Theorem n0scut 28232

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

Assertion

Ref	Expression
n0scut	⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))

Proof of Theorem n0scut

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

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑦 = 0s → 𝑦 = 0s )
2		oveq1 7396	. . . . 5 ⊢ (𝑦 = 0s → (𝑦 -s 1s ) = ( 0s -s 1s ))
3	2	sneqd 4603	. . . 4 ⊢ (𝑦 = 0s → {(𝑦 -s 1s )} = {( 0s -s 1s )})
4	3	oveq1d 7404	. . 3 ⊢ (𝑦 = 0s → ({(𝑦 -s 1s )} |s ∅) = ({( 0s -s 1s )} |s ∅))
5	1, 4	eqeq12d 2746	. 2 ⊢ (𝑦 = 0s → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ 0s = ({( 0s -s 1s )} |s ∅)))
6		id 22	. . 3 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
7		oveq1 7396	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 -s 1s ) = (𝑥 -s 1s ))
8	7	sneqd 4603	. . . 4 ⊢ (𝑦 = 𝑥 → {(𝑦 -s 1s )} = {(𝑥 -s 1s )})
9	8	oveq1d 7404	. . 3 ⊢ (𝑦 = 𝑥 → ({(𝑦 -s 1s )} |s ∅) = ({(𝑥 -s 1s )} |s ∅))
10	6, 9	eqeq12d 2746	. 2 ⊢ (𝑦 = 𝑥 → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ 𝑥 = ({(𝑥 -s 1s )} |s ∅)))
11		id 22	. . 3 ⊢ (𝑦 = (𝑥 +s 1s ) → 𝑦 = (𝑥 +s 1s ))
12		oveq1 7396	. . . . 5 ⊢ (𝑦 = (𝑥 +s 1s ) → (𝑦 -s 1s ) = ((𝑥 +s 1s ) -s 1s ))
13	12	sneqd 4603	. . . 4 ⊢ (𝑦 = (𝑥 +s 1s ) → {(𝑦 -s 1s )} = {((𝑥 +s 1s ) -s 1s )})
14	13	oveq1d 7404	. . 3 ⊢ (𝑦 = (𝑥 +s 1s ) → ({(𝑦 -s 1s )} |s ∅) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
15	11, 14	eqeq12d 2746	. 2 ⊢ (𝑦 = (𝑥 +s 1s ) → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅)))
16		id 22	. . 3 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
17		oveq1 7396	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 -s 1s ) = (𝐴 -s 1s ))
18	17	sneqd 4603	. . . 4 ⊢ (𝑦 = 𝐴 → {(𝑦 -s 1s )} = {(𝐴 -s 1s )})
19	18	oveq1d 7404	. . 3 ⊢ (𝑦 = 𝐴 → ({(𝑦 -s 1s )} |s ∅) = ({(𝐴 -s 1s )} |s ∅))
20	16, 19	eqeq12d 2746	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ 𝐴 = ({(𝐴 -s 1s )} |s ∅)))
21		0sno 27744	. . . . . . 7 ⊢ 0s ∈ No
22		1sno 27745	. . . . . . 7 ⊢ 1s ∈ No
23		subscl 27972	. . . . . . 7 ⊢ (( 0s ∈ No ∧ 1s ∈ No ) → ( 0s -s 1s ) ∈ No )
24	21, 22, 23	mp2an 692	. . . . . 6 ⊢ ( 0s -s 1s ) ∈ No
25	24	a1i 11	. . . . 5 ⊢ (⊤ → ( 0s -s 1s ) ∈ No )
26	21	a1i 11	. . . . . 6 ⊢ (⊤ → 0s ∈ No )
27	26	sltm1d 28011	. . . . 5 ⊢ (⊤ → ( 0s -s 1s ) <s 0s )
28	25, 27	cutneg 27751	. . . 4 ⊢ (⊤ → ({( 0s -s 1s )} |s ∅) = 0s )
29	28	mptru 1547	. . 3 ⊢ ({( 0s -s 1s )} |s ∅) = 0s
30	29	eqcomi 2739	. 2 ⊢ 0s = ({( 0s -s 1s )} |s ∅)
31		ovex 7422	. . . . . . . . . . 11 ⊢ (𝑥 -s 1s ) ∈ V
32		oveq1 7396	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑥 -s 1s ) → (𝑏 +s 1s ) = ((𝑥 -s 1s ) +s 1s ))
33	32	eqeq2d 2741	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑥 -s 1s ) → (𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = ((𝑥 -s 1s ) +s 1s )))
34	31, 33	rexsn 4648	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = ((𝑥 -s 1s ) +s 1s ))
35		n0sno 28222	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
36		npcans 27985	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
37	35, 22, 36	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ0s → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
38	37	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
39	38	eqeq2d 2741	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑎 = ((𝑥 -s 1s ) +s 1s ) ↔ 𝑎 = 𝑥))
40	34, 39	bitrid 283	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = 𝑥))
41	40	alrimiv 1927	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ∀𝑎(∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = 𝑥))
42		absn 4611	. . . . . . . 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 234	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {𝑎 ∣ ∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s )} = {𝑥})
44	21	elexi 3473	. . . . . . . . . . 11 ⊢ 0s ∈ V
45		oveq2 7397	. . . . . . . . . . . 12 ⊢ (𝑏 = 0s → (𝑥 +s 𝑏) = (𝑥 +s 0s ))
46	45	eqeq2d 2741	. . . . . . . . . . 11 ⊢ (𝑏 = 0s → (𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = (𝑥 +s 0s )))
47	44, 46	rexsn 4648	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = (𝑥 +s 0s ))
48	35	addsridd 27878	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ0s → (𝑥 +s 0s ) = 𝑥)
49	48	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 +s 0s ) = 𝑥)
50	49	eqeq2d 2741	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑎 = (𝑥 +s 0s ) ↔ 𝑎 = 𝑥))
51	47, 50	bitrid 283	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = 𝑥))
52	51	alrimiv 1927	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ∀𝑎(∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = 𝑥))
53		absn 4611	. . . . . . . 8 ⊢ ({𝑎 ∣ ∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏)} = {𝑥} ↔ ∀𝑎(∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = 𝑥))
54	52, 53	sylibr 234	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {𝑎 ∣ ∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏)} = {𝑥})
55	43, 54	uneq12d 4134	. . . . . 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 4122	. . . . . 6 ⊢ ({𝑥} ∪ {𝑥}) = {𝑥}
57	55, 56	eqtrdi 2781	. . . . 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 4325	. . . . . . . . 9 ⊢ ¬ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )
59	58	abf 4371	. . . . . . . 8 ⊢ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} = ∅
60		rex0 4325	. . . . . . . . 9 ⊢ ¬ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)
61	60	abf 4371	. . . . . . . 8 ⊢ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)} = ∅
62	59, 61	uneq12i 4131	. . . . . . 7 ⊢ ({𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)}) = (∅ ∪ ∅)
63		un0 4359	. . . . . . 7 ⊢ (∅ ∪ ∅) = ∅
64	62, 63	eqtri 2753	. . . . . 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 7407	. . . 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 27972	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → (𝑥 -s 1s ) ∈ No )
68	35, 22, 67	sylancl 586	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → (𝑥 -s 1s ) ∈ No )
69	68	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 -s 1s ) ∈ No )
70	31	snelpw 5407	. . . . . . 7 ⊢ ((𝑥 -s 1s ) ∈ No ↔ {(𝑥 -s 1s )} ∈ 𝒫 No )
71	69, 70	sylib 218	. . . . . 6 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {(𝑥 -s 1s )} ∈ 𝒫 No )
72		nulssgt 27716	. . . . . 6 ⊢ ({(𝑥 -s 1s )} ∈ 𝒫 No → {(𝑥 -s 1s )} <<s ∅)
73	71, 72	syl 17	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {(𝑥 -s 1s )} <<s ∅)
74	44	snelpw 5407	. . . . . . 7 ⊢ ( 0s ∈ No ↔ { 0s } ∈ 𝒫 No )
75	21, 74	mpbi 230	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
76		nulssgt 27716	. . . . . 6 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
77	75, 76	mp1i 13	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → { 0s } <<s ∅)
78		simpr 484	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → 𝑥 = ({(𝑥 -s 1s )} |s ∅))
79		df-1s 27743	. . . . . 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 27915	. . . 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 480	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → 𝑥 ∈ No )
83		pncans 27982	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
84	82, 22, 83	sylancl 586	. . . . . 6 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
85	84	sneqd 4603	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {((𝑥 +s 1s ) -s 1s )} = {𝑥})
86	85	oveq1d 7404	. . . 4 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ({((𝑥 +s 1s ) -s 1s )} |s ∅) = ({𝑥} |s ∅))
87	66, 81, 86	3eqtr4d 2775	. . 3 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
88	87	ex 412	. 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 28231	1 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  {cab 2708  ∃wrex 3054   ∪ cun 3914  ∅c0 4298  𝒫 cpw 4565  {csn 4591   class class class wbr 5109  (class class class)co 7389   No csur 27557   <<s csslt 27698   |s cscut 27700   0_s c0s 27740   1_s c1s 27741   +_s cadds 27872   -_s csubs 27932  ℕ_0scnn0s 28212

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-2o 8437  df-nadd 8632  df-no 27560  df-slt 27561  df-bday 27562  df-sle 27663  df-sslt 27699  df-scut 27701  df-0s 27742  df-1s 27743  df-made 27761  df-old 27762  df-left 27764  df-right 27765  df-norec 27851  df-norec2 27862  df-adds 27873  df-negs 27933  df-subs 27934  df-n0s 28214

This theorem is referenced by:  n0scut2  28233  n0ons  28234  n0sfincut  28252  zscut  28301  addhalfcut  28340