Theorem dchrmulcl 27095

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 27094	. 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 ∘f · 𝑌))
8		mulcl 11200	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
9	8	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10		eqid 2731	. . . . 5 ⊢ (Base‘𝑍) = (Base‘𝑍)
11	1, 2, 3, 10, 5	dchrf 27088	. . . 4 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
12	1, 2, 3, 10, 6	dchrf 27088	. . . 4 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ)
13		fvexd 6906	. . . 4 ⊢ (𝜑 → (Base‘𝑍) ∈ V)
14		inidm 4218	. . . 4 ⊢ ((Base‘𝑍) ∩ (Base‘𝑍)) = (Base‘𝑍)
15	9, 11, 12, 13, 13, 14	off 7692	. . 3 ⊢ (𝜑 → (𝑋 ∘f · 𝑌):(Base‘𝑍)⟶ℂ)
16		eqid 2731	. . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
17	10, 16	unitcl 20273	. . . . . . 7 ⊢ (𝑥 ∈ (Unit‘𝑍) → 𝑥 ∈ (Base‘𝑍))
18	10, 16	unitcl 20273	. . . . . . 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 27086	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
21	5, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	1, 2, 10, 16, 21, 3	dchrelbas2 27083	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
23	5, 22	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
24	23	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
25		eqid 2731	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
26	25, 10	mgpbas 20041	. . . . . . . . . . . 12 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
27		eqid 2731	. . . . . . . . . . . . 13 ⊢ (.r‘𝑍) = (.r‘𝑍)
28	25, 27	mgpplusg 20039	. . . . . . . . . . . 12 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍))
29		eqid 2731	. . . . . . . . . . . . 13 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
30		cnfldmul 21239	. . . . . . . . . . . . 13 ⊢ · = (.r‘ℂfld)
31	29, 30	mgpplusg 20039	. . . . . . . . . . . 12 ⊢ · = (+g‘(mulGrp‘ℂfld))
32	26, 28, 31	mhmlin 18721	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘(𝑥(.r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
33	32	3expb 1119	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (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 27083	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌 ∈ 𝐷 ↔ (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
36	6, 35	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
37	36	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
38	26, 28, 31	mhmlin 18721	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘(𝑥(.r‘𝑍)𝑦)) = ((𝑌‘𝑥) · (𝑌‘𝑦)))
39	38	3expb 1119	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (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 7430	. . . . . . . 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 7086	. . . . . . . . . 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 7083	. . . . . . . . . 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 7086	. . . . . . . . . 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 7083	. . . . . . . . . 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 11433	. . . . . . . 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 2771	. . . . . . 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 6718	. . . . . . . . 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 6718	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍))
56	55	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑌 Fn (Base‘𝑍))
57		fvexd 6906	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (Base‘𝑍) ∈ V)
58	21	nnnn0d 12539	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
59	2	zncrng 21410	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing)
60		crngring 20146	. . . . . . . . . 10 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
61	58, 59, 60	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ Ring)
62	10, 27	ringcl 20151	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑥(.r‘𝑍)𝑦) ∈ (Base‘𝑍))
63	62	3expb 1119	. . . . . . . . 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 7691	. . . . . . . 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 6906	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (Base‘𝑍) ∈ V)
70		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑥 ∈ (Base‘𝑍))
71		fnfvof 7691	. . . . . . . . . 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 7691	. . . . . . . . 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 7430	. . . . . . 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 2781	. . . . . 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 3199	. . . 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 2731	. . . . . . . 8 ⊢ (1r‘𝑍) = (1r‘𝑍)
82	10, 81	ringidcl 20161	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (1r‘𝑍) ∈ (Base‘𝑍))
83	61, 82	syl 17	. . . . . 6 ⊢ (𝜑 → (1r‘𝑍) ∈ (Base‘𝑍))
84		fnfvof 7691	. . . . . 6 ⊢ (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ (1r‘𝑍) ∈ (Base‘𝑍))) → ((𝑋 ∘f · 𝑌)‘(1r‘𝑍)) = ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))))
85	53, 55, 13, 83, 84	syl22anc 836	. . . . 5 ⊢ (𝜑 → ((𝑋 ∘f · 𝑌)‘(1r‘𝑍)) = ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))))
86	25, 81	ringidval 20084	. . . . . . . . 9 ⊢ (1r‘𝑍) = (0g‘(mulGrp‘𝑍))
87		cnfld1 21259	. . . . . . . . . 10 ⊢ 1 = (1r‘ℂfld)
88	29, 87	ringidval 20084	. . . . . . . . 9 ⊢ 1 = (0g‘(mulGrp‘ℂfld))
89	86, 88	mhm0 18722	. . . . . . . 8 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑋‘(1r‘𝑍)) = 1)
90	24, 89	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑋‘(1r‘𝑍)) = 1)
91	86, 88	mhm0 18722	. . . . . . . 8 ⊢ (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑌‘(1r‘𝑍)) = 1)
92	37, 91	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑌‘(1r‘𝑍)) = 1)
93	90, 92	oveq12d 7430	. . . . . 6 ⊢ (𝜑 → ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))) = (1 · 1))
94		1t1e1 12381	. . . . . 6 ⊢ (1 · 1) = 1
95	93, 94	eqtrdi 2787	. . . . 5 ⊢ (𝜑 → ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))) = 1)
96	85, 95	eqtrd 2771	. . . 4 ⊢ (𝜑 → ((𝑋 ∘f · 𝑌)‘(1r‘𝑍)) = 1)
97	72	neeq1d 2999	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋 ∘f · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋‘𝑥) · (𝑌‘𝑥)) ≠ 0))
98	42, 47	mulne0bd 11872	. . . . . . 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 3247	. . . . . . 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 3145	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(((𝑋 ∘f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
105	80, 96, 104	3jca 1127	. . 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)‘(1r‘𝑍)) = 1 ∧ ∀𝑥 ∈ (Base‘𝑍)(((𝑋 ∘f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
106	1, 2, 10, 16, 21, 3	dchrelbas3 27084	. . 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)‘(1r‘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 2832	1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  Vcvv 3473   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412   ∘_f cof 7672  ℂcc 11114  0cc0 11116  1c1 11117   · cmul 11121  ℕcn 12219  ℕ₀cn0 12479  Basecbs 17151  +_gcplusg 17204  ._rcmulr 17205   MndHom cmhm 18709  mulGrpcmgp 20035  1_rcur 20082  Ringcrg 20134  CRingccrg 20135  Unitcui 20253  ℂ_fldccnfld 21233  ℤ/nℤczn 21362  DChrcdchr 27078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-addf 11195  ax-mulf 11196

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-tpos 8217  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-ec 8711  df-qs 8715  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-starv 17219  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-unif 17227  df-0g 17394  df-imas 17461  df-qus 17462  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-mhm 18711  df-grp 18864  df-minusg 18865  df-sbg 18866  df-subg 19046  df-nsg 19047  df-eqg 19048  df-cmn 19698  df-abl 19699  df-mgp 20036  df-rng 20054  df-ur 20083  df-ring 20136  df-cring 20137  df-oppr 20232  df-dvdsr 20255  df-unit 20256  df-subrng 20442  df-subrg 20467  df-lmod 20704  df-lss 20775  df-lsp 20815  df-sra 21019  df-rgmod 21020  df-lidl 21021  df-rsp 21022  df-2idl 21095  df-cnfld 21234  df-zring 21307  df-zn 21366  df-dchr 27079

This theorem is referenced by:  dchrabl  27100  dchrinv  27107