Theorem cncrng 21597

Description: The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.) Avoid ax-mulf 11199. (Revised by GG, 31-Mar-2025.)

Assertion

Ref	Expression
cncrng	⊢ ℂfld ∈ CRing

Proof of Theorem cncrng

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

Step	Hyp	Ref	Expression
1		cnfldbas 21580	. . . 4 ⊢ ℂ = (Base‘ℂfld)
2	1	a1i 11	. . 3 ⊢ (⊤ → ℂ = (Base‘ℂfld))
3		cnfldadd 21582	. . . 4 ⊢ + = (+g‘ℂfld)
4	3	a1i 11	. . 3 ⊢ (⊤ → + = (+g‘ℂfld))
5		mpocnfldmul 21583	. . . 4 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (.r‘ℂfld)
6	5	a1i 11	. . 3 ⊢ (⊤ → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (.r‘ℂfld))
7		addcl 11201	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
8		addass 11206	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9		0cn 11217	. . . . 5 ⊢ 0 ∈ ℂ
10		addlid 11412	. . . . 5 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥)
11		negcl 11476	. . . . 5 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ)
12		id 23	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
13	11, 12	addcomd 11431	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = (𝑥 + -𝑥))
14		negid 11524	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0)
15	13, 14	eqtrd 2800	. . . . 5 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = 0)
16	1, 3, 7, 8, 9, 10, 11, 15	isgrpi 19074	. . . 4 ⊢ ℂfld ∈ Grp
17	16	a1i 11	. . 3 ⊢ (⊤ → ℂfld ∈ Grp)
18		mpomulf 11214	. . . . 5 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)):(ℂ × ℂ)⟶ℂ
19	18	fovcl 7547	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) ∈ ℂ)
20	19	3adant1 1148	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) ∈ ℂ)
21		mulass 11207	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
22		mulcl 11203	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
23		ovmpot 7580	. . . . . . 7 ⊢ (((𝑥 · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 · 𝑦)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = ((𝑥 · 𝑦) · 𝑧))
24	22, 23	stoic3 1809	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 · 𝑦)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = ((𝑥 · 𝑦) · 𝑧))
25		simp1 1154	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
26		mulcl 11203	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 · 𝑧) ∈ ℂ)
27	26	3adant1 1148	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 · 𝑧) ∈ ℂ)
28		ovmpot 7580	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ (𝑦 · 𝑧) ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑦 · 𝑧)) = (𝑥 · (𝑦 · 𝑧)))
29	25, 27, 28	syl2anc 596	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑦 · 𝑧)) = (𝑥 · (𝑦 · 𝑧)))
30	21, 24, 29	3eqtr4d 2810	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 · 𝑦)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑦 · 𝑧)))
31		ovmpot 7580	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) = (𝑥 · 𝑦))
32	31	3adant3 1150	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) = (𝑥 · 𝑦))
33	32	oveq1d 7434	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = ((𝑥 · 𝑦)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧))
34		ovmpot 7580	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = (𝑦 · 𝑧))
35	34	3adant1 1148	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = (𝑦 · 𝑧))
36	35	oveq2d 7435	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧)) = (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑦 · 𝑧)))
37	30, 33, 36	3eqtr4d 2810	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧)))
38	37	adantl 487	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧)))
39		adddi 11208	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
40		addcl 11201	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 + 𝑧) ∈ ℂ)
41	40	3adant1 1148	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 + 𝑧) ∈ ℂ)
42		ovmpot 7580	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (𝑦 + 𝑧) ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑦 + 𝑧)) = (𝑥 · (𝑦 + 𝑧)))
43	25, 41, 42	syl2anc 596	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑦 + 𝑧)) = (𝑥 · (𝑦 + 𝑧)))
44		ovmpot 7580	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = (𝑥 · 𝑧))
45	44	3adant2 1149	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = (𝑥 · 𝑧))
46	32, 45	oveq12d 7437	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) + (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
47	39, 43, 46	3eqtr4d 2810	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑦 + 𝑧)) = ((𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) + (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧)))
48	47	adantl 487	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑦 + 𝑧)) = ((𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) + (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧)))
49		adddir 11216	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
50		ovmpot 7580	. . . . . 6 ⊢ (((𝑥 + 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = ((𝑥 + 𝑦) · 𝑧))
51	7, 50	stoic3 1809	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = ((𝑥 + 𝑦) · 𝑧))
52	45, 35	oveq12d 7437	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) + (𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧)) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
53	49, 51, 52	3eqtr4d 2810	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = ((𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) + (𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧)))
54	53	adantl 487	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 + 𝑦)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) = ((𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧) + (𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑧)))
55		1cnd 11221	. . 3 ⊢ (⊤ → 1 ∈ ℂ)
56		ax-1cn 11177	. . . . . 6 ⊢ 1 ∈ ℂ
57		ovmpot 7580	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑥) = (1 · 𝑥))
58	56, 57	mpan 703	. . . . 5 ⊢ (𝑥 ∈ ℂ → (1(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑥) = (1 · 𝑥))
59		mullid 11226	. . . . 5 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥)
60	58, 59	eqtrd 2800	. . . 4 ⊢ (𝑥 ∈ ℂ → (1(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑥) = 𝑥)
61	60	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (1(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑥) = 𝑥)
62		ovmpot 7580	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))1) = (𝑥 · 1))
63	56, 62	mpan2 704	. . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))1) = (𝑥 · 1))
64		mulrid 11225	. . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 · 1) = 𝑥)
65	63, 64	eqtrd 2800	. . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))1) = 𝑥)
66	65	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))1) = 𝑥)
67		mulcom 11205	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
68		ovmpot 7580	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑥) = (𝑦 · 𝑥))
69	68	ancoms 464	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑥) = (𝑦 · 𝑥))
70	67, 31, 69	3eqtr4d 2810	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) = (𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑥))
71	70	3adant1 1148	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) = (𝑦(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑥))
72	2, 4, 6, 17, 20, 38, 48, 54, 55, 61, 66, 71	iscrngd 20425	. 2 ⊢ (⊤ → ℂfld ∈ CRing)
73	72	mptru 1577	1 ⊢ ℂfld ∈ CRing

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  ℂcc 11117  0cc0 11119  1c1 11120   + caddc 11122   · cmul 11124  -cneg 11461  Basecbs 17295  +_gcplusg 17336  ._rcmulr 17337  Grpcgrp 19048  CRingccrg 20364  ℂ_fldccnfld 21576

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11175  ax-resscn 11176  ax-1cn 11177  ax-icn 11178  ax-addcl 11179  ax-addrcl 11180  ax-mulcl 11181  ax-mulrcl 11182  ax-mulcom 11183  ax-addass 11184  ax-mulass 11185  ax-distr 11186  ax-i2m1 11187  ax-1ne0 11188  ax-1rid 11189  ax-rnegex 11190  ax-rrecex 11191  ax-cnre 11192  ax-pre-lttri 11193  ax-pre-lttrn 11194  ax-pre-ltadd 11195  ax-pre-mulgt0 11196  ax-addf 11198

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  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 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11264  df-mnf 11265  df-xr 11266  df-ltxr 11267  df-le 11268  df-sub 11462  df-neg 11463  df-nn 12253  df-2 12322  df-3 12323  df-4 12324  df-5 12325  df-6 12326  df-7 12327  df-8 12328  df-9 12329  df-n0 12524  df-z 12611  df-dec 12732  df-uz 12883  df-fz 13556  df-struct 17233  df-sets 17250  df-slot 17268  df-ndx 17280  df-base 17296  df-plusg 17349  df-mulr 17350  df-starv 17351  df-tset 17355  df-ple 17356  df-ds 17358  df-unif 17359  df-0g 17520  df-mgm 18724  df-sgrp 18813  df-mnd 18829  df-grp 19051  df-cmn 19900  df-mgp 20265  df-ring 20365  df-cring 20366  df-cnfld 21577

This theorem is used by:  cnring  21598  cnmgpabl  21632  zringcrng  21652  zring0  21662  re0g  21816  refld  21823  smadiadetr  22886  plypf1  26424  amgmlem  27209  amgm  27210  wilthlem2  27288  wilthlem3  27289  qfld  33686  gzcrng  33729  cnfldfld  33730  ccfldextrr  34104  2sqr3minply  34238  cos9thpiminplylem6  34245  cos9thpiminply  34246  2zrng0  49085  amgmwlem  50726  amgmlemALT  50727