Theorem zsoring 28795

Description: The surreal integers form an ordered ring. Note that we have to restrict the operations here since No is a proper class. (Contributed by Scott Fenton, 23-Dec-2025.)

Hypotheses

Ref	Expression
zsoring.1	⊢ ℤs = (Base‘𝐾)
zsoring.2	⊢ ( +s ↾ (ℤs × ℤs)) = (+g‘𝐾)
zsoring.3	⊢ ( ·s ↾ (ℤs × ℤs)) = (.r‘𝐾)
zsoring.4	⊢ ( ≤s ∩ (ℤs × ℤs)) = (le‘𝐾)
zsoring.5	⊢ 0s = (0g‘𝐾)

Assertion

Ref	Expression
zsoring	⊢ 𝐾 ∈ oRing

Proof of Theorem zsoring

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

Step	Hyp	Ref	Expression
1		zsoring.1	. . . 4 ⊢ ℤs = (Base‘𝐾)
2		zsoring.2	. . . 4 ⊢ ( +s ↾ (ℤs × ℤs)) = (+g‘𝐾)
3		ovres 7586	. . . . 5 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑥( +s ↾ (ℤs × ℤs))𝑦) = (𝑥 +s 𝑦))
4		zaddscl 28780	. . . . 5 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑥 +s 𝑦) ∈ ℤs)
5	3, 4	eqeltrd 2861	. . . 4 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑥( +s ↾ (ℤs × ℤs))𝑦) ∈ ℤs)
6		zno 28768	. . . . . 6 ⊢ (𝑥 ∈ ℤs → 𝑥 ∈ No )
7		zno 28768	. . . . . 6 ⊢ (𝑦 ∈ ℤs → 𝑦 ∈ No )
8		zno 28768	. . . . . 6 ⊢ (𝑧 ∈ ℤs → 𝑧 ∈ No )
9		addsass 28391	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧)))
10	6, 7, 8, 9	syl3an 1178	. . . . 5 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧)))
11	3	3adant3 1150	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( +s ↾ (ℤs × ℤs))𝑦) = (𝑥 +s 𝑦))
12	11	oveq1d 7435	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))𝑧) = ((𝑥 +s 𝑦)( +s ↾ (ℤs × ℤs))𝑧))
13	4	3adant3 1150	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 +s 𝑦) ∈ ℤs)
14		simp3 1156	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑧 ∈ ℤs)
15	13, 14	ovresd 7587	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 +s 𝑦)( +s ↾ (ℤs × ℤs))𝑧) = ((𝑥 +s 𝑦) +s 𝑧))
16	12, 15	eqtrd 2796	. . . . 5 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))𝑧) = ((𝑥 +s 𝑦) +s 𝑧))
17		ovres 7586	. . . . . . . 8 ⊢ ((𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑦( +s ↾ (ℤs × ℤs))𝑧) = (𝑦 +s 𝑧))
18	17	3adant1 1148	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑦( +s ↾ (ℤs × ℤs))𝑧) = (𝑦 +s 𝑧))
19	18	oveq2d 7436	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( +s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)) = (𝑥( +s ↾ (ℤs × ℤs))(𝑦 +s 𝑧)))
20		simp1 1154	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑥 ∈ ℤs)
21		zaddscl 28780	. . . . . . . 8 ⊢ ((𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑦 +s 𝑧) ∈ ℤs)
22	21	3adant1 1148	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑦 +s 𝑧) ∈ ℤs)
23	20, 22	ovresd 7587	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( +s ↾ (ℤs × ℤs))(𝑦 +s 𝑧)) = (𝑥 +s (𝑦 +s 𝑧)))
24	19, 23	eqtrd 2796	. . . . 5 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( +s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)) = (𝑥 +s (𝑦 +s 𝑧)))
25	10, 16, 24	3eqtr4d 2806	. . . 4 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))𝑧) = (𝑥( +s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)))
26		0zs 28774	. . . 4 ⊢ 0s ∈ ℤs
27		ovres 7586	. . . . . 6 ⊢ (( 0s ∈ ℤs ∧ 𝑥 ∈ ℤs) → ( 0s ( +s ↾ (ℤs × ℤs))𝑥) = ( 0s +s 𝑥))
28	26, 27	mpan 703	. . . . 5 ⊢ (𝑥 ∈ ℤs → ( 0s ( +s ↾ (ℤs × ℤs))𝑥) = ( 0s +s 𝑥))
29		addslid 28354	. . . . . 6 ⊢ (𝑥 ∈ No → ( 0s +s 𝑥) = 𝑥)
30	6, 29	syl 18	. . . . 5 ⊢ (𝑥 ∈ ℤs → ( 0s +s 𝑥) = 𝑥)
31	28, 30	eqtrd 2796	. . . 4 ⊢ (𝑥 ∈ ℤs → ( 0s ( +s ↾ (ℤs × ℤs))𝑥) = 𝑥)
32		znegscl 28778	. . . 4 ⊢ (𝑥 ∈ ℤs → ( -us ‘𝑥) ∈ ℤs)
33		id 23	. . . . . 6 ⊢ (𝑥 ∈ ℤs → 𝑥 ∈ ℤs)
34	32, 33	ovresd 7587	. . . . 5 ⊢ (𝑥 ∈ ℤs → (( -us ‘𝑥)( +s ↾ (ℤs × ℤs))𝑥) = (( -us ‘𝑥) +s 𝑥))
35	32	znod 28769	. . . . . 6 ⊢ (𝑥 ∈ ℤs → ( -us ‘𝑥) ∈ No )
36	35, 6	addscomd 28353	. . . . 5 ⊢ (𝑥 ∈ ℤs → (( -us ‘𝑥) +s 𝑥) = (𝑥 +s ( -us ‘𝑥)))
37	6	negsidd 28428	. . . . 5 ⊢ (𝑥 ∈ ℤs → (𝑥 +s ( -us ‘𝑥)) = 0s )
38	34, 36, 37	3eqtrd 2800	. . . 4 ⊢ (𝑥 ∈ ℤs → (( -us ‘𝑥)( +s ↾ (ℤs × ℤs))𝑥) = 0s )
39	1, 2, 5, 25, 26, 31, 32, 38	isgrpi 19170	. . 3 ⊢ 𝐾 ∈ Grp
40		ovres 7586	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))𝑦) = (𝑥 ·s 𝑦))
41		simpl 488	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → 𝑥 ∈ ℤs)
42		simpr 490	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → 𝑦 ∈ ℤs)
43	41, 42	zmulscld 28783	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑥 ·s 𝑦) ∈ ℤs)
44	40, 43	eqeltrd 2861	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∈ ℤs)
45		mulsass 28552	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ((𝑥 ·s 𝑦) ·s 𝑧) = (𝑥 ·s (𝑦 ·s 𝑧)))
46	6, 7, 8, 45	syl3an 1178	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 ·s 𝑦) ·s 𝑧) = (𝑥 ·s (𝑦 ·s 𝑧)))
47	40	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))𝑦) = (𝑥 ·s 𝑦))
48	47	oveq1d 7435	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = ((𝑥 ·s 𝑦)( ·s ↾ (ℤs × ℤs))𝑧))
49		simp2 1155	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑦 ∈ ℤs)
50	20, 49	zmulscld 28783	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 ·s 𝑦) ∈ ℤs)
51	50, 14	ovresd 7587	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 ·s 𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = ((𝑥 ·s 𝑦) ·s 𝑧))
52	48, 51	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = ((𝑥 ·s 𝑦) ·s 𝑧))
53		ovres 7586	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑦( ·s ↾ (ℤs × ℤs))𝑧) = (𝑦 ·s 𝑧))
54	53	3adant1 1148	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑦( ·s ↾ (ℤs × ℤs))𝑧) = (𝑦 ·s 𝑧))
55	54	oveq2d 7436	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧)) = (𝑥( ·s ↾ (ℤs × ℤs))(𝑦 ·s 𝑧)))
56	49, 14	zmulscld 28783	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑦 ·s 𝑧) ∈ ℤs)
57	20, 56	ovresd 7587	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))(𝑦 ·s 𝑧)) = (𝑥 ·s (𝑦 ·s 𝑧)))
58	55, 57	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧)) = (𝑥 ·s (𝑦 ·s 𝑧)))
59	46, 52, 58	3eqtr4d 2806	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = (𝑥( ·s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧)))
60	59	3expa 1136	. . . . . . 7 ⊢ (((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) ∧ 𝑧 ∈ ℤs) → ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = (𝑥( ·s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧)))
61	60	ralrimiva 3155	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → ∀𝑧 ∈ ℤs ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = (𝑥( ·s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧)))
62	44, 61	jca 521	. . . . 5 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → ((𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∈ ℤs ∧ ∀𝑧 ∈ ℤs ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = (𝑥( ·s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧))))
63	62	rgen2 3203	. . . 4 ⊢ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs ((𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∈ ℤs ∧ ∀𝑧 ∈ ℤs ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = (𝑥( ·s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧)))
64		1zs 28777	. . . . 5 ⊢ 1s ∈ ℤs
65		ovres 7586	. . . . . . . . 9 ⊢ (( 1s ∈ ℤs ∧ 𝑥 ∈ ℤs) → ( 1s ( ·s ↾ (ℤs × ℤs))𝑥) = ( 1s ·s 𝑥))
66	64, 65	mpan 703	. . . . . . . 8 ⊢ (𝑥 ∈ ℤs → ( 1s ( ·s ↾ (ℤs × ℤs))𝑥) = ( 1s ·s 𝑥))
67	6	mulslidd 28529	. . . . . . . 8 ⊢ (𝑥 ∈ ℤs → ( 1s ·s 𝑥) = 𝑥)
68	66, 67	eqtrd 2796	. . . . . . 7 ⊢ (𝑥 ∈ ℤs → ( 1s ( ·s ↾ (ℤs × ℤs))𝑥) = 𝑥)
69		ovres 7586	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤs ∧ 1s ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs)) 1s ) = (𝑥 ·s 1s ))
70	64, 69	mpan2 704	. . . . . . . 8 ⊢ (𝑥 ∈ ℤs → (𝑥( ·s ↾ (ℤs × ℤs)) 1s ) = (𝑥 ·s 1s ))
71	6	mulsridd 28500	. . . . . . . 8 ⊢ (𝑥 ∈ ℤs → (𝑥 ·s 1s ) = 𝑥)
72	70, 71	eqtrd 2796	. . . . . . 7 ⊢ (𝑥 ∈ ℤs → (𝑥( ·s ↾ (ℤs × ℤs)) 1s ) = 𝑥)
73	68, 72	jca 521	. . . . . 6 ⊢ (𝑥 ∈ ℤs → (( 1s ( ·s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( ·s ↾ (ℤs × ℤs)) 1s ) = 𝑥))
74	73	rgen 3079	. . . . 5 ⊢ ∀𝑥 ∈ ℤs (( 1s ( ·s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( ·s ↾ (ℤs × ℤs)) 1s ) = 𝑥)
75		oveq1 7427	. . . . . . . 8 ⊢ (𝑦 = 1s → (𝑦( ·s ↾ (ℤs × ℤs))𝑥) = ( 1s ( ·s ↾ (ℤs × ℤs))𝑥))
76	75	eqeq1d 2763	. . . . . . 7 ⊢ (𝑦 = 1s → ((𝑦( ·s ↾ (ℤs × ℤs))𝑥) = 𝑥 ↔ ( 1s ( ·s ↾ (ℤs × ℤs))𝑥) = 𝑥))
77	76	ovanraleqv 7444	. . . . . 6 ⊢ (𝑦 = 1s → (∀𝑥 ∈ ℤs ((𝑦( ·s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( ·s ↾ (ℤs × ℤs))𝑦) = 𝑥) ↔ ∀𝑥 ∈ ℤs (( 1s ( ·s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( ·s ↾ (ℤs × ℤs)) 1s ) = 𝑥)))
78	77	rspcev 3577	. . . . 5 ⊢ (( 1s ∈ ℤs ∧ ∀𝑥 ∈ ℤs (( 1s ( ·s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( ·s ↾ (ℤs × ℤs)) 1s ) = 𝑥)) → ∃𝑦 ∈ ℤs ∀𝑥 ∈ ℤs ((𝑦( ·s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( ·s ↾ (ℤs × ℤs))𝑦) = 𝑥))
79	64, 74, 78	mp2an 705	. . . 4 ⊢ ∃𝑦 ∈ ℤs ∀𝑥 ∈ ℤs ((𝑦( ·s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( ·s ↾ (ℤs × ℤs))𝑦) = 𝑥)
80		eqid 2761	. . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
81	80, 1	mgpbas 20365	. . . . 5 ⊢ ℤs = (Base‘(mulGrp‘𝐾))
82		zsoring.3	. . . . . 6 ⊢ ( ·s ↾ (ℤs × ℤs)) = (.r‘𝐾)
83	80, 82	mgpplusg 20364	. . . . 5 ⊢ ( ·s ↾ (ℤs × ℤs)) = (+g‘(mulGrp‘𝐾))
84	81, 83	ismnd 18926	. . . 4 ⊢ ((mulGrp‘𝐾) ∈ Mnd ↔ (∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs ((𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∈ ℤs ∧ ∀𝑧 ∈ ℤs ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = (𝑥( ·s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧))) ∧ ∃𝑦 ∈ ℤs ∀𝑥 ∈ ℤs ((𝑦( ·s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( ·s ↾ (ℤs × ℤs))𝑦) = 𝑥)))
85	63, 79, 84	mpbir2an 724	. . 3 ⊢ (mulGrp‘𝐾) ∈ Mnd
86		addsdi 28541	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → (𝑥 ·s (𝑦 +s 𝑧)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)))
87	6, 7, 8, 86	syl3an 1178	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 ·s (𝑦 +s 𝑧)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)))
88	18	oveq2d 7436	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)) = (𝑥( ·s ↾ (ℤs × ℤs))(𝑦 +s 𝑧)))
89	20, 22	ovresd 7587	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))(𝑦 +s 𝑧)) = (𝑥 ·s (𝑦 +s 𝑧)))
90	88, 89	eqtrd 2796	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)) = (𝑥 ·s (𝑦 +s 𝑧)))
91		ovres 7586	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))𝑧) = (𝑥 ·s 𝑧))
92	91	3adant2 1149	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))𝑧) = (𝑥 ·s 𝑧))
93	47, 92	oveq12d 7438	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))(𝑥( ·s ↾ (ℤs × ℤs))𝑧)) = ((𝑥 ·s 𝑦)( +s ↾ (ℤs × ℤs))(𝑥 ·s 𝑧)))
94	20, 14	zmulscld 28783	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 ·s 𝑧) ∈ ℤs)
95	50, 94	ovresd 7587	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 ·s 𝑦)( +s ↾ (ℤs × ℤs))(𝑥 ·s 𝑧)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)))
96	93, 95	eqtrd 2796	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))(𝑥( ·s ↾ (ℤs × ℤs))𝑧)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)))
97	87, 90, 96	3eqtr4d 2806	. . . . 5 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ·s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)) = ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))(𝑥( ·s ↾ (ℤs × ℤs))𝑧)))
98	20	znod 28769	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑥 ∈ No )
99	49	znod 28769	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑦 ∈ No )
100	14	znod 28769	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑧 ∈ No )
101	98, 99, 100	addsdird 28543	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 +s 𝑦) ·s 𝑧) = ((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑧)))
102	11	oveq1d 7435	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = ((𝑥 +s 𝑦)( ·s ↾ (ℤs × ℤs))𝑧))
103	13, 14	ovresd 7587	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 +s 𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = ((𝑥 +s 𝑦) ·s 𝑧))
104	102, 103	eqtrd 2796	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = ((𝑥 +s 𝑦) ·s 𝑧))
105	92, 54	oveq12d 7438	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ·s ↾ (ℤs × ℤs))𝑧)( +s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧)) = ((𝑥 ·s 𝑧)( +s ↾ (ℤs × ℤs))(𝑦 ·s 𝑧)))
106	94, 56	ovresd 7587	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 ·s 𝑧)( +s ↾ (ℤs × ℤs))(𝑦 ·s 𝑧)) = ((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑧)))
107	105, 106	eqtrd 2796	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ·s ↾ (ℤs × ℤs))𝑧)( +s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧)) = ((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑧)))
108	101, 104, 107	3eqtr4d 2806	. . . . 5 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = ((𝑥( ·s ↾ (ℤs × ℤs))𝑧)( +s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧)))
109	97, 108	jca 521	. . . 4 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ·s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)) = ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))(𝑥( ·s ↾ (ℤs × ℤs))𝑧)) ∧ ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = ((𝑥( ·s ↾ (ℤs × ℤs))𝑧)( +s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧))))
110	109	rgen3 3208	. . 3 ⊢ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs ∀𝑧 ∈ ℤs ((𝑥( ·s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)) = ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))(𝑥( ·s ↾ (ℤs × ℤs))𝑧)) ∧ ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = ((𝑥( ·s ↾ (ℤs × ℤs))𝑧)( +s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧)))
111	1, 80, 2, 82	isring 20463	. . 3 ⊢ (𝐾 ∈ Ring ↔ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs ∀𝑧 ∈ ℤs ((𝑥( ·s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)) = ((𝑥( ·s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))(𝑥( ·s ↾ (ℤs × ℤs))𝑧)) ∧ ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( ·s ↾ (ℤs × ℤs))𝑧) = ((𝑥( ·s ↾ (ℤs × ℤs))𝑧)( +s ↾ (ℤs × ℤs))(𝑦( ·s ↾ (ℤs × ℤs))𝑧)))))
112	39, 85, 110, 111	mpbir3an 1360	. 2 ⊢ 𝐾 ∈ Ring
113	25	3expa 1136	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) ∧ 𝑧 ∈ ℤs) → ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))𝑧) = (𝑥( +s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)))
114	113	ralrimiva 3155	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → ∀𝑧 ∈ ℤs ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))𝑧) = (𝑥( +s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)))
115	5, 114	jca 521	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → ((𝑥( +s ↾ (ℤs × ℤs))𝑦) ∈ ℤs ∧ ∀𝑧 ∈ ℤs ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))𝑧) = (𝑥( +s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧))))
116	115	rgen2 3203	. . . . 5 ⊢ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs ((𝑥( +s ↾ (ℤs × ℤs))𝑦) ∈ ℤs ∧ ∀𝑧 ∈ ℤs ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))𝑧) = (𝑥( +s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)))
117		ovres 7586	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 0s ∈ ℤs) → (𝑥( +s ↾ (ℤs × ℤs)) 0s ) = (𝑥 +s 0s ))
118	26, 117	mpan2 704	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤs → (𝑥( +s ↾ (ℤs × ℤs)) 0s ) = (𝑥 +s 0s ))
119	6	addsridd 28351	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤs → (𝑥 +s 0s ) = 𝑥)
120	118, 119	eqtrd 2796	. . . . . . . 8 ⊢ (𝑥 ∈ ℤs → (𝑥( +s ↾ (ℤs × ℤs)) 0s ) = 𝑥)
121	31, 120	jca 521	. . . . . . 7 ⊢ (𝑥 ∈ ℤs → (( 0s ( +s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( +s ↾ (ℤs × ℤs)) 0s ) = 𝑥))
122	121	rgen 3079	. . . . . 6 ⊢ ∀𝑥 ∈ ℤs (( 0s ( +s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( +s ↾ (ℤs × ℤs)) 0s ) = 𝑥)
123		oveq1 7427	. . . . . . . . 9 ⊢ (𝑦 = 0s → (𝑦( +s ↾ (ℤs × ℤs))𝑥) = ( 0s ( +s ↾ (ℤs × ℤs))𝑥))
124	123	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑦 = 0s → ((𝑦( +s ↾ (ℤs × ℤs))𝑥) = 𝑥 ↔ ( 0s ( +s ↾ (ℤs × ℤs))𝑥) = 𝑥))
125	124	ovanraleqv 7444	. . . . . . 7 ⊢ (𝑦 = 0s → (∀𝑥 ∈ ℤs ((𝑦( +s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( +s ↾ (ℤs × ℤs))𝑦) = 𝑥) ↔ ∀𝑥 ∈ ℤs (( 0s ( +s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( +s ↾ (ℤs × ℤs)) 0s ) = 𝑥)))
126	125	rspcev 3577	. . . . . 6 ⊢ (( 0s ∈ ℤs ∧ ∀𝑥 ∈ ℤs (( 0s ( +s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( +s ↾ (ℤs × ℤs)) 0s ) = 𝑥)) → ∃𝑦 ∈ ℤs ∀𝑥 ∈ ℤs ((𝑦( +s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( +s ↾ (ℤs × ℤs))𝑦) = 𝑥))
127	26, 122, 126	mp2an 705	. . . . 5 ⊢ ∃𝑦 ∈ ℤs ∀𝑥 ∈ ℤs ((𝑦( +s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( +s ↾ (ℤs × ℤs))𝑦) = 𝑥)
128	1, 2	ismnd 18926	. . . . 5 ⊢ (𝐾 ∈ Mnd ↔ (∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs ((𝑥( +s ↾ (ℤs × ℤs))𝑦) ∈ ℤs ∧ ∀𝑧 ∈ ℤs ((𝑥( +s ↾ (ℤs × ℤs))𝑦)( +s ↾ (ℤs × ℤs))𝑧) = (𝑥( +s ↾ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧))) ∧ ∃𝑦 ∈ ℤs ∀𝑥 ∈ ℤs ((𝑦( +s ↾ (ℤs × ℤs))𝑥) = 𝑥 ∧ (𝑥( +s ↾ (ℤs × ℤs))𝑦) = 𝑥)))
129	116, 127, 128	mpbir2an 724	. . . 4 ⊢ 𝐾 ∈ Mnd
130	39	elexi 3473	. . . . . 6 ⊢ 𝐾 ∈ V
131		lesid 28124	. . . . . . . . . . 11 ⊢ (𝑥 ∈ No → 𝑥 ≤s 𝑥)
132	6, 131	syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤs → 𝑥 ≤s 𝑥)
133		brxp 5700	. . . . . . . . . . . 12 ⊢ (𝑥(ℤs × ℤs)𝑥 ↔ (𝑥 ∈ ℤs ∧ 𝑥 ∈ ℤs))
134	133	biimpri 231	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑥 ∈ ℤs) → 𝑥(ℤs × ℤs)𝑥)
135	134	anidms 577	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤs → 𝑥(ℤs × ℤs)𝑥)
136		brin 5157	. . . . . . . . . 10 ⊢ (𝑥( ≤s ∩ (ℤs × ℤs))𝑥 ↔ (𝑥 ≤s 𝑥 ∧ 𝑥(ℤs × ℤs)𝑥))
137	132, 135, 136	sylanbrc 595	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤs → 𝑥( ≤s ∩ (ℤs × ℤs))𝑥)
138	137	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑥( ≤s ∩ (ℤs × ℤs))𝑥)
139		brin 5157	. . . . . . . . . . 11 ⊢ (𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ↔ (𝑥 ≤s 𝑦 ∧ 𝑥(ℤs × ℤs)𝑦))
140		brxp 5700	. . . . . . . . . . . . . 14 ⊢ (𝑥(ℤs × ℤs)𝑦 ↔ (𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs))
141	140	biimpri 231	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → 𝑥(ℤs × ℤs)𝑦)
142	141	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑥(ℤs × ℤs)𝑦)
143	142	biantrud 541	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 ≤s 𝑦 ↔ (𝑥 ≤s 𝑦 ∧ 𝑥(ℤs × ℤs)𝑦)))
144	139, 143	bitr4id 293	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ↔ 𝑥 ≤s 𝑦))
145		brin 5157	. . . . . . . . . . . 12 ⊢ (𝑦( ≤s ∩ (ℤs × ℤs))𝑥 ↔ (𝑦 ≤s 𝑥 ∧ 𝑦(ℤs × ℤs)𝑥))
146		brxp 5700	. . . . . . . . . . . . . . 15 ⊢ (𝑦(ℤs × ℤs)𝑥 ↔ (𝑦 ∈ ℤs ∧ 𝑥 ∈ ℤs))
147	146	biimpri 231	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℤs ∧ 𝑥 ∈ ℤs) → 𝑦(ℤs × ℤs)𝑥)
148	147	ancoms 464	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → 𝑦(ℤs × ℤs)𝑥)
149	148	biantrud 541	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑦 ≤s 𝑥 ↔ (𝑦 ≤s 𝑥 ∧ 𝑦(ℤs × ℤs)𝑥)))
150	145, 149	bitr4id 293	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑦( ≤s ∩ (ℤs × ℤs))𝑥 ↔ 𝑦 ≤s 𝑥))
151	150	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑦( ≤s ∩ (ℤs × ℤs))𝑥 ↔ 𝑦 ≤s 𝑥))
152	144, 151	anbi12d 644	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∧ 𝑦( ≤s ∩ (ℤs × ℤs))𝑥) ↔ (𝑥 ≤s 𝑦 ∧ 𝑦 ≤s 𝑥)))
153		lestri3 28112	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (𝑥 = 𝑦 ↔ (𝑥 ≤s 𝑦 ∧ 𝑦 ≤s 𝑥)))
154	6, 7, 153	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑥 = 𝑦 ↔ (𝑥 ≤s 𝑦 ∧ 𝑦 ≤s 𝑥)))
155	154	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 = 𝑦 ↔ (𝑥 ≤s 𝑦 ∧ 𝑦 ≤s 𝑥)))
156	155	biimprd 251	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 ≤s 𝑦 ∧ 𝑦 ≤s 𝑥) → 𝑥 = 𝑦))
157	152, 156	sylbid 243	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∧ 𝑦( ≤s ∩ (ℤs × ℤs))𝑥) → 𝑥 = 𝑦))
158		lestr 28119	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ((𝑥 ≤s 𝑦 ∧ 𝑦 ≤s 𝑧) → 𝑥 ≤s 𝑧))
159	6, 7, 8, 158	syl3an 1178	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 ≤s 𝑦 ∧ 𝑦 ≤s 𝑧) → 𝑥 ≤s 𝑧))
160	141	biantrud 541	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑥 ≤s 𝑦 ↔ (𝑥 ≤s 𝑦 ∧ 𝑥(ℤs × ℤs)𝑦)))
161	139, 160	bitr4id 293	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ↔ 𝑥 ≤s 𝑦))
162	161	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ↔ 𝑥 ≤s 𝑦))
163		brin 5157	. . . . . . . . . . 11 ⊢ (𝑦( ≤s ∩ (ℤs × ℤs))𝑧 ↔ (𝑦 ≤s 𝑧 ∧ 𝑦(ℤs × ℤs)𝑧))
164		brxp 5700	. . . . . . . . . . . . . 14 ⊢ (𝑦(ℤs × ℤs)𝑧 ↔ (𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs))
165	164	biimpri 231	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑦(ℤs × ℤs)𝑧)
166	165	3adant1 1148	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑦(ℤs × ℤs)𝑧)
167	166	biantrud 541	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑦 ≤s 𝑧 ↔ (𝑦 ≤s 𝑧 ∧ 𝑦(ℤs × ℤs)𝑧)))
168	163, 167	bitr4id 293	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑦( ≤s ∩ (ℤs × ℤs))𝑧 ↔ 𝑦 ≤s 𝑧))
169	162, 168	anbi12d 644	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∧ 𝑦( ≤s ∩ (ℤs × ℤs))𝑧) ↔ (𝑥 ≤s 𝑦 ∧ 𝑦 ≤s 𝑧)))
170		brin 5157	. . . . . . . . . 10 ⊢ (𝑥( ≤s ∩ (ℤs × ℤs))𝑧 ↔ (𝑥 ≤s 𝑧 ∧ 𝑥(ℤs × ℤs)𝑧))
171		brxp 5700	. . . . . . . . . . . . 13 ⊢ (𝑥(ℤs × ℤs)𝑧 ↔ (𝑥 ∈ ℤs ∧ 𝑧 ∈ ℤs))
172	171	biimpri 231	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑥(ℤs × ℤs)𝑧)
173	172	3adant2 1149	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → 𝑥(ℤs × ℤs)𝑧)
174	173	biantrud 541	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 ≤s 𝑧 ↔ (𝑥 ≤s 𝑧 ∧ 𝑥(ℤs × ℤs)𝑧)))
175	170, 174	bitr4id 293	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ≤s ∩ (ℤs × ℤs))𝑧 ↔ 𝑥 ≤s 𝑧))
176	159, 169, 175	3imtr4d 297	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∧ 𝑦( ≤s ∩ (ℤs × ℤs))𝑧) → 𝑥( ≤s ∩ (ℤs × ℤs))𝑧))
177	138, 157, 176	3jca 1146	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ≤s ∩ (ℤs × ℤs))𝑥 ∧ ((𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∧ 𝑦( ≤s ∩ (ℤs × ℤs))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∧ 𝑦( ≤s ∩ (ℤs × ℤs))𝑧) → 𝑥( ≤s ∩ (ℤs × ℤs))𝑧)))
178	177	rgen3 3208	. . . . . 6 ⊢ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs ∀𝑧 ∈ ℤs (𝑥( ≤s ∩ (ℤs × ℤs))𝑥 ∧ ((𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∧ 𝑦( ≤s ∩ (ℤs × ℤs))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∧ 𝑦( ≤s ∩ (ℤs × ℤs))𝑧) → 𝑥( ≤s ∩ (ℤs × ℤs))𝑧))
179		zsoring.4	. . . . . . 7 ⊢ ( ≤s ∩ (ℤs × ℤs)) = (le‘𝐾)
180	1, 179	ispos 18488	. . . . . 6 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs ∀𝑧 ∈ ℤs (𝑥( ≤s ∩ (ℤs × ℤs))𝑥 ∧ ((𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∧ 𝑦( ≤s ∩ (ℤs × ℤs))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∧ 𝑦( ≤s ∩ (ℤs × ℤs))𝑧) → 𝑥( ≤s ∩ (ℤs × ℤs))𝑧))))
181	130, 178, 180	mpbir2an 724	. . . . 5 ⊢ 𝐾 ∈ Poset
182		lestric 28125	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (𝑥 ≤s 𝑦 ∨ 𝑦 ≤s 𝑥))
183	6, 7, 182	syl2an 608	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑥 ≤s 𝑦 ∨ 𝑦 ≤s 𝑥))
184	161, 150	orbi12d 932	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → ((𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∨ 𝑦( ≤s ∩ (ℤs × ℤs))𝑥) ↔ (𝑥 ≤s 𝑦 ∨ 𝑦 ≤s 𝑥)))
185	183, 184	mpbird 260	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs) → (𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∨ 𝑦( ≤s ∩ (ℤs × ℤs))𝑥))
186	185	rgen2 3203	. . . . 5 ⊢ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs (𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∨ 𝑦( ≤s ∩ (ℤs × ℤs))𝑥)
187	1, 179	istos 18590	. . . . 5 ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs (𝑥( ≤s ∩ (ℤs × ℤs))𝑦 ∨ 𝑦( ≤s ∩ (ℤs × ℤs))𝑥)))
188	181, 186, 187	mpbir2an 724	. . . 4 ⊢ 𝐾 ∈ Toset
189		leadds1 28375	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → (𝑥 ≤s 𝑦 ↔ (𝑥 +s 𝑧) ≤s (𝑦 +s 𝑧)))
190	6, 7, 8, 189	syl3an 1178	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 ≤s 𝑦 ↔ (𝑥 +s 𝑧) ≤s (𝑦 +s 𝑧)))
191	190	biimpd 232	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 ≤s 𝑦 → (𝑥 +s 𝑧) ≤s (𝑦 +s 𝑧)))
192	20, 14	ovresd 7587	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( +s ↾ (ℤs × ℤs))𝑧) = (𝑥 +s 𝑧))
193	49, 14	ovresd 7587	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑦( +s ↾ (ℤs × ℤs))𝑧) = (𝑦 +s 𝑧))
194	192, 193	breq12d 5116	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( +s ↾ (ℤs × ℤs))𝑧)( ≤s ∩ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧) ↔ (𝑥 +s 𝑧)( ≤s ∩ (ℤs × ℤs))(𝑦 +s 𝑧)))
195		brin 5157	. . . . . . . 8 ⊢ ((𝑥 +s 𝑧)( ≤s ∩ (ℤs × ℤs))(𝑦 +s 𝑧) ↔ ((𝑥 +s 𝑧) ≤s (𝑦 +s 𝑧) ∧ (𝑥 +s 𝑧)(ℤs × ℤs)(𝑦 +s 𝑧)))
196		zaddscl 28780	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 +s 𝑧) ∈ ℤs)
197	196	3adant2 1149	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 +s 𝑧) ∈ ℤs)
198		brxp 5700	. . . . . . . . . 10 ⊢ ((𝑥 +s 𝑧)(ℤs × ℤs)(𝑦 +s 𝑧) ↔ ((𝑥 +s 𝑧) ∈ ℤs ∧ (𝑦 +s 𝑧) ∈ ℤs))
199	197, 22, 198	sylanbrc 595	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥 +s 𝑧)(ℤs × ℤs)(𝑦 +s 𝑧))
200	199	biantrud 541	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 +s 𝑧) ≤s (𝑦 +s 𝑧) ↔ ((𝑥 +s 𝑧) ≤s (𝑦 +s 𝑧) ∧ (𝑥 +s 𝑧)(ℤs × ℤs)(𝑦 +s 𝑧))))
201	195, 200	bitr4id 293	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥 +s 𝑧)( ≤s ∩ (ℤs × ℤs))(𝑦 +s 𝑧) ↔ (𝑥 +s 𝑧) ≤s (𝑦 +s 𝑧)))
202	194, 201	bitrd 282	. . . . . 6 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → ((𝑥( +s ↾ (ℤs × ℤs))𝑧)( ≤s ∩ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧) ↔ (𝑥 +s 𝑧) ≤s (𝑦 +s 𝑧)))
203	191, 144, 202	3imtr4d 297	. . . . 5 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℤs ∧ 𝑧 ∈ ℤs) → (𝑥( ≤s ∩ (ℤs × ℤs))𝑦 → (𝑥( +s ↾ (ℤs × ℤs))𝑧)( ≤s ∩ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧)))
204	203	rgen3 3208	. . . 4 ⊢ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs ∀𝑧 ∈ ℤs (𝑥( ≤s ∩ (ℤs × ℤs))𝑦 → (𝑥( +s ↾ (ℤs × ℤs))𝑧)( ≤s ∩ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧))
205	1, 2, 179	isomnd 20337	. . . 4 ⊢ (𝐾 ∈ oMnd ↔ (𝐾 ∈ Mnd ∧ 𝐾 ∈ Toset ∧ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs ∀𝑧 ∈ ℤs (𝑥( ≤s ∩ (ℤs × ℤs))𝑦 → (𝑥( +s ↾ (ℤs × ℤs))𝑧)( ≤s ∩ (ℤs × ℤs))(𝑦( +s ↾ (ℤs × ℤs))𝑧))))
206	129, 188, 204, 205	mpbir3an 1360	. . 3 ⊢ 𝐾 ∈ oMnd
207		isogrp 20338	. . 3 ⊢ (𝐾 ∈ oGrp ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ oMnd))
208	39, 206, 207	mpbir2an 724	. 2 ⊢ 𝐾 ∈ oGrp
209		simplr 781	. . . . . . . 8 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → 𝑥 ∈ ℤs)
210	209	znod 28769	. . . . . . 7 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → 𝑥 ∈ No )
211		simprr 785	. . . . . . . 8 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → 𝑦 ∈ ℤs)
212	211	znod 28769	. . . . . . 7 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → 𝑦 ∈ No )
213		simpll 779	. . . . . . 7 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → 0s ≤s 𝑥)
214		simprl 783	. . . . . . 7 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → 0s ≤s 𝑦)
215	210, 212, 213, 214	mulsge0d 28532	. . . . . 6 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → 0s ≤s (𝑥 ·s 𝑦))
216	209, 211	ovresd 7587	. . . . . 6 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → (𝑥( ·s ↾ (ℤs × ℤs))𝑦) = (𝑥 ·s 𝑦))
217	215, 216	breqtrrd 5133	. . . . 5 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → 0s ≤s (𝑥( ·s ↾ (ℤs × ℤs))𝑦))
218	209, 211	zmulscld 28783	. . . . . 6 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → (𝑥 ·s 𝑦) ∈ ℤs)
219	216, 218	eqeltrd 2861	. . . . 5 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∈ ℤs)
220	217, 219	jca 521	. . . 4 ⊢ ((( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)) → ( 0s ≤s (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∧ (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∈ ℤs))
221		brin 5157	. . . . . 6 ⊢ ( 0s ( ≤s ∩ (ℤs × ℤs))𝑥 ↔ ( 0s ≤s 𝑥 ∧ 0s (ℤs × ℤs)𝑥))
222		brxp 5700	. . . . . . . 8 ⊢ ( 0s (ℤs × ℤs)𝑥 ↔ ( 0s ∈ ℤs ∧ 𝑥 ∈ ℤs))
223	26, 222	mpbiran 722	. . . . . . 7 ⊢ ( 0s (ℤs × ℤs)𝑥 ↔ 𝑥 ∈ ℤs)
224	223	anbi2i 635	. . . . . 6 ⊢ (( 0s ≤s 𝑥 ∧ 0s (ℤs × ℤs)𝑥) ↔ ( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs))
225	221, 224	bitri 278	. . . . 5 ⊢ ( 0s ( ≤s ∩ (ℤs × ℤs))𝑥 ↔ ( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs))
226		brin 5157	. . . . . 6 ⊢ ( 0s ( ≤s ∩ (ℤs × ℤs))𝑦 ↔ ( 0s ≤s 𝑦 ∧ 0s (ℤs × ℤs)𝑦))
227		brxp 5700	. . . . . . . 8 ⊢ ( 0s (ℤs × ℤs)𝑦 ↔ ( 0s ∈ ℤs ∧ 𝑦 ∈ ℤs))
228	26, 227	mpbiran 722	. . . . . . 7 ⊢ ( 0s (ℤs × ℤs)𝑦 ↔ 𝑦 ∈ ℤs)
229	228	anbi2i 635	. . . . . 6 ⊢ (( 0s ≤s 𝑦 ∧ 0s (ℤs × ℤs)𝑦) ↔ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs))
230	226, 229	bitri 278	. . . . 5 ⊢ ( 0s ( ≤s ∩ (ℤs × ℤs))𝑦 ↔ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs))
231	225, 230	anbi12i 640	. . . 4 ⊢ (( 0s ( ≤s ∩ (ℤs × ℤs))𝑥 ∧ 0s ( ≤s ∩ (ℤs × ℤs))𝑦) ↔ (( 0s ≤s 𝑥 ∧ 𝑥 ∈ ℤs) ∧ ( 0s ≤s 𝑦 ∧ 𝑦 ∈ ℤs)))
232		brin 5157	. . . . 5 ⊢ ( 0s ( ≤s ∩ (ℤs × ℤs))(𝑥( ·s ↾ (ℤs × ℤs))𝑦) ↔ ( 0s ≤s (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∧ 0s (ℤs × ℤs)(𝑥( ·s ↾ (ℤs × ℤs))𝑦)))
233		brxp 5700	. . . . . . 7 ⊢ ( 0s (ℤs × ℤs)(𝑥( ·s ↾ (ℤs × ℤs))𝑦) ↔ ( 0s ∈ ℤs ∧ (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∈ ℤs))
234	26, 233	mpbiran 722	. . . . . 6 ⊢ ( 0s (ℤs × ℤs)(𝑥( ·s ↾ (ℤs × ℤs))𝑦) ↔ (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∈ ℤs)
235	234	anbi2i 635	. . . . 5 ⊢ (( 0s ≤s (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∧ 0s (ℤs × ℤs)(𝑥( ·s ↾ (ℤs × ℤs))𝑦)) ↔ ( 0s ≤s (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∧ (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∈ ℤs))
236	232, 235	bitri 278	. . . 4 ⊢ ( 0s ( ≤s ∩ (ℤs × ℤs))(𝑥( ·s ↾ (ℤs × ℤs))𝑦) ↔ ( 0s ≤s (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∧ (𝑥( ·s ↾ (ℤs × ℤs))𝑦) ∈ ℤs))
237	220, 231, 236	3imtr4i 295	. . 3 ⊢ (( 0s ( ≤s ∩ (ℤs × ℤs))𝑥 ∧ 0s ( ≤s ∩ (ℤs × ℤs))𝑦) → 0s ( ≤s ∩ (ℤs × ℤs))(𝑥( ·s ↾ (ℤs × ℤs))𝑦))
238	237	rgen2w 3082	. 2 ⊢ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs (( 0s ( ≤s ∩ (ℤs × ℤs))𝑥 ∧ 0s ( ≤s ∩ (ℤs × ℤs))𝑦) → 0s ( ≤s ∩ (ℤs × ℤs))(𝑥( ·s ↾ (ℤs × ℤs))𝑦))
239		zsoring.5	. . 3 ⊢ 0s = (0g‘𝐾)
240	1, 239, 82, 179	isorng 21118	. 2 ⊢ (𝐾 ∈ oRing ↔ (𝐾 ∈ Ring ∧ 𝐾 ∈ oGrp ∧ ∀𝑥 ∈ ℤs ∀𝑦 ∈ ℤs (( 0s ( ≤s ∩ (ℤs × ℤs))𝑥 ∧ 0s ( ≤s ∩ (ℤs × ℤs))𝑦) → 0s ( ≤s ∩ (ℤs × ℤs))(𝑥( ·s ↾ (ℤs × ℤs))𝑦))))
241	112, 208, 238, 240	mpbir3an 1360	1 ⊢ 𝐾 ∈ oRing

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∩ cin 3898   class class class wbr 5103   × cxp 5649   ↾ cres 5653  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  ._rcmulr 17429  lecple 17435  0_gc0g 17610  Posetcpo 18481  Tosetctos 18588  Mndcmnd 18923  Grpcgrp 19144  oMndcomnd 20333  oGrpcogrp 20334  mulGrpcmgp 20360  Ringcrg 20459  oRingcorng 21114   No csur 27997   ≤s cles 28101   0_s c0s 28191   1_s c1s 28192   +_s cadds 28345   -u_s cnegs 28405   ·_s cmuls 28492  ℤ_sczs 28764

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 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

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-nel 3063  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 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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-nadd 8675  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-plusg 17441  df-0g 17612  df-poset 18487  df-toset 18589  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-omnd 20335  df-ogrp 20336  df-mgp 20361  df-ring 20461  df-orng 21116  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-muls 28493  df-n0s 28700  df-nns 28701  df-zs 28765

This theorem is used by: (None)