Theorem n0subs 28742

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 5107	. . . . . 6 ⊢ (𝑥 = 0s → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 0s ))
2		oveq1 7425	. . . . . . 7 ⊢ (𝑥 = 0s → (𝑥 -s 𝑧) = ( 0s -s 𝑧))
3	2	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = 0s → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ ( 0s -s 𝑧) ∈ ℕ0s))
4	1, 3	imbi12d 347	. . . . 5 ⊢ (𝑥 = 0s → ((𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s)))
5	4	ralbidv 3186	. . . 4 ⊢ (𝑥 = 0s → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s)))
6		breq2 5107	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 𝑦))
7		oveq1 7425	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 -s 𝑧) = (𝑦 -s 𝑧))
8	7	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ (𝑦 -s 𝑧) ∈ ℕ0s))
9	6, 8	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s)))
10	9	ralbidv 3186	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s)))
11		breq2 5107	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s (𝑦 +s 1s )))
12		oveq1 7425	. . . . . . 7 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 𝑧) = ((𝑦 +s 1s ) -s 𝑧))
13	12	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))
14	11, 13	imbi12d 347	. . . . 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 3186	. . . 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 5107	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 𝑁))
17		oveq1 7425	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 -s 𝑧) = (𝑁 -s 𝑧))
18	17	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ (𝑁 -s 𝑧) ∈ ℕ0s))
19	16, 18	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s)))
20	19	ralbidv 3186	. . . 4 ⊢ (𝑥 = 𝑁 → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s)))
21		n0sge0 28717	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → 0s ≤s 𝑧)
22	21	biantrud 541	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
23		n0no 28702	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → 𝑧 ∈ No )
24		0no 28188	. . . . . . . 8 ⊢ 0s ∈ No
25		lestri3 28105	. . . . . . . 8 ⊢ ((𝑧 ∈ No ∧ 0s ∈ No ) → (𝑧 = 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
26	23, 24, 25	sylancl 598	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → (𝑧 = 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
27	22, 26	bitr4d 285	. . . . . 6 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s ↔ 𝑧 = 0s ))
28		oveq2 7426	. . . . . . 7 ⊢ (𝑧 = 0s → ( 0s -s 𝑧) = ( 0s -s 0s ))
29		subsid 28448	. . . . . . . . 9 ⊢ ( 0s ∈ No → ( 0s -s 0s ) = 0s )
30	24, 29	ax-mp 5	. . . . . . . 8 ⊢ ( 0s -s 0s ) = 0s
31		0n0s 28708	. . . . . . . 8 ⊢ 0s ∈ ℕ0s
32	30, 31	eqeltri 2857	. . . . . . 7 ⊢ ( 0s -s 0s ) ∈ ℕ0s
33	28, 32	eqeltrdi 2869	. . . . . 6 ⊢ (𝑧 = 0s → ( 0s -s 𝑧) ∈ ℕ0s)
34	27, 33	biimtrdi 256	. . . . 5 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s))
35	34	rgen 3079	. . . 4 ⊢ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s)
36		breq1 5106	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ≤s 𝑦 ↔ 𝑥 ≤s 𝑦))
37		oveq2 7426	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 -s 𝑧) = (𝑦 -s 𝑥))
38	37	eleq1d 2846	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑦 -s 𝑧) ∈ ℕ0s ↔ (𝑦 -s 𝑥) ∈ ℕ0s))
39	36, 38	imbi12d 347	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s) ↔ (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s)))
40	39	cbvralvw 3241	. . . . 5 ⊢ (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s))
41		n0no 28702	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No )
42		peano2no 28363	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ No → (𝑦 +s 1s ) ∈ No )
43		subsid1 28447	. . . . . . . . . . . 12 ⊢ ((𝑦 +s 1s ) ∈ No → ((𝑦 +s 1s ) -s 0s ) = (𝑦 +s 1s ))
44	41, 42, 43	3syl 19	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 0s ) = (𝑦 +s 1s ))
45		peano2n0s 28709	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
46	44, 45	eqeltrd 2861	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 0s ) ∈ ℕ0s)
47		oveq2 7426	. . . . . . . . . . 11 ⊢ (𝑧 = 0s → ((𝑦 +s 1s ) -s 𝑧) = ((𝑦 +s 1s ) -s 0s ))
48	47	eleq1d 2846	. . . . . . . . . 10 ⊢ (𝑧 = 0s → (((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 0s ) ∈ ℕ0s))
49	46, 48	syl5ibrcom 250	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → (𝑧 = 0s → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))
50	49	2a1dd 52	. . . . . . . 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 486	. . . . . . 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 5106	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 -s 1s ) → (𝑥 ≤s 𝑦 ↔ (𝑧 -s 1s ) ≤s 𝑦))
53		oveq2 7426	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑧 -s 1s ) → (𝑦 -s 𝑥) = (𝑦 -s (𝑧 -s 1s )))
54	53	eleq1d 2846	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 -s 1s ) → ((𝑦 -s 𝑥) ∈ ℕ0s ↔ (𝑦 -s (𝑧 -s 1s )) ∈ ℕ0s))
55	52, 54	imbi12d 347	. . . . . . . . 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 487	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 𝑧 ∈ No )
58		1no 28189	. . . . . . . . . . . 12 ⊢ 1s ∈ No
59	58	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 1s ∈ No )
60	41	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 𝑦 ∈ No )
61	57, 59, 60	lesubaddsd 28472	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑧 -s 1s ) ≤s 𝑦 ↔ 𝑧 ≤s (𝑦 +s 1s )))
62	60, 57, 59	subsubs2d 28474	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 -s (𝑧 -s 1s )) = (𝑦 +s ( 1s -s 𝑧)))
63	60, 59, 57	addsubsassd 28460	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑦 +s 1s ) -s 𝑧) = (𝑦 +s ( 1s -s 𝑧)))
64	62, 63	eqtr4d 2799	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 -s (𝑧 -s 1s )) = ((𝑦 +s 1s ) -s 𝑧))
65	64	eleq1d 2846	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑦 -s (𝑧 -s 1s )) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))
66	61, 65	imbi12d 347	. . . . . . . . 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 232	. . . . . . . 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 79	. . . . . . 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 28741	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → (𝑧 = 0s ∨ (𝑧 -s 1s ) ∈ ℕ0s))
70	69	adantl 487	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑧 = 0s ∨ (𝑧 -s 1s ) ∈ ℕ0s))
71	51, 68, 70	mpjaod 874	. . . . . 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 3163	. . . . 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 245	. . . 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 28712	. . 3 ⊢ (𝑁 ∈ ℕ0s → ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s))
75		breq1 5106	. . . . 5 ⊢ (𝑧 = 𝑀 → (𝑧 ≤s 𝑁 ↔ 𝑀 ≤s 𝑁))
76		oveq2 7426	. . . . . 6 ⊢ (𝑧 = 𝑀 → (𝑁 -s 𝑧) = (𝑁 -s 𝑀))
77	76	eleq1d 2846	. . . . 5 ⊢ (𝑧 = 𝑀 → ((𝑁 -s 𝑧) ∈ ℕ0s ↔ (𝑁 -s 𝑀) ∈ ℕ0s))
78	75, 77	imbi12d 347	. . . 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 605	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 → (𝑁 -s 𝑀) ∈ ℕ0s))
81		n0sge0 28717	. . 3 ⊢ ((𝑁 -s 𝑀) ∈ ℕ0s → 0s ≤s (𝑁 -s 𝑀))
82		n0no 28702	. . . . 5 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ No )
83	82	adantl 487	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ No )
84		n0no 28702	. . . . 5 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
85	84	adantr 486	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑀 ∈ No )
86	83, 85	subsge0d 28479	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ( 0s ≤s (𝑁 -s 𝑀) ↔ 𝑀 ≤s 𝑁))
87	81, 86	imbitrid 247	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑁 -s 𝑀) ∈ ℕ0s → 𝑀 ≤s 𝑁))
88	80, 87	impbid 215	1 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕ0s))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3077   class class class wbr 5103  (class class class)co 7418   No csur 27990   ≤s cles 28094   0_s c0s 28184   1_s c1s 28185   +_s cadds 28338   -_s csubs 28399  ℕ_0scn0s 28691

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-nadd 8668  df-no 27993  df-lts 27994  df-bday 27995  df-les 28095  df-slts 28137  df-cuts 28139  df-0s 28186  df-1s 28187  df-made 28206  df-old 28207  df-left 28209  df-right 28210  df-norec 28317  df-norec2 28328  df-adds 28339  df-negs 28400  df-subs 28401  df-n0s 28693

This theorem is used by:  n0subs2  28743  elzn0s  28777  eln0zs  28779  bdayfinbndlem1  28846