Theorem n0cut 28505

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 7419	. . . . 5 ⊢ (𝑦 = 0s → (𝑦 -s 1s ) = ( 0s -s 1s ))
3	2	sneqd 4602	. . . 4 ⊢ (𝑦 = 0s → {(𝑦 -s 1s )} = {( 0s -s 1s )})
4	3	oveq1d 7427	. . 3 ⊢ (𝑦 = 0s → ({(𝑦 -s 1s )} |s ∅) = ({( 0s -s 1s )} |s ∅))
5	1, 4	eqeq12d 2779	. 2 ⊢ (𝑦 = 0s → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ 0s = ({( 0s -s 1s )} |s ∅)))
6		id 23	. . 3 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
7		oveq1 7419	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 -s 1s ) = (𝑥 -s 1s ))
8	7	sneqd 4602	. . . 4 ⊢ (𝑦 = 𝑥 → {(𝑦 -s 1s )} = {(𝑥 -s 1s )})
9	8	oveq1d 7427	. . 3 ⊢ (𝑦 = 𝑥 → ({(𝑦 -s 1s )} |s ∅) = ({(𝑥 -s 1s )} |s ∅))
10	6, 9	eqeq12d 2779	. 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 7419	. . . . 5 ⊢ (𝑦 = (𝑥 +s 1s ) → (𝑦 -s 1s ) = ((𝑥 +s 1s ) -s 1s ))
13	12	sneqd 4602	. . . 4 ⊢ (𝑦 = (𝑥 +s 1s ) → {(𝑦 -s 1s )} = {((𝑥 +s 1s ) -s 1s )})
14	13	oveq1d 7427	. . 3 ⊢ (𝑦 = (𝑥 +s 1s ) → ({(𝑦 -s 1s )} |s ∅) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
15	11, 14	eqeq12d 2779	. 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 7419	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 -s 1s ) = (𝐴 -s 1s ))
18	17	sneqd 4602	. . . 4 ⊢ (𝑦 = 𝐴 → {(𝑦 -s 1s )} = {(𝐴 -s 1s )})
19	18	oveq1d 7427	. . 3 ⊢ (𝑦 = 𝐴 → ({(𝑦 -s 1s )} |s ∅) = ({(𝐴 -s 1s )} |s ∅))
20	16, 19	eqeq12d 2779	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ 𝐴 = ({(𝐴 -s 1s )} |s ∅)))
21		0no 27980	. . . . . . 7 ⊢ 0s ∈ No
22		1no 27981	. . . . . . 7 ⊢ 1s ∈ No
23		subscl 28233	. . . . . . 7 ⊢ (( 0s ∈ No ∧ 1s ∈ No ) → ( 0s -s 1s ) ∈ No )
24	21, 22, 23	mp2an 704	. . . . . 6 ⊢ ( 0s -s 1s ) ∈ No
25	24	a1i 11	. . . . 5 ⊢ (⊤ → ( 0s -s 1s ) ∈ No )
26	21	a1i 11	. . . . . 6 ⊢ (⊤ → 0s ∈ No )
27	26	ltsm1d 28273	. . . . 5 ⊢ (⊤ → ( 0s -s 1s ) <s 0s )
28	25, 27	cutneg 27987	. . . 4 ⊢ (⊤ → ({( 0s -s 1s )} |s ∅) = 0s )
29	28	mptru 1577	. . 3 ⊢ ({( 0s -s 1s )} |s ∅) = 0s
30	29	eqcomi 2772	. 2 ⊢ 0s = ({( 0s -s 1s )} |s ∅)
31		ovex 7445	. . . . . . . . . . 11 ⊢ (𝑥 -s 1s ) ∈ V
32		oveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑥 -s 1s ) → (𝑏 +s 1s ) = ((𝑥 -s 1s ) +s 1s ))
33	32	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑥 -s 1s ) → (𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = ((𝑥 -s 1s ) +s 1s )))
34	31, 33	rexsn 4649	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = ((𝑥 -s 1s ) +s 1s ))
35		n0no 28494	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
36		npcans 28246	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
37	35, 22, 36	sylancl 597	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ0s → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
38	37	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
39	38	eqeq2d 2774	. . . . . . . . . 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 1957	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ∀𝑎(∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = 𝑥))
42		absn 4610	. . . . . . . 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 3477	. . . . . . . . . . 11 ⊢ 0s ∈ V
45		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑏 = 0s → (𝑥 +s 𝑏) = (𝑥 +s 0s ))
46	45	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (𝑏 = 0s → (𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = (𝑥 +s 0s )))
47	44, 46	rexsn 4649	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = (𝑥 +s 0s ))
48	35	addsridd 28136	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ0s → (𝑥 +s 0s ) = 𝑥)
49	48	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 +s 0s ) = 𝑥)
50	49	eqeq2d 2774	. . . . . . . . . 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 1957	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ∀𝑎(∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = 𝑥))
53		absn 4610	. . . . . . . 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 4124	. . . . . 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 4112	. . . . . 6 ⊢ ({𝑥} ∪ {𝑥}) = {𝑥}
57	55, 56	eqtrdi 2814	. . . . 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 4316	. . . . . . . . 9 ⊢ ¬ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )
59	58	abf 4372	. . . . . . . 8 ⊢ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} = ∅
60		rex0 4316	. . . . . . . . 9 ⊢ ¬ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)
61	60	abf 4372	. . . . . . . 8 ⊢ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)} = ∅
62	59, 61	uneq12i 4121	. . . . . . 7 ⊢ ({𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)}) = (∅ ∪ ∅)
63		un0 4352	. . . . . . 7 ⊢ (∅ ∪ ∅) = ∅
64	62, 63	eqtri 2786	. . . . . 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 7430	. . . 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 28233	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → (𝑥 -s 1s ) ∈ No )
68	35, 22, 67	sylancl 597	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → (𝑥 -s 1s ) ∈ No )
69	68	adantr 485	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 -s 1s ) ∈ No )
70	31	snelpw 5428	. . . . . . 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 27947	. . . . . 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 5428	. . . . . . 7 ⊢ ( 0s ∈ No ↔ { 0s } ∈ 𝒫 No )
75	21, 74	mpbi 233	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
76		nulsgts 27947	. . . . . 6 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
77	75, 76	mp1i 14	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → { 0s } <<s ∅)
78		simpr 489	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → 𝑥 = ({(𝑥 -s 1s )} |s ∅))
79		df-1s 27979	. . . . . 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 28173	. . . 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 485	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → 𝑥 ∈ No )
83		pncans 28243	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
84	82, 22, 83	sylancl 597	. . . . . 6 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
85	84	sneqd 4602	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {((𝑥 +s 1s ) -s 1s )} = {𝑥})
86	85	oveq1d 7427	. . . 4 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ({((𝑥 +s 1s ) -s 1s )} |s ∅) = ({𝑥} |s ∅))
87	66, 81, 86	3eqtr4d 2808	. . 3 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
88	87	ex 417	. 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 28504	1 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  ⊤wtru 1571   ∈ wcel 2143  {cab 2741  ∃wrex 3089   ∪ cun 3904  ∅c0 4287  𝒫 cpw 4563  {csn 4590   class class class wbr 5110  (class class class)co 7412   No csur 27782   <<s cslts 27928   |s ccuts 27930   0_s c0s 27976   1_s c1s 27977   +_s cadds 28130   -_s csubs 28191  ℕ_0scn0s 28483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27785  df-lts 27786  df-bday 27787  df-les 27887  df-slts 27929  df-cuts 27931  df-0s 27978  df-1s 27979  df-made 27998  df-old 27999  df-left 28001  df-right 28002  df-norec 28109  df-norec2 28120  df-adds 28131  df-negs 28192  df-subs 28193  df-n0s 28485

This theorem is referenced by:  n0cut2  28506  n0on  28507  n0fincut  28526  zcuts  28578  addhalfcut  28630