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Theorem dchrmulcl 27086

Description: Closure of the group operation on Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)

**Hypotheses**
Ref	Expression
dchrmhm.g	⊢ 𝐺 = (DChr‘𝑁)
dchrmhm.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrmhm.b	⊢ 𝐷 = (Base‘𝐺)
dchrmul.t	⊢ · = (+_g‘𝐺)
dchrmul.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrmul.y	⊢ (𝜑 → 𝑌 ∈ 𝐷)

**Assertion**
Ref	Expression
dchrmulcl	⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)

Proof of Theorem dchrmulcl Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrmhm.g	. . 3 ⊢ 𝐺 = (DChr‘𝑁)
2		dchrmhm.z	. . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
3		dchrmhm.b	. . 3 ⊢ 𝐷 = (Base‘𝐺)
4		dchrmul.t	. . 3 ⊢ · = (+_g‘𝐺)
5		dchrmul.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
6		dchrmul.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
7	1, 2, 3, 4, 5, 6	dchrmul 27085	. 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 ∘_f · 𝑌))
8		mulcl 11189	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
9	8	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10		eqid 2724	. . . . 5 ⊢ (Base‘𝑍) = (Base‘𝑍)
11	1, 2, 3, 10, 5	dchrf 27079	. . . 4 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
12	1, 2, 3, 10, 6	dchrf 27079	. . . 4 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ)
13		fvexd 6896	. . . 4 ⊢ (𝜑 → (Base‘𝑍) ∈ V)
14		inidm 4210	. . . 4 ⊢ ((Base‘𝑍) ∩ (Base‘𝑍)) = (Base‘𝑍)
15	9, 11, 12, 13, 13, 14	off 7681	. . 3 ⊢ (𝜑 → (𝑋 ∘_f · 𝑌):(Base‘𝑍)⟶ℂ)
16		eqid 2724	. . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
17	10, 16	unitcl 20262	. . . . . . 7 ⊢ (𝑥 ∈ (Unit‘𝑍) → 𝑥 ∈ (Base‘𝑍))
18	10, 16	unitcl 20262	. . . . . . 7 ⊢ (𝑦 ∈ (Unit‘𝑍) → 𝑦 ∈ (Base‘𝑍))
19	17, 18	anim12i 612	. . . . . 6 ⊢ ((𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍)) → (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)))
20	1, 3	dchrrcl 27077	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
21	5, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	1, 2, 10, 16, 21, 3	dchrelbas2 27074	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
23	5, 22	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
24	23	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
25		eqid 2724	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
26	25, 10	mgpbas 20030	. . . . . . . . . . . 12 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
27		eqid 2724	. . . . . . . . . . . . 13 ⊢ (._r‘𝑍) = (._r‘𝑍)
28	25, 27	mgpplusg 20028	. . . . . . . . . . . 12 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
29		eqid 2724	. . . . . . . . . . . . 13 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
30		cnfldmul 21229	. . . . . . . . . . . . 13 ⊢ · = (._r‘ℂ_fld)
31	29, 30	mgpplusg 20028	. . . . . . . . . . . 12 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
32	26, 28, 31	mhmlin 18710	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
33	32	3expb 1117	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
34	24, 33	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
35	1, 2, 10, 16, 21, 3	dchrelbas2 27074	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌 ∈ 𝐷 ↔ (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
36	6, 35	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
37	36	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
38	26, 28, 31	mhmlin 18710	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘(𝑥(._r‘𝑍)𝑦)) = ((𝑌‘𝑥) · (𝑌‘𝑦)))
39	38	3expb 1117	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(._r‘𝑍)𝑦)) = ((𝑌‘𝑥) · (𝑌‘𝑦)))
40	37, 39	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(._r‘𝑍)𝑦)) = ((𝑌‘𝑥) · (𝑌‘𝑦)))
41	34, 40	oveq12d 7419	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(._r‘𝑍)𝑦)) · (𝑌‘(𝑥(._r‘𝑍)𝑦))) = (((𝑋‘𝑥) · (𝑋‘𝑦)) · ((𝑌‘𝑥) · (𝑌‘𝑦))))
42	11	ffvelcdmda 7076	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
43	42	adantrr 714	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘𝑥) ∈ ℂ)
44		simpr 484	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → 𝑦 ∈ (Base‘𝑍))
45		ffvelcdm 7073	. . . . . . . . . 10 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘𝑦) ∈ ℂ)
46	11, 44, 45	syl2an 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘𝑦) ∈ ℂ)
47	12	ffvelcdmda 7076	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑌‘𝑥) ∈ ℂ)
48	47	adantrr 714	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘𝑥) ∈ ℂ)
49		ffvelcdm 7073	. . . . . . . . . 10 ⊢ ((𝑌:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘𝑦) ∈ ℂ)
50	12, 44, 49	syl2an 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘𝑦) ∈ ℂ)
51	43, 46, 48, 50	mul4d 11422	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋‘𝑥) · (𝑋‘𝑦)) · ((𝑌‘𝑥) · (𝑌‘𝑦))) = (((𝑋‘𝑥) · (𝑌‘𝑥)) · ((𝑋‘𝑦) · (𝑌‘𝑦))))
52	41, 51	eqtrd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(._r‘𝑍)𝑦)) · (𝑌‘(𝑥(._r‘𝑍)𝑦))) = (((𝑋‘𝑥) · (𝑌‘𝑥)) · ((𝑋‘𝑦) · (𝑌‘𝑦))))
53	11	ffnd 6708	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍))
54	53	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑋 Fn (Base‘𝑍))
55	12	ffnd 6708	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍))
56	55	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑌 Fn (Base‘𝑍))
57		fvexd 6896	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (Base‘𝑍) ∈ V)
58	21	nnnn0d 12528	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
59	2	zncrng 21400	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
60		crngring 20135	. . . . . . . . . 10 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
61	58, 59, 60	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ Ring)
62	10, 27	ringcl 20140	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑥(._r‘𝑍)𝑦) ∈ (Base‘𝑍))
63	62	3expb 1117	. . . . . . . . 9 ⊢ ((𝑍 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(._r‘𝑍)𝑦) ∈ (Base‘𝑍))
64	61, 63	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(._r‘𝑍)𝑦) ∈ (Base‘𝑍))
65		fnfvof 7680	. . . . . . . 8 ⊢ (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ (𝑥(._r‘𝑍)𝑦) ∈ (Base‘𝑍))) → ((𝑋 ∘_f · 𝑌)‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘(𝑥(._r‘𝑍)𝑦)) · (𝑌‘(𝑥(._r‘𝑍)𝑦))))
66	54, 56, 57, 64, 65	syl22anc 836	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘_f · 𝑌)‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘(𝑥(._r‘𝑍)𝑦)) · (𝑌‘(𝑥(._r‘𝑍)𝑦))))
67	53	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑋 Fn (Base‘𝑍))
68	55	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑌 Fn (Base‘𝑍))
69		fvexd 6896	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (Base‘𝑍) ∈ V)
70		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑥 ∈ (Base‘𝑍))
71		fnfvof 7680	. . . . . . . . . 10 ⊢ (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ 𝑥 ∈ (Base‘𝑍))) → ((𝑋 ∘_f · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
72	67, 68, 69, 70, 71	syl22anc 836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((𝑋 ∘_f · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
73	72	adantrr 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘_f · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
74		simprr 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑦 ∈ (Base‘𝑍))
75		fnfvof 7680	. . . . . . . . 9 ⊢ (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘_f · 𝑌)‘𝑦) = ((𝑋‘𝑦) · (𝑌‘𝑦)))
76	54, 56, 57, 74, 75	syl22anc 836	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘_f · 𝑌)‘𝑦) = ((𝑋‘𝑦) · (𝑌‘𝑦)))
77	73, 76	oveq12d 7419	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋 ∘_f · 𝑌)‘𝑥) · ((𝑋 ∘_f · 𝑌)‘𝑦)) = (((𝑋‘𝑥) · (𝑌‘𝑥)) · ((𝑋‘𝑦) · (𝑌‘𝑦))))
78	52, 66, 77	3eqtr4d 2774	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘_f · 𝑌)‘(𝑥(._r‘𝑍)𝑦)) = (((𝑋 ∘_f · 𝑌)‘𝑥) · ((𝑋 ∘_f · 𝑌)‘𝑦)))
79	19, 78	sylan2 592	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍))) → ((𝑋 ∘_f · 𝑌)‘(𝑥(._r‘𝑍)𝑦)) = (((𝑋 ∘_f · 𝑌)‘𝑥) · ((𝑋 ∘_f · 𝑌)‘𝑦)))
80	79	ralrimivva 3192	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)((𝑋 ∘_f · 𝑌)‘(𝑥(._r‘𝑍)𝑦)) = (((𝑋 ∘_f · 𝑌)‘𝑥) · ((𝑋 ∘_f · 𝑌)‘𝑦)))
81		eqid 2724	. . . . . . . 8 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
82	10, 81	ringidcl 20150	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) ∈ (Base‘𝑍))
83	61, 82	syl 17	. . . . . 6 ⊢ (𝜑 → (1_r‘𝑍) ∈ (Base‘𝑍))
84		fnfvof 7680	. . . . . 6 ⊢ (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ (1_r‘𝑍) ∈ (Base‘𝑍))) → ((𝑋 ∘_f · 𝑌)‘(1_r‘𝑍)) = ((𝑋‘(1_r‘𝑍)) · (𝑌‘(1_r‘𝑍))))
85	53, 55, 13, 83, 84	syl22anc 836	. . . . 5 ⊢ (𝜑 → ((𝑋 ∘_f · 𝑌)‘(1_r‘𝑍)) = ((𝑋‘(1_r‘𝑍)) · (𝑌‘(1_r‘𝑍))))
86	25, 81	ringidval 20073	. . . . . . . . 9 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
87		cnfld1 21249	. . . . . . . . . 10 ⊢ 1 = (1_r‘ℂ_fld)
88	29, 87	ringidval 20073	. . . . . . . . 9 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
89	86, 88	mhm0 18711	. . . . . . . 8 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → (𝑋‘(1_r‘𝑍)) = 1)
90	24, 89	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑋‘(1_r‘𝑍)) = 1)
91	86, 88	mhm0 18711	. . . . . . . 8 ⊢ (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → (𝑌‘(1_r‘𝑍)) = 1)
92	37, 91	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑌‘(1_r‘𝑍)) = 1)
93	90, 92	oveq12d 7419	. . . . . 6 ⊢ (𝜑 → ((𝑋‘(1_r‘𝑍)) · (𝑌‘(1_r‘𝑍))) = (1 · 1))
94		1t1e1 12370	. . . . . 6 ⊢ (1 · 1) = 1
95	93, 94	eqtrdi 2780	. . . . 5 ⊢ (𝜑 → ((𝑋‘(1_r‘𝑍)) · (𝑌‘(1_r‘𝑍))) = 1)
96	85, 95	eqtrd 2764	. . . 4 ⊢ (𝜑 → ((𝑋 ∘_f · 𝑌)‘(1_r‘𝑍)) = 1)
97	72	neeq1d 2992	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋 ∘_f · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋‘𝑥) · (𝑌‘𝑥)) ≠ 0))
98	42, 47	mulne0bd 11861	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋‘𝑥) ≠ 0 ∧ (𝑌‘𝑥) ≠ 0) ↔ ((𝑋‘𝑥) · (𝑌‘𝑥)) ≠ 0))
99	97, 98	bitr4d 282	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋 ∘_f · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋‘𝑥) ≠ 0 ∧ (𝑌‘𝑥) ≠ 0)))
100	23	simprd 495	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
101	100	r19.21bi 3240	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
102	101	adantrd 491	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋‘𝑥) ≠ 0 ∧ (𝑌‘𝑥) ≠ 0) → 𝑥 ∈ (Unit‘𝑍)))
103	99, 102	sylbid 239	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋 ∘_f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
104	103	ralrimiva 3138	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(((𝑋 ∘_f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
105	80, 96, 104	3jca 1125	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)((𝑋 ∘_f · 𝑌)‘(𝑥(._r‘𝑍)𝑦)) = (((𝑋 ∘_f · 𝑌)‘𝑥) · ((𝑋 ∘_f · 𝑌)‘𝑦)) ∧ ((𝑋 ∘_f · 𝑌)‘(1_r‘𝑍)) = 1 ∧ ∀𝑥 ∈ (Base‘𝑍)(((𝑋 ∘_f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
106	1, 2, 10, 16, 21, 3	dchrelbas3 27075	. . 3 ⊢ (𝜑 → ((𝑋 ∘_f · 𝑌) ∈ 𝐷 ↔ ((𝑋 ∘_f · 𝑌):(Base‘𝑍)⟶ℂ ∧ (∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)((𝑋 ∘_f · 𝑌)‘(𝑥(._r‘𝑍)𝑦)) = (((𝑋 ∘_f · 𝑌)‘𝑥) · ((𝑋 ∘_f · 𝑌)‘𝑦)) ∧ ((𝑋 ∘_f · 𝑌)‘(1_r‘𝑍)) = 1 ∧ ∀𝑥 ∈ (Base‘𝑍)(((𝑋 ∘_f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))))
107	15, 105, 106	mpbir2and 710	. 2 ⊢ (𝜑 → (𝑋 ∘_f · 𝑌) ∈ 𝐷)
108	7, 107	eqeltrd 2825	1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 Vcvv 3466 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ∘_f cof 7661 ℂcc 11103 0cc0 11105 1c1 11106 · cmul 11110 ℕcn 12208 ℕ₀cn0 12468 Basecbs 17140 +_gcplusg 17193 ._rcmulr 17194 MndHom cmhm 18698 mulGrpcmgp 20024 1_rcur 20071 Ringcrg 20123 CRingccrg 20124 Unitcui 20242 ℂ_fldccnfld 21223 ℤ/nℤczn 21352 DChrcdchr 27069

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-addf 11184 ax-mulf 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-ec 8700 df-qs 8704 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-inf 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-imas 17450 df-qus 17451 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-mhm 18700 df-grp 18853 df-minusg 18854 df-sbg 18855 df-subg 19035 df-nsg 19036 df-eqg 19037 df-cmn 19687 df-abl 19688 df-mgp 20025 df-rng 20043 df-ur 20072 df-ring 20125 df-cring 20126 df-oppr 20221 df-dvdsr 20244 df-unit 20245 df-subrng 20431 df-subrg 20456 df-lmod 20693 df-lss 20764 df-lsp 20804 df-sra 21006 df-rgmod 21007 df-lidl 21052 df-rsp 21053 df-2idl 21092 df-cnfld 21224 df-zring 21297 df-zn 21356 df-dchr 27070

This theorem is referenced by: dchrabl 27091 dchrinv 27098

W3C validator