Theorem expghmap 14645

Description: Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.) (Revised by Jim Kingdon, 11-Sep-2025.)

Hypotheses

Ref	Expression
expghm.m	⊢ 𝑀 = (mulGrp‘ℂfld)
expghmap.u	⊢ 𝑈 = (𝑀 ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0})

Assertion

Ref	Expression
expghmap	⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈))

Distinct variable group:   𝑥,𝐴,𝑧

Allowed substitution hints:   𝑈(𝑥,𝑧)   𝑀(𝑥,𝑧)

Proof of Theorem expghmap

Dummy variables 𝑟 𝑠 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		expclzaplem 10831	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})
2	1	3expa 1229	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})
3	2	fmpttd 5805	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)):ℤ⟶{𝑧 ∈ ℂ ∣ 𝑧 # 0})
4		expaddzap 10851	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝐴↑(𝑢 + 𝑣)) = ((𝐴↑𝑢) · (𝐴↑𝑣)))
5		eqid 2230	. . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) = (𝑥 ∈ ℤ ↦ (𝐴↑𝑥))
6		oveq2 6031	. . . . . 6 ⊢ (𝑥 = (𝑢 + 𝑣) → (𝐴↑𝑥) = (𝐴↑(𝑢 + 𝑣)))
7		zaddcl 9524	. . . . . . 7 ⊢ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) → (𝑢 + 𝑣) ∈ ℤ)
8	7	adantl 277	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝑢 + 𝑣) ∈ ℤ)
9		simpll 527	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → 𝐴 ∈ ℂ)
10		simplr 529	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → 𝐴 # 0)
11	9, 10, 8	expclzapd 10946	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝐴↑(𝑢 + 𝑣)) ∈ ℂ)
12	5, 6, 8, 11	fvmptd3 5743	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (𝐴↑(𝑢 + 𝑣)))
13		oveq2 6031	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝐴↑𝑥) = (𝐴↑𝑢))
14		simprl 531	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → 𝑢 ∈ ℤ)
15	9, 10, 14	expclzapd 10946	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝐴↑𝑢) ∈ ℂ)
16	5, 13, 14, 15	fvmptd3 5743	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) = (𝐴↑𝑢))
17		oveq2 6031	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (𝐴↑𝑥) = (𝐴↑𝑣))
18		simprr 533	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → 𝑣 ∈ ℤ)
19	9, 10, 18	expclzapd 10946	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝐴↑𝑣) ∈ ℂ)
20	5, 17, 18, 19	fvmptd3 5743	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣) = (𝐴↑𝑣))
21	16, 20	oveq12d 6041	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)) = ((𝐴↑𝑢) · (𝐴↑𝑣)))
22	4, 12, 21	3eqtr4d 2273	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)))
23	22	ralrimivva 2613	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ∀𝑢 ∈ ℤ ∀𝑣 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)))
24		simplr 529	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → 𝑢 ∈ ℤ)
25	15	anassrs 400	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → (𝐴↑𝑢) ∈ ℂ)
26	5, 13, 24, 25	fvmptd3 5743	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) = (𝐴↑𝑢))
27	26, 25	eqeltrd 2307	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) ∈ ℂ)
28		simpr 110	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → 𝑣 ∈ ℤ)
29	19	anassrs 400	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → (𝐴↑𝑣) ∈ ℂ)
30	5, 17, 28, 29	fvmptd3 5743	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣) = (𝐴↑𝑣))
31	30, 29	eqeltrd 2307	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣) ∈ ℂ)
32	27, 31	mulcld 8205	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)) ∈ ℂ)
33		oveq1 6030	. . . . . . . 8 ⊢ (𝑟 = ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) → (𝑟 · 𝑠) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · 𝑠))
34		oveq2 6031	. . . . . . . 8 ⊢ (𝑠 = ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣) → (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · 𝑠) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)))
35		eqid 2230	. . . . . . . 8 ⊢ (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠)) = (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠))
36	33, 34, 35	ovmpog 6161	. . . . . . 7 ⊢ ((((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) ∈ ℂ ∧ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣) ∈ ℂ ∧ (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)) ∈ ℂ) → (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢)(𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠))((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)))
37	27, 31, 32, 36	syl3anc 1273	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢)(𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠))((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)))
38	37	eqeq2d 2242	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢)(𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠))((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)) ↔ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣))))
39	38	ralbidva 2527	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) → (∀𝑣 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢)(𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠))((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)) ↔ ∀𝑣 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣))))
40	39	ralbidva 2527	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (∀𝑢 ∈ ℤ ∀𝑣 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢)(𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠))((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)) ↔ ∀𝑢 ∈ ℤ ∀𝑣 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣))))
41	23, 40	mpbird 167	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ∀𝑢 ∈ ℤ ∀𝑣 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢)(𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠))((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)))
42		zringgrp 14633	. . . 4 ⊢ ℤring ∈ Grp
43		cnring 14608	. . . . 5 ⊢ ℂfld ∈ Ring
44		cnfldui 14627	. . . . . 6 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Unit‘ℂfld)
45		expghmap.u	. . . . . . 7 ⊢ 𝑈 = (𝑀 ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0})
46		expghm.m	. . . . . . . 8 ⊢ 𝑀 = (mulGrp‘ℂfld)
47	46	oveq1i 6033	. . . . . . 7 ⊢ (𝑀 ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0}) = ((mulGrp‘ℂfld) ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0})
48	45, 47	eqtri 2251	. . . . . 6 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0})
49	44, 48	unitgrp 14154	. . . . 5 ⊢ (ℂfld ∈ Ring → 𝑈 ∈ Grp)
50	43, 49	ax-mp 5	. . . 4 ⊢ 𝑈 ∈ Grp
51	42, 50	pm3.2i 272	. . 3 ⊢ (ℤring ∈ Grp ∧ 𝑈 ∈ Grp)
52		zringbas 14634	. . . 4 ⊢ ℤ = (Base‘ℤring)
53	45	a1i 9	. . . . . 6 ⊢ (⊤ → 𝑈 = (𝑀 ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0}))
54		cnfldbas 14598	. . . . . . . 8 ⊢ ℂ = (Base‘ℂfld)
55	46, 54	mgpbasg 13963	. . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂ = (Base‘𝑀))
56	43, 55	mp1i 10	. . . . . 6 ⊢ (⊤ → ℂ = (Base‘𝑀))
57	46	mgpex 13962	. . . . . . 7 ⊢ (ℂfld ∈ Ring → 𝑀 ∈ V)
58	43, 57	mp1i 10	. . . . . 6 ⊢ (⊤ → 𝑀 ∈ V)
59		apsscn 8832	. . . . . . 7 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ⊆ ℂ
60	59	a1i 9	. . . . . 6 ⊢ (⊤ → {𝑧 ∈ ℂ ∣ 𝑧 # 0} ⊆ ℂ)
61	53, 56, 58, 60	ressbas2d 13174	. . . . 5 ⊢ (⊤ → {𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Base‘𝑈))
62	61	mptru 1406	. . . 4 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Base‘𝑈)
63		zringplusg 14635	. . . 4 ⊢ + = (+g‘ℤring)
64		mpocnfldmul 14601	. . . . . . . 8 ⊢ (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠)) = (.r‘ℂfld)
65	46, 64	mgpplusgg 13961	. . . . . . 7 ⊢ (ℂfld ∈ Ring → (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠)) = (+g‘𝑀))
66	43, 65	mp1i 10	. . . . . 6 ⊢ (⊤ → (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠)) = (+g‘𝑀))
67		cnex 8161	. . . . . . . 8 ⊢ ℂ ∈ V
68	67	rabex 4235	. . . . . . 7 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∈ V
69	68	a1i 9	. . . . . 6 ⊢ (⊤ → {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∈ V)
70	53, 66, 69, 58	ressplusgd 13235	. . . . 5 ⊢ (⊤ → (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠)) = (+g‘𝑈))
71	70	mptru 1406	. . . 4 ⊢ (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠)) = (+g‘𝑈)
72	52, 62, 63, 71	isghm 13853	. . 3 ⊢ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈) ↔ ((ℤring ∈ Grp ∧ 𝑈 ∈ Grp) ∧ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥)):ℤ⟶{𝑧 ∈ ℂ ∣ 𝑧 # 0} ∧ ∀𝑢 ∈ ℤ ∀𝑣 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢)(𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠))((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)))))
73	51, 72	mpbiran 948	. 2 ⊢ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈) ↔ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥)):ℤ⟶{𝑧 ∈ ℂ ∣ 𝑧 # 0} ∧ ∀𝑢 ∈ ℤ ∀𝑣 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢)(𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠))((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣))))
74	3, 41, 73	sylanbrc 417	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397  ⊤wtru 1398   ∈ wcel 2201  ∀wral 2509  {crab 2513  Vcvv 2801   ⊆ wss 3199   class class class wbr 4089   ↦ cmpt 4151  ⟶wf 5324  ‘cfv 5328  (class class class)co 6023   ∈ cmpo 6025  ℂcc 8035  0cc0 8037   + caddc 8040   · cmul 8042   # cap 8766  ℤcz 9484  ↑cexp 10806  Basecbs 13105   ↾_s cress 13106  +_gcplusg 13183  Grpcgrp 13606   GrpHom cghm 13850  mulGrpcmgp 13957  Ringcrg 14033  ℂ_fldccnfld 14594  ℤ_ringczring 14628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155  ax-addf 8159  ax-mulf 8160

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-tp 3678  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-tpos 6416  df-recs 6476  df-frec 6562  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-z 9485  df-dec 9617  df-uz 9761  df-rp 9894  df-fz 10249  df-seqfrec 10716  df-exp 10807  df-cj 11425  df-abs 11582  df-struct 13107  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112  df-iress 13113  df-plusg 13196  df-mulr 13197  df-starv 13198  df-tset 13202  df-ple 13203  df-ds 13205  df-unif 13206  df-0g 13364  df-topgen 13366  df-mgm 13462  df-sgrp 13508  df-mnd 13523  df-grp 13609  df-minusg 13610  df-subg 13780  df-ghm 13851  df-cmn 13896  df-abl 13897  df-mgp 13958  df-ur 13997  df-srg 14001  df-ring 14035  df-cring 14036  df-oppr 14105  df-dvdsr 14126  df-unit 14127  df-subrg 14257  df-bl 14584  df-mopn 14585  df-fg 14587  df-metu 14588  df-cnfld 14595  df-zring 14629

This theorem is referenced by:  lgseisenlem4  15831