30, 31	eqeltri 2824	. . . . . . 7 ⊢ ( 0s -s 0s ) ∈ ℕ0s
33	28, 32	eqeltrdi 2836	. . . . . 6 ⊢ (𝑧 = 0s → ( 0s -s 𝑧) ∈ ℕ0s)
34	27, 33	biimtrdi 253	. . . . 5 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s))
35	34	rgen 3046	. . . 4 ⊢ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s)
36		breq1 5095	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ≤s 𝑦 ↔ 𝑥 ≤s 𝑦))
37		oveq2 7357	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 -s 𝑧) = (𝑦 -s 𝑥))
38	37	eleq1d 2813	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑦 -s 𝑧) ∈ ℕ0s ↔ (𝑦 -s 𝑥) ∈ ℕ0s))
39	36, 38	imbi12d 344	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s) ↔ (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s)))
40	39	cbvralvw 3207	. . . . 5 ⊢ (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s))
41		n0sno 28221	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No )
42		peano2no 27896	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ No → (𝑦 +s 1s ) ∈ No )
43		subsid1 27977	. . . . . . . . . . . 12 ⊢ ((𝑦 +s 1s ) ∈ No → ((𝑦 +s 1s ) -s 0s ) = (𝑦 +s 1s ))
44	41, 42, 43	3syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 0s ) = (𝑦 +s 1s ))
45		peano2n0s 28228	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
46	44, 45	eqeltrd 2828	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 0s ) ∈ ℕ0s)
47		oveq2 7357	. . . . . . . . . . 11 ⊢ (𝑧 = 0s → ((𝑦 +s 1s ) -s 𝑧) = ((𝑦 +s 1s ) -s 0s ))
48	47	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑧 = 0s → (((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 0s ) ∈ ℕ0s))
49	46, 48	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → (𝑧 = 0s → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))
50	49	2a1dd 51	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ0s → (𝑧 = 0s → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s) → (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))))
51	50	adantr 480	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑧 = 0s → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s) → (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))))
52		breq1 5095	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 -s 1s ) → (𝑥 ≤s 𝑦 ↔ (𝑧 -s 1s ) ≤s 𝑦))
53		oveq2 7357	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑧 -s 1s ) → (𝑦 -s 𝑥) = (𝑦 -s (𝑧 -s 1s )))
54	53	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 -s 1s ) → ((𝑦 -s 𝑥) ∈ ℕ0s ↔ (𝑦 -s (𝑧 -s 1s )) ∈ ℕ0s))
55	52, 54	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = (𝑧 -s 1s ) → ((𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s) ↔ ((𝑧 -s 1s ) ≤s 𝑦 → (𝑦 -s (𝑧 -s 1s )) ∈ ℕ0s)))
56	55	rspcv 3573	. . . . . . . 8 ⊢ ((𝑧 -s 1s ) ∈ ℕ0s → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s) → ((𝑧 -s 1s ) ≤s 𝑦 → (𝑦 -s (𝑧 -s 1s )) ∈ ℕ0s)))
57	23	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 𝑧 ∈ No )
58		1sno 27741	. . . . . . . . . . . 12 ⊢ 1s ∈ No
59	58	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 1s ∈ No )
60	41	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 𝑦 ∈ No )
61	57, 59, 60	slesubaddd 28002	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑧 -s 1s ) ≤s 𝑦 ↔ 𝑧 ≤s (𝑦 +s 1s )))
62	60, 57, 59	subsubs2d 28004	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 -s (𝑧 -s 1s )) = (𝑦 +s ( 1s -s 𝑧)))
63	60, 59, 57	addsubsassd 27990	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑦 +s 1s ) -s 𝑧) = (𝑦 +s ( 1s -s 𝑧)))
64	62, 63	eqtr4d 2767	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 -s (𝑧 -s 1s )) = ((𝑦 +s 1s ) -s 𝑧))
65	64	eleq1d 2813	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑦 -s (𝑧 -s 1s )) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))
66	61, 65	imbi12d 344	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (((𝑧 -s 1s ) ≤s 𝑦 → (𝑦 -s (𝑧 -s 1s )) ∈ ℕ0s) ↔ (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s)))
67	66	biimpd 229	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (((𝑧 -s 1s ) ≤s 𝑦 → (𝑦 -s (𝑧 -s 1s )) ∈ ℕ0s) → (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s)))
68	56, 67	syl9r 78	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑧 -s 1s ) ∈ ℕ0s → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s) → (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))))
69		n0s0m1 28257	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → (𝑧 = 0s ∨ (𝑧 -s 1s ) ∈ ℕ0s))
70	69	adantl 481	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑧 = 0s ∨ (𝑧 -s 1s ) ∈ ℕ0s))
71	51, 68, 70	mpjaod 860	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s) → (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s)))
72	71	ralrimdva 3129	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s) → ∀𝑧 ∈ ℕ0s (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s)))
73	40, 72	biimtrid 242	. . . 4 ⊢ (𝑦 ∈ ℕ0s → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s) → ∀𝑧 ∈ ℕ0s (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s)))
74	5, 10, 15, 20, 35, 73	n0sind 28230	. . 3 ⊢ (𝑁 ∈ ℕ0s → ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s))
75		breq1 5095	. . . . 5 ⊢ (𝑧 = 𝑀 → (𝑧 ≤s 𝑁 ↔ 𝑀 ≤s 𝑁))
76		oveq2 7357	. . . . . 6 ⊢ (𝑧 = 𝑀 → (𝑁 -s 𝑧) = (𝑁 -s 𝑀))
77	76	eleq1d 2813	. . . . 5 ⊢ (𝑧 = 𝑀 → ((𝑁 -s 𝑧) ∈ ℕ0s ↔ (𝑁 -s 𝑀) ∈ ℕ0s))
78	75, 77	imbi12d 344	. . . 4 ⊢ (𝑧 = 𝑀 → ((𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s) ↔ (𝑀 ≤s 𝑁 → (𝑁 -s 𝑀) ∈ ℕ0s)))
79	78	rspcva 3575	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s)) → (𝑀 ≤s 𝑁 → (𝑁 -s 𝑀) ∈ ℕ0s))
80	74, 79	sylan2 593	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 → (𝑁 -s 𝑀) ∈ ℕ0s))
81		n0sge0 28235	. . 3 ⊢ ((𝑁 -s 𝑀) ∈ ℕ0s → 0s ≤s (𝑁 -s 𝑀))
82		n0sno 28221	. . . . 5 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ No )
83	82	adantl 481	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ No )
84		n0sno 28221	. . . . 5 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
85	84	adantr 480	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑀 ∈ No )
86	83, 85	subsge0d 28008	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ( 0s ≤s (𝑁 -s 𝑀) ↔ 𝑀 ≤s 𝑁))
87	81, 86	imbitrid 244	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑁 -s 𝑀) ∈ ℕ0s → 𝑀 ≤s 𝑁))
88	80, 87	impbid 212	1 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕ0s))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5092  (class class class)co 7349   No csur 27549   ≤s csle 27654   0_s c0s 27736   1_s c1s 27737   +_s cadds 27871   -_s csubs 27931  ℕ_0scnn0s 28211

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-nadd 8584  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694  df-0s 27738  df-1s 27739  df-made 27757  df-old 27758  df-left 27760  df-right 27761  df-norec 27850  df-norec2 27861  df-adds 27872  df-negs 27932  df-subs 27933  df-n0s 28213

This theorem is referenced by:  n0subs2  28259  elzn0s  28291  eln0zs  28293