Theorem dchrmulcl 27160

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 27159	. 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 ∘f · 𝑌))
8		mulcl 11152	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
9	8	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10		eqid 2729	. . . . 5 ⊢ (Base‘𝑍) = (Base‘𝑍)
11	1, 2, 3, 10, 5	dchrf 27153	. . . 4 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
12	1, 2, 3, 10, 6	dchrf 27153	. . . 4 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ)
13		fvexd 6873	. . . 4 ⊢ (𝜑 → (Base‘𝑍) ∈ V)
14		inidm 4190	. . . 4 ⊢ ((Base‘𝑍) ∩ (Base‘𝑍)) = (Base‘𝑍)
15	9, 11, 12, 13, 13, 14	off 7671	. . 3 ⊢ (𝜑 → (𝑋 ∘f · 𝑌):(Base‘𝑍)⟶ℂ)
16		eqid 2729	. . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
17	10, 16	unitcl 20284	. . . . . . 7 ⊢ (𝑥 ∈ (Unit‘𝑍) → 𝑥 ∈ (Base‘𝑍))
18	10, 16	unitcl 20284	. . . . . . 7 ⊢ (𝑦 ∈ (Unit‘𝑍) → 𝑦 ∈ (Base‘𝑍))
19	17, 18	anim12i 613	. . . . . 6 ⊢ ((𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍)) → (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)))
20	1, 3	dchrrcl 27151	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
21	5, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	1, 2, 10, 16, 21, 3	dchrelbas2 27148	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
23	5, 22	mpbid 232	. . . . . . . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
26	25, 10	mgpbas 20054	. . . . . . . . . . . 12 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
27		eqid 2729	. . . . . . . . . . . . 13 ⊢ (.r‘𝑍) = (.r‘𝑍)
28	25, 27	mgpplusg 20053	. . . . . . . . . . . 12 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍))
29		eqid 2729	. . . . . . . . . . . . 13 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
30		cnfldmul 21272	. . . . . . . . . . . . 13 ⊢ · = (.r‘ℂfld)
31	29, 30	mgpplusg 20053	. . . . . . . . . . . 12 ⊢ · = (+g‘(mulGrp‘ℂfld))
32	26, 28, 31	mhmlin 18720	. . . . . . . . . . 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 1120	. . . . . . . . . 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 580	. . . . . . . . 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 27148	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌 ∈ 𝐷 ↔ (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
36	6, 35	mpbid 232	. . . . . . . . . . 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 18720	. . . . . . . . . . 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 1120	. . . . . . . . . 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 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r‘𝑍)𝑦)) = ((𝑌‘𝑥) · (𝑌‘𝑦)))
41	34, 40	oveq12d 7405	. . . . . . . 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 7056	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
43	42	adantrr 717	. . . . . . . . 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 7053	. . . . . . . . . 10 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘𝑦) ∈ ℂ)
46	11, 44, 45	syl2an 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘𝑦) ∈ ℂ)
47	12	ffvelcdmda 7056	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑌‘𝑥) ∈ ℂ)
48	47	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘𝑥) ∈ ℂ)
49		ffvelcdm 7053	. . . . . . . . . 10 ⊢ ((𝑌:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘𝑦) ∈ ℂ)
50	12, 44, 49	syl2an 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘𝑦) ∈ ℂ)
51	43, 46, 48, 50	mul4d 11386	. . . . . . . 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 6689	. . . . . . . . 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 6689	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍))
56	55	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑌 Fn (Base‘𝑍))
57		fvexd 6873	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (Base‘𝑍) ∈ V)
58	21	nnnn0d 12503	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
59	2	zncrng 21454	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing)
60		crngring 20154	. . . . . . . . . 10 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
61	58, 59, 60	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ Ring)
62	10, 27	ringcl 20159	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑥(.r‘𝑍)𝑦) ∈ (Base‘𝑍))
63	62	3expb 1120	. . . . . . . . 9 ⊢ ((𝑍 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r‘𝑍)𝑦) ∈ (Base‘𝑍))
64	61, 63	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r‘𝑍)𝑦) ∈ (Base‘𝑍))
65		fnfvof 7670	. . . . . . . 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 838	. . . . . . 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 6873	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (Base‘𝑍) ∈ V)
70		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑥 ∈ (Base‘𝑍))
71		fnfvof 7670	. . . . . . . . . 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 838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((𝑋 ∘f · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
73	72	adantrr 717	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘f · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
74		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑦 ∈ (Base‘𝑍))
75		fnfvof 7670	. . . . . . . . 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 838	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘f · 𝑌)‘𝑦) = ((𝑋‘𝑦) · (𝑌‘𝑦)))
77	73, 76	oveq12d 7405	. . . . . . 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 593	. . . . 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 3180	. . . 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 2729	. . . . . . . 8 ⊢ (1r‘𝑍) = (1r‘𝑍)
82	10, 81	ringidcl 20174	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (1r‘𝑍) ∈ (Base‘𝑍))
83	61, 82	syl 17	. . . . . 6 ⊢ (𝜑 → (1r‘𝑍) ∈ (Base‘𝑍))
84		fnfvof 7670	. . . . . 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 838	. . . . 5 ⊢ (𝜑 → ((𝑋 ∘f · 𝑌)‘(1r‘𝑍)) = ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))))
86	25, 81	ringidval 20092	. . . . . . . . 9 ⊢ (1r‘𝑍) = (0g‘(mulGrp‘𝑍))
87		cnfld1 21305	. . . . . . . . . 10 ⊢ 1 = (1r‘ℂfld)
88	29, 87	ringidval 20092	. . . . . . . . 9 ⊢ 1 = (0g‘(mulGrp‘ℂfld))
89	86, 88	mhm0 18721	. . . . . . . 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 18721	. . . . . . . 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 7405	. . . . . 6 ⊢ (𝜑 → ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))) = (1 · 1))
94		1t1e1 12343	. . . . . 6 ⊢ (1 · 1) = 1
95	93, 94	eqtrdi 2780	. . . . 5 ⊢ (𝜑 → ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))) = 1)
96	85, 95	eqtrd 2764	. . . 4 ⊢ (𝜑 → ((𝑋 ∘f · 𝑌)‘(1r‘𝑍)) = 1)
97	72	neeq1d 2984	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋 ∘f · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋‘𝑥) · (𝑌‘𝑥)) ≠ 0))
98	42, 47	mulne0bd 11829	. . . . . . 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 3229	. . . . . . 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 240	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋 ∘f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
104	103	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(((𝑋 ∘f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
105	80, 96, 104	3jca 1128	. . 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 27149	. . 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 713	. 2 ⊢ (𝜑 → (𝑋 ∘f · 𝑌) ∈ 𝐷)
108	7, 107	eqeltrd 2828	1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3447   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651  ℂcc 11066  0cc0 11068  1c1 11069   · cmul 11073  ℕcn 12186  ℕ₀cn0 12442  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221   MndHom cmhm 18708  mulGrpcmgp 20049  1_rcur 20090  Ringcrg 20142  CRingccrg 20143  Unitcui 20264  ℂ_fldccnfld 21264  ℤ/nℤczn 21412  DChrcdchr 27143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-addf 11147  ax-mulf 11148

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-ec 8673  df-qs 8677  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-0g 17404  df-imas 17471  df-qus 17472  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-grp 18868  df-minusg 18869  df-sbg 18870  df-subg 19055  df-nsg 19056  df-eqg 19057  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-cring 20145  df-oppr 20246  df-dvdsr 20266  df-unit 20267  df-subrng 20455  df-subrg 20479  df-lmod 20768  df-lss 20838  df-lsp 20878  df-sra 21080  df-rgmod 21081  df-lidl 21118  df-rsp 21119  df-2idl 21160  df-cnfld 21265  df-zring 21357  df-zn 21416  df-dchr 27144

This theorem is referenced by:  dchrabl  27165  dchrinv  27172