Theorem n0cut 28607

Description: A cut form for non-negative surreal integers. (Contributed by Scott Fenton, 2-Apr-2025.)

Assertion

Ref	Expression
n0cut	⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))

Proof of Theorem n0cut

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

Step	Hyp	Ref	Expression
1		id 23	. . 3 ⊢ (𝑦 = 0s → 𝑦 = 0s )
2		oveq1 7424	. . . . 5 ⊢ (𝑦 = 0s → (𝑦 -s 1s ) = ( 0s -s 1s ))
3	2	sneqd 4599	. . . 4 ⊢ (𝑦 = 0s → {(𝑦 -s 1s )} = {( 0s -s 1s )})
4	3	oveq1d 7432	. . 3 ⊢ (𝑦 = 0s → ({(𝑦 -s 1s )} |s ∅) = ({( 0s -s 1s )} |s ∅))
5	1, 4	eqeq12d 2778	. 2 ⊢ (𝑦 = 0s → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ 0s = ({( 0s -s 1s )} |s ∅)))
6		id 23	. . 3 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
7		oveq1 7424	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 -s 1s ) = (𝑥 -s 1s ))
8	7	sneqd 4599	. . . 4 ⊢ (𝑦 = 𝑥 → {(𝑦 -s 1s )} = {(𝑥 -s 1s )})
9	8	oveq1d 7432	. . 3 ⊢ (𝑦 = 𝑥 → ({(𝑦 -s 1s )} |s ∅) = ({(𝑥 -s 1s )} |s ∅))
10	6, 9	eqeq12d 2778	. 2 ⊢ (𝑦 = 𝑥 → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ 𝑥 = ({(𝑥 -s 1s )} |s ∅)))
11		id 23	. . 3 ⊢ (𝑦 = (𝑥 +s 1s ) → 𝑦 = (𝑥 +s 1s ))
12		oveq1 7424	. . . . 5 ⊢ (𝑦 = (𝑥 +s 1s ) → (𝑦 -s 1s ) = ((𝑥 +s 1s ) -s 1s ))
13	12	sneqd 4599	. . . 4 ⊢ (𝑦 = (𝑥 +s 1s ) → {(𝑦 -s 1s )} = {((𝑥 +s 1s ) -s 1s )})
14	13	oveq1d 7432	. . 3 ⊢ (𝑦 = (𝑥 +s 1s ) → ({(𝑦 -s 1s )} |s ∅) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
15	11, 14	eqeq12d 2778	. 2 ⊢ (𝑦 = (𝑥 +s 1s ) → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅)))
16		id 23	. . 3 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
17		oveq1 7424	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 -s 1s ) = (𝐴 -s 1s ))
18	17	sneqd 4599	. . . 4 ⊢ (𝑦 = 𝐴 → {(𝑦 -s 1s )} = {(𝐴 -s 1s )})
19	18	oveq1d 7432	. . 3 ⊢ (𝑦 = 𝐴 → ({(𝑦 -s 1s )} |s ∅) = ({(𝐴 -s 1s )} |s ∅))
20	16, 19	eqeq12d 2778	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ 𝐴 = ({(𝐴 -s 1s )} |s ∅)))
21		0no 28082	. . . . . . 7 ⊢ 0s ∈ No
22		1no 28083	. . . . . . 7 ⊢ 1s ∈ No
23		subscl 28335	. . . . . . 7 ⊢ (( 0s ∈ No ∧ 1s ∈ No ) → ( 0s -s 1s ) ∈ No )
24	21, 22, 23	mp2an 705	. . . . . 6 ⊢ ( 0s -s 1s ) ∈ No
25	24	a1i 11	. . . . 5 ⊢ (⊤ → ( 0s -s 1s ) ∈ No )
26	21	a1i 11	. . . . . 6 ⊢ (⊤ → 0s ∈ No )
27	26	ltsm1d 28375	. . . . 5 ⊢ (⊤ → ( 0s -s 1s ) <s 0s )
28	25, 27	cutneg 28089	. . . 4 ⊢ (⊤ → ({( 0s -s 1s )} |s ∅) = 0s )
29	28	mptru 1577	. . 3 ⊢ ({( 0s -s 1s )} |s ∅) = 0s
30	29	eqcomi 2771	. 2 ⊢ 0s = ({( 0s -s 1s )} |s ∅)
31		ovex 7450	. . . . . . . . . . 11 ⊢ (𝑥 -s 1s ) ∈ V
32		oveq1 7424	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑥 -s 1s ) → (𝑏 +s 1s ) = ((𝑥 -s 1s ) +s 1s ))
33	32	eqeq2d 2773	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑥 -s 1s ) → (𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = ((𝑥 -s 1s ) +s 1s )))
34	31, 33	rexsn 4646	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = ((𝑥 -s 1s ) +s 1s ))
35		n0no 28596	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
36		npcans 28348	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
37	35, 22, 36	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ0s → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
38	37	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
39	38	eqeq2d 2773	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑎 = ((𝑥 -s 1s ) +s 1s ) ↔ 𝑎 = 𝑥))
40	34, 39	bitrid 286	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = 𝑥))
41	40	alrimiv 1960	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ∀𝑎(∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = 𝑥))
42		absn 4607	. . . . . . . 8 ⊢ ({𝑎 ∣ ∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s )} = {𝑥} ↔ ∀𝑎(∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = 𝑥))
43	41, 42	sylibr 237	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {𝑎 ∣ ∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s )} = {𝑥})
44	21	elexi 3475	. . . . . . . . . . 11 ⊢ 0s ∈ V
45		oveq2 7425	. . . . . . . . . . . 12 ⊢ (𝑏 = 0s → (𝑥 +s 𝑏) = (𝑥 +s 0s ))
46	45	eqeq2d 2773	. . . . . . . . . . 11 ⊢ (𝑏 = 0s → (𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = (𝑥 +s 0s )))
47	44, 46	rexsn 4646	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = (𝑥 +s 0s ))
48	35	addsridd 28238	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ0s → (𝑥 +s 0s ) = 𝑥)
49	48	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 +s 0s ) = 𝑥)
50	49	eqeq2d 2773	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑎 = (𝑥 +s 0s ) ↔ 𝑎 = 𝑥))
51	47, 50	bitrid 286	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = 𝑥))
52	51	alrimiv 1960	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ∀𝑎(∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = 𝑥))
53		absn 4607	. . . . . . . 8 ⊢ ({𝑎 ∣ ∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏)} = {𝑥} ↔ ∀𝑎(∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = 𝑥))
54	52, 53	sylibr 237	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {𝑎 ∣ ∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏)} = {𝑥})
55	43, 54	uneq12d 4119	. . . . . 6 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ({𝑎 ∣ ∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏)}) = ({𝑥} ∪ {𝑥}))
56		unidm 4107	. . . . . 6 ⊢ ({𝑥} ∪ {𝑥}) = {𝑥}
57	55, 56	eqtrdi 2813	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ({𝑎 ∣ ∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏)}) = {𝑥})
58		rex0 4311	. . . . . . . . 9 ⊢ ¬ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )
59	58	abf 4367	. . . . . . . 8 ⊢ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} = ∅
60		rex0 4311	. . . . . . . . 9 ⊢ ¬ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)
61	60	abf 4367	. . . . . . . 8 ⊢ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)} = ∅
62	59, 61	uneq12i 4116	. . . . . . 7 ⊢ ({𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)}) = (∅ ∪ ∅)
63		un0 4347	. . . . . . 7 ⊢ (∅ ∪ ∅) = ∅
64	62, 63	eqtri 2785	. . . . . 6 ⊢ ({𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)}) = ∅
65	64	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ({𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)}) = ∅)
66	57, 65	oveq12d 7435	. . . 4 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (({𝑎 ∣ ∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏)}) |s ({𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)})) = ({𝑥} |s ∅))
67		subscl 28335	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → (𝑥 -s 1s ) ∈ No )
68	35, 22, 67	sylancl 598	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → (𝑥 -s 1s ) ∈ No )
69	68	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 -s 1s ) ∈ No )
70	31	snelpw 5424	. . . . . . 7 ⊢ ((𝑥 -s 1s ) ∈ No ↔ {(𝑥 -s 1s )} ∈ 𝒫 No )
71	69, 70	sylib 221	. . . . . 6 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {(𝑥 -s 1s )} ∈ 𝒫 No )
72		nulsgts 28049	. . . . . 6 ⊢ ({(𝑥 -s 1s )} ∈ 𝒫 No → {(𝑥 -s 1s )} <<s ∅)
73	71, 72	syl 18	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {(𝑥 -s 1s )} <<s ∅)
74	44	snelpw 5424	. . . . . . 7 ⊢ ( 0s ∈ No ↔ { 0s } ∈ 𝒫 No )
75	21, 74	mpbi 233	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
76		nulsgts 28049	. . . . . 6 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
77	75, 76	mp1i 14	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → { 0s } <<s ∅)
78		simpr 490	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → 𝑥 = ({(𝑥 -s 1s )} |s ∅))
79		df-1s 28081	. . . . . 6 ⊢ 1s = ({ 0s } |s ∅)
80	79	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → 1s = ({ 0s } |s ∅))
81	73, 77, 78, 80	addsunif 28275	. . . 4 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 +s 1s ) = (({𝑎 ∣ ∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏)}) |s ({𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)})))
82	35	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → 𝑥 ∈ No )
83		pncans 28345	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
84	82, 22, 83	sylancl 598	. . . . . 6 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
85	84	sneqd 4599	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {((𝑥 +s 1s ) -s 1s )} = {𝑥})
86	85	oveq1d 7432	. . . 4 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ({((𝑥 +s 1s ) -s 1s )} |s ∅) = ({𝑥} |s ∅))
87	66, 81, 86	3eqtr4d 2807	. . 3 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
88	87	ex 418	. 2 ⊢ (𝑥 ∈ ℕ0s → (𝑥 = ({(𝑥 -s 1s )} |s ∅) → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅)))
89	5, 10, 15, 20, 30, 88	n0sind 28606	1 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145  {cab 2740  ∃wrex 3088   ∪ cun 3900  ∅c0 4282  𝒫 cpw 4560  {csn 4587   class class class wbr 5107  (class class class)co 7417   No csur 27884   <<s cslts 28030   |s ccuts 28032   0_s c0s 28078   1_s c1s 28079   +_s cadds 28232   -_s csubs 28293  ℕ_0scn0s 28585

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-nadd 8658  df-no 27887  df-lts 27888  df-bday 27889  df-les 27989  df-slts 28031  df-cuts 28033  df-0s 28080  df-1s 28081  df-made 28100  df-old 28101  df-left 28103  df-right 28104  df-norec 28211  df-norec2 28222  df-adds 28233  df-negs 28294  df-subs 28295  df-n0s 28587

This theorem is used by:  n0cut2  28608  n0on  28609  n0fincut  28628  zcuts  28680  addhalfcut  28732