Theorem expghmap 14804

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 10932	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})
2	1	3expa 1230	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})
3	2	fmpttd 5834	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)):ℤ⟶{𝑧 ∈ ℂ ∣ 𝑧 # 0})
4		expaddzap 10952	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝐴↑(𝑢 + 𝑣)) = ((𝐴↑𝑢) · (𝐴↑𝑣)))
5		eqid 2234	. . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) = (𝑥 ∈ ℤ ↦ (𝐴↑𝑥))
6		oveq2 6060	. . . . . 6 ⊢ (𝑥 = (𝑢 + 𝑣) → (𝐴↑𝑥) = (𝐴↑(𝑢 + 𝑣)))
7		zaddcl 9622	. . . . . . 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 11048	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝐴↑(𝑢 + 𝑣)) ∈ ℂ)
12	5, 6, 8, 11	fvmptd3 5773	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑢 + 𝑣)) = (𝐴↑(𝑢 + 𝑣)))
13		oveq2 6060	. . . . . . 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 11048	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝐴↑𝑢) ∈ ℂ)
16	5, 13, 14, 15	fvmptd3 5773	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) = (𝐴↑𝑢))
17		oveq2 6060	. . . . . . 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 11048	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝐴↑𝑣) ∈ ℂ)
20	5, 17, 18, 19	fvmptd3 5773	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣) = (𝐴↑𝑣))
21	16, 20	oveq12d 6070	. . . . 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 2277	. . . 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 2626	. . 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 5773	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) = (𝐴↑𝑢))
27	26, 25	eqeltrd 2311	. . . . . . 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 5773	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣) = (𝐴↑𝑣))
31	30, 29	eqeltrd 2311	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣) ∈ ℂ)
32	27, 31	mulcld 8299	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑢 ∈ ℤ) ∧ 𝑣 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑣)) ∈ ℂ)
33		oveq1 6059	. . . . . . . 8 ⊢ (𝑟 = ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) → (𝑟 · 𝑠) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑢) · 𝑠))
34		oveq2 6060	. . . . . . . 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 2234	. . . . . . . 8 ⊢ (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠)) = (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠))
36	33, 34, 35	ovmpog 6190	. . . . . . 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 1274	. . . . . 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 2246	. . . . 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 2540	. . . 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 2540	. . 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 14792	. . . 4 ⊢ ℤring ∈ Grp
43		cnring 14767	. . . . 5 ⊢ ℂfld ∈ Ring
44		cnfldui 14786	. . . . . 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 6062	. . . . . . 7 ⊢ (𝑀 ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0}) = ((mulGrp‘ℂfld) ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0})
48	45, 47	eqtri 2255	. . . . . 6 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0})
49	44, 48	unitgrp 14283	. . . . 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 14793	. . . 4 ⊢ ℤ = (Base‘ℤring)
53	45	a1i 9	. . . . . 6 ⊢ (⊤ → 𝑈 = (𝑀 ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0}))
54		cnfldbas 14757	. . . . . . . 8 ⊢ ℂ = (Base‘ℂfld)
55	46, 54	mgpbasg 14091	. . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂ = (Base‘𝑀))
56	43, 55	mp1i 10	. . . . . 6 ⊢ (⊤ → ℂ = (Base‘𝑀))
57	46	mgpex 14090	. . . . . . 7 ⊢ (ℂfld ∈ Ring → 𝑀 ∈ V)
58	43, 57	mp1i 10	. . . . . 6 ⊢ (⊤ → 𝑀 ∈ V)
59		apsscn 8926	. . . . . . 7 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ⊆ ℂ
60	59	a1i 9	. . . . . 6 ⊢ (⊤ → {𝑧 ∈ ℂ ∣ 𝑧 # 0} ⊆ ℂ)
61	53, 56, 58, 60	ressbas2d 13302	. . . . 5 ⊢ (⊤ → {𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Base‘𝑈))
62	61	mptru 1407	. . . 4 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Base‘𝑈)
63		zringplusg 14794	. . . 4 ⊢ + = (+g‘ℤring)
64		mpocnfldmul 14760	. . . . . . . 8 ⊢ (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠)) = (.r‘ℂfld)
65	46, 64	mgpplusgg 14089	. . . . . . 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 8256	. . . . . . . 8 ⊢ ℂ ∈ V
68	67	rabex 4258	. . . . . . 7 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∈ V
69	68	a1i 9	. . . . . 6 ⊢ (⊤ → {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∈ V)
70	53, 66, 69, 58	ressplusgd 13363	. . . . 5 ⊢ (⊤ → (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠)) = (+g‘𝑈))
71	70	mptru 1407	. . . 4 ⊢ (𝑟 ∈ ℂ, 𝑠 ∈ ℂ ↦ (𝑟 · 𝑠)) = (+g‘𝑈)
72	52, 62, 63, 71	isghm 13981	. . 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 949	. 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 1398  ⊤wtru 1399   ∈ wcel 2205  ∀wral 2522  {crab 2526  Vcvv 2815   ⊆ wss 3213   class class class wbr 4111   ↦ cmpt 4173  ⟶wf 5350  ‘cfv 5354  (class class class)co 6052   ∈ cmpo 6054  ℂcc 8130  0cc0 8132   + caddc 8135   · cmul 8137   # cap 8860  ℤcz 9582  ↑cexp 10907  Basecbs 13233   ↾_s cress 13234  +_gcplusg 13311  Grpcgrp 13734   GrpHom cghm 13978  mulGrpcmgp 14085  Ringcrg 14161  ℂ_fldccnfld 14753  ℤ_ringczring 14787

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-addf 8254  ax-mulf 8255

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-tp 3699  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-tpos 6478  df-recs 6538  df-frec 6624  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-z 9583  df-dec 9716  df-uz 9860  df-rp 9993  df-fz 10349  df-seqfrec 10817  df-exp 10908  df-cj 11535  df-abs 11692  df-struct 13235  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-iress 13241  df-plusg 13324  df-mulr 13325  df-starv 13326  df-tset 13330  df-ple 13331  df-ds 13333  df-unif 13334  df-0g 13492  df-topgen 13494  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-grp 13737  df-minusg 13738  df-subg 13908  df-ghm 13979  df-cmn 14024  df-abl 14025  df-mgp 14086  df-ur 14125  df-srg 14129  df-ring 14163  df-cring 14164  df-oppr 14233  df-dvdsr 14255  df-unit 14256  df-subrg 14387  df-bl 14743  df-mopn 14744  df-fg 14746  df-metu 14747  df-cnfld 14754  df-zring 14788

This theorem is referenced by:  lgseisenlem4  15995