Theorem n0scut 28356

Description: A cut form for surreal naturals. (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 7455	. . . . 5 ⊢ (𝑦 = 0s → (𝑦 -s 1s ) = ( 0s -s 1s ))
3	2	sneqd 4660	. . . 4 ⊢ (𝑦 = 0s → {(𝑦 -s 1s )} = {( 0s -s 1s )})
4	3	oveq1d 7463	. . 3 ⊢ (𝑦 = 0s → ({(𝑦 -s 1s )} |s ∅) = ({( 0s -s 1s )} |s ∅))
5	1, 4	eqeq12d 2756	. 2 ⊢ (𝑦 = 0s → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ 0s = ({( 0s -s 1s )} |s ∅)))
6		id 22	. . 3 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
7		oveq1 7455	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 -s 1s ) = (𝑥 -s 1s ))
8	7	sneqd 4660	. . . 4 ⊢ (𝑦 = 𝑥 → {(𝑦 -s 1s )} = {(𝑥 -s 1s )})
9	8	oveq1d 7463	. . 3 ⊢ (𝑦 = 𝑥 → ({(𝑦 -s 1s )} |s ∅) = ({(𝑥 -s 1s )} |s ∅))
10	6, 9	eqeq12d 2756	. 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 7455	. . . . 5 ⊢ (𝑦 = (𝑥 +s 1s ) → (𝑦 -s 1s ) = ((𝑥 +s 1s ) -s 1s ))
13	12	sneqd 4660	. . . 4 ⊢ (𝑦 = (𝑥 +s 1s ) → {(𝑦 -s 1s )} = {((𝑥 +s 1s ) -s 1s )})
14	13	oveq1d 7463	. . 3 ⊢ (𝑦 = (𝑥 +s 1s ) → ({(𝑦 -s 1s )} |s ∅) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
15	11, 14	eqeq12d 2756	. 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 7455	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 -s 1s ) = (𝐴 -s 1s ))
18	17	sneqd 4660	. . . 4 ⊢ (𝑦 = 𝐴 → {(𝑦 -s 1s )} = {(𝐴 -s 1s )})
19	18	oveq1d 7463	. . 3 ⊢ (𝑦 = 𝐴 → ({(𝑦 -s 1s )} |s ∅) = ({(𝐴 -s 1s )} |s ∅))
20	16, 19	eqeq12d 2756	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 = ({(𝑦 -s 1s )} |s ∅) ↔ 𝐴 = ({(𝐴 -s 1s )} |s ∅)))
21		0sno 27889	. . . . . . . 8 ⊢ 0s ∈ No
22		1sno 27890	. . . . . . . 8 ⊢ 1s ∈ No
23		subscl 28110	. . . . . . . 8 ⊢ (( 0s ∈ No ∧ 1s ∈ No ) → ( 0s -s 1s ) ∈ No )
24	21, 22, 23	mp2an 691	. . . . . . 7 ⊢ ( 0s -s 1s ) ∈ No
25	24	a1i 11	. . . . . 6 ⊢ (⊤ → ( 0s -s 1s ) ∈ No )
26	21	a1i 11	. . . . . 6 ⊢ (⊤ → 0s ∈ No )
27		0slt1s 27892	. . . . . . . 8 ⊢ 0s <s 1s
28		addslid 28019	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
29	22, 28	ax-mp 5	. . . . . . . 8 ⊢ ( 0s +s 1s ) = 1s
30	27, 29	breqtrri 5193	. . . . . . 7 ⊢ 0s <s ( 0s +s 1s )
31	22	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1s ∈ No )
32	26, 31, 26	sltsubaddd 28137	. . . . . . 7 ⊢ (⊤ → (( 0s -s 1s ) <s 0s ↔ 0s <s ( 0s +s 1s )))
33	30, 32	mpbiri 258	. . . . . 6 ⊢ (⊤ → ( 0s -s 1s ) <s 0s )
34	25, 26, 33	ssltsn 27855	. . . . 5 ⊢ (⊤ → {( 0s -s 1s )} <<s { 0s })
35	21	elexi 3511	. . . . . . . . 9 ⊢ 0s ∈ V
36	35	snelpw 5465	. . . . . . . 8 ⊢ ( 0s ∈ No ↔ { 0s } ∈ 𝒫 No )
37	21, 36	mpbi 230	. . . . . . 7 ⊢ { 0s } ∈ 𝒫 No
38		nulssgt 27861	. . . . . . 7 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
39	37, 38	ax-mp 5	. . . . . 6 ⊢ { 0s } <<s ∅
40	39	a1i 11	. . . . 5 ⊢ (⊤ → { 0s } <<s ∅)
41	34, 40	cuteq0 27895	. . . 4 ⊢ (⊤ → ({( 0s -s 1s )} |s ∅) = 0s )
42	41	mptru 1544	. . 3 ⊢ ({( 0s -s 1s )} |s ∅) = 0s
43	42	eqcomi 2749	. 2 ⊢ 0s = ({( 0s -s 1s )} |s ∅)
44		ovex 7481	. . . . . . . . . . 11 ⊢ (𝑥 -s 1s ) ∈ V
45		oveq1 7455	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑥 -s 1s ) → (𝑏 +s 1s ) = ((𝑥 -s 1s ) +s 1s ))
46	45	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑥 -s 1s ) → (𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = ((𝑥 -s 1s ) +s 1s )))
47	44, 46	rexsn 4706	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = ((𝑥 -s 1s ) +s 1s ))
48		n0sno 28346	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
49		npcans 28123	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
50	48, 22, 49	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ0s → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
51	50	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ((𝑥 -s 1s ) +s 1s ) = 𝑥)
52	51	eqeq2d 2751	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑎 = ((𝑥 -s 1s ) +s 1s ) ↔ 𝑎 = 𝑥))
53	47, 52	bitrid 283	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = 𝑥))
54	53	alrimiv 1926	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ∀𝑎(∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = 𝑥))
55		absn 4667	. . . . . . . 8 ⊢ ({𝑎 ∣ ∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s )} = {𝑥} ↔ ∀𝑎(∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s ) ↔ 𝑎 = 𝑥))
56	54, 55	sylibr 234	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {𝑎 ∣ ∃𝑏 ∈ {(𝑥 -s 1s )}𝑎 = (𝑏 +s 1s )} = {𝑥})
57		oveq2 7456	. . . . . . . . . . . 12 ⊢ (𝑏 = 0s → (𝑥 +s 𝑏) = (𝑥 +s 0s ))
58	57	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (𝑏 = 0s → (𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = (𝑥 +s 0s )))
59	35, 58	rexsn 4706	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = (𝑥 +s 0s ))
60	48	addsridd 28016	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ0s → (𝑥 +s 0s ) = 𝑥)
61	60	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 +s 0s ) = 𝑥)
62	61	eqeq2d 2751	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑎 = (𝑥 +s 0s ) ↔ 𝑎 = 𝑥))
63	59, 62	bitrid 283	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = 𝑥))
64	63	alrimiv 1926	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ∀𝑎(∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = 𝑥))
65		absn 4667	. . . . . . . 8 ⊢ ({𝑎 ∣ ∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏)} = {𝑥} ↔ ∀𝑎(∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏) ↔ 𝑎 = 𝑥))
66	64, 65	sylibr 234	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {𝑎 ∣ ∃𝑏 ∈ { 0s }𝑎 = (𝑥 +s 𝑏)} = {𝑥})
67	56, 66	uneq12d 4192	. . . . . 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}))
68		unidm 4180	. . . . . 6 ⊢ ({𝑥} ∪ {𝑥}) = {𝑥}
69	67, 68	eqtrdi 2796	. . . . 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})
70		rex0 4383	. . . . . . . . 9 ⊢ ¬ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )
71	70	abf 4429	. . . . . . . 8 ⊢ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} = ∅
72		rex0 4383	. . . . . . . . 9 ⊢ ¬ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)
73	72	abf 4429	. . . . . . . 8 ⊢ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)} = ∅
74	71, 73	uneq12i 4189	. . . . . . 7 ⊢ ({𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)}) = (∅ ∪ ∅)
75		un0 4417	. . . . . . 7 ⊢ (∅ ∪ ∅) = ∅
76	74, 75	eqtri 2768	. . . . . 6 ⊢ ({𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)}) = ∅
77	76	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ({𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑏 +s 1s )} ∪ {𝑎 ∣ ∃𝑏 ∈ ∅ 𝑎 = (𝑥 +s 𝑏)}) = ∅)
78	69, 77	oveq12d 7466	. . . 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 ∅))
79		subscl 28110	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → (𝑥 -s 1s ) ∈ No )
80	48, 22, 79	sylancl 585	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → (𝑥 -s 1s ) ∈ No )
81	80	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 -s 1s ) ∈ No )
82	44	snelpw 5465	. . . . . . 7 ⊢ ((𝑥 -s 1s ) ∈ No ↔ {(𝑥 -s 1s )} ∈ 𝒫 No )
83	81, 82	sylib 218	. . . . . 6 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {(𝑥 -s 1s )} ∈ 𝒫 No )
84		nulssgt 27861	. . . . . 6 ⊢ ({(𝑥 -s 1s )} ∈ 𝒫 No → {(𝑥 -s 1s )} <<s ∅)
85	83, 84	syl 17	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {(𝑥 -s 1s )} <<s ∅)
86	39	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → { 0s } <<s ∅)
87		simpr 484	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → 𝑥 = ({(𝑥 -s 1s )} |s ∅))
88		df-1s 27888	. . . . . 6 ⊢ 1s = ({ 0s } |s ∅)
89	88	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → 1s = ({ 0s } |s ∅))
90	85, 86, 87, 89	addsunif 28053	. . . 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)})))
91	48	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → 𝑥 ∈ No )
92		pncans 28120	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
93	91, 22, 92	sylancl 585	. . . . . 6 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
94	93	sneqd 4660	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → {((𝑥 +s 1s ) -s 1s )} = {𝑥})
95	94	oveq1d 7463	. . . 4 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → ({((𝑥 +s 1s ) -s 1s )} |s ∅) = ({𝑥} |s ∅))
96	78, 90, 95	3eqtr4d 2790	. . 3 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑥 = ({(𝑥 -s 1s )} |s ∅)) → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
97	96	ex 412	. 2 ⊢ (𝑥 ∈ ℕ0s → (𝑥 = ({(𝑥 -s 1s )} |s ∅) → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅)))
98	5, 10, 15, 20, 43, 97	n0sind 28355	1 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  ⊤wtru 1538   ∈ wcel 2108  {cab 2717  ∃wrex 3076   ∪ cun 3974  ∅c0 4352  𝒫 cpw 4622  {csn 4648   class class class wbr 5166  (class class class)co 7448   No csur 27702   <s cslt 27703   <<s csslt 27843   |s cscut 27845   0_s c0s 27885   1_s c1s 27886   +_s cadds 28010   -_s csubs 28070  ℕ_0scnn0s 28336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-n0s 28338

This theorem is referenced by:  n0ons  28357  n0sbday  28372  zscut  28411  addhalfcut  28437