Theorem dchrmulcl 27200

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 27199	. 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 ∘f · 𝑌))
8		mulcl 11111	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
9	8	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10		eqid 2737	. . . . 5 ⊢ (Base‘𝑍) = (Base‘𝑍)
11	1, 2, 3, 10, 5	dchrf 27193	. . . 4 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
12	1, 2, 3, 10, 6	dchrf 27193	. . . 4 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ)
13		fvexd 6847	. . . 4 ⊢ (𝜑 → (Base‘𝑍) ∈ V)
14		inidm 4168	. . . 4 ⊢ ((Base‘𝑍) ∩ (Base‘𝑍)) = (Base‘𝑍)
15	9, 11, 12, 13, 13, 14	off 7640	. . 3 ⊢ (𝜑 → (𝑋 ∘f · 𝑌):(Base‘𝑍)⟶ℂ)
16		eqid 2737	. . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
17	10, 16	unitcl 20313	. . . . . . 7 ⊢ (𝑥 ∈ (Unit‘𝑍) → 𝑥 ∈ (Base‘𝑍))
18	10, 16	unitcl 20313	. . . . . . 7 ⊢ (𝑦 ∈ (Unit‘𝑍) → 𝑦 ∈ (Base‘𝑍))
19	17, 18	anim12i 614	. . . . . 6 ⊢ ((𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍)) → (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)))
20	1, 3	dchrrcl 27191	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
21	5, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	1, 2, 10, 16, 21, 3	dchrelbas2 27188	. . . . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
26	25, 10	mgpbas 20084	. . . . . . . . . . . 12 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
27		eqid 2737	. . . . . . . . . . . . 13 ⊢ (.r‘𝑍) = (.r‘𝑍)
28	25, 27	mgpplusg 20083	. . . . . . . . . . . 12 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍))
29		eqid 2737	. . . . . . . . . . . . 13 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
30		cnfldmul 21319	. . . . . . . . . . . . 13 ⊢ · = (.r‘ℂfld)
31	29, 30	mgpplusg 20083	. . . . . . . . . . . 12 ⊢ · = (+g‘(mulGrp‘ℂfld))
32	26, 28, 31	mhmlin 18719	. . . . . . . . . . 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 1121	. . . . . . . . . 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 581	. . . . . . . . 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 27188	. . . . . . . . . . . 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 18719	. . . . . . . . . . 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 1121	. . . . . . . . . 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 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r‘𝑍)𝑦)) = ((𝑌‘𝑥) · (𝑌‘𝑦)))
41	34, 40	oveq12d 7376	. . . . . . . 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 7028	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
43	42	adantrr 718	. . . . . . . . 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 7025	. . . . . . . . . 10 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘𝑦) ∈ ℂ)
46	11, 44, 45	syl2an 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘𝑦) ∈ ℂ)
47	12	ffvelcdmda 7028	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑌‘𝑥) ∈ ℂ)
48	47	adantrr 718	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘𝑥) ∈ ℂ)
49		ffvelcdm 7025	. . . . . . . . . 10 ⊢ ((𝑌:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘𝑦) ∈ ℂ)
50	12, 44, 49	syl2an 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘𝑦) ∈ ℂ)
51	43, 46, 48, 50	mul4d 11346	. . . . . . . 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 2772	. . . . . . 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 6661	. . . . . . . . 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 6661	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍))
56	55	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑌 Fn (Base‘𝑍))
57		fvexd 6847	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (Base‘𝑍) ∈ V)
58	21	nnnn0d 12463	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
59	2	zncrng 21501	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing)
60		crngring 20184	. . . . . . . . . 10 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
61	58, 59, 60	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ Ring)
62	10, 27	ringcl 20189	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑥(.r‘𝑍)𝑦) ∈ (Base‘𝑍))
63	62	3expb 1121	. . . . . . . . 9 ⊢ ((𝑍 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r‘𝑍)𝑦) ∈ (Base‘𝑍))
64	61, 63	sylan 581	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r‘𝑍)𝑦) ∈ (Base‘𝑍))
65		fnfvof 7639	. . . . . . . 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 839	. . . . . . 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 6847	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (Base‘𝑍) ∈ V)
70		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑥 ∈ (Base‘𝑍))
71		fnfvof 7639	. . . . . . . . . 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 839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((𝑋 ∘f · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
73	72	adantrr 718	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘f · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
74		simprr 773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑦 ∈ (Base‘𝑍))
75		fnfvof 7639	. . . . . . . . 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 839	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘f · 𝑌)‘𝑦) = ((𝑋‘𝑦) · (𝑌‘𝑦)))
77	73, 76	oveq12d 7376	. . . . . . 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 2782	. . . . . 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 594	. . . . 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 3181	. . . 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 2737	. . . . . . . 8 ⊢ (1r‘𝑍) = (1r‘𝑍)
82	10, 81	ringidcl 20204	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (1r‘𝑍) ∈ (Base‘𝑍))
83	61, 82	syl 17	. . . . . 6 ⊢ (𝜑 → (1r‘𝑍) ∈ (Base‘𝑍))
84		fnfvof 7639	. . . . . 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 839	. . . . 5 ⊢ (𝜑 → ((𝑋 ∘f · 𝑌)‘(1r‘𝑍)) = ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))))
86	25, 81	ringidval 20122	. . . . . . . . 9 ⊢ (1r‘𝑍) = (0g‘(mulGrp‘𝑍))
87		cnfld1 21350	. . . . . . . . . 10 ⊢ 1 = (1r‘ℂfld)
88	29, 87	ringidval 20122	. . . . . . . . 9 ⊢ 1 = (0g‘(mulGrp‘ℂfld))
89	86, 88	mhm0 18720	. . . . . . . 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 18720	. . . . . . . 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 7376	. . . . . 6 ⊢ (𝜑 → ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))) = (1 · 1))
94		1t1e1 12303	. . . . . 6 ⊢ (1 · 1) = 1
95	93, 94	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))) = 1)
96	85, 95	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((𝑋 ∘f · 𝑌)‘(1r‘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 11789	. . . . . . 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 3230	. . . . . . 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 3130	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(((𝑋 ∘f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
105	80, 96, 104	3jca 1129	. . 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 27189	. . 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 714	. 2 ⊢ (𝜑 → (𝑋 ∘f · 𝑌) ∈ 𝐷)
108	7, 107	eqeltrd 2837	1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3430   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358   ∘_f cof 7620  ℂcc 11025  0cc0 11027  1c1 11028   · cmul 11032  ℕcn 12146  ℕ₀cn0 12402  Basecbs 17137  +_gcplusg 17178  ._rcmulr 17179   MndHom cmhm 18707  mulGrpcmgp 20079  1_rcur 20120  Ringcrg 20172  CRingccrg 20173  Unitcui 20293  ℂ_fldccnfld 21311  ℤ/nℤczn 21459  DChrcdchr 27183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-addf 11106  ax-mulf 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8167  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-ec 8636  df-qs 8640  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9346  df-inf 9347  df-pnf 11169  df-mnf 11170  df-xr 11171  df-ltxr 11172  df-le 11173  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-z 12490  df-dec 12609  df-uz 12753  df-fz 13425  df-struct 17075  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17138  df-ress 17159  df-plusg 17191  df-mulr 17192  df-starv 17193  df-sca 17194  df-vsca 17195  df-ip 17196  df-tset 17197  df-ple 17198  df-ds 17200  df-unif 17201  df-0g 17362  df-imas 17430  df-qus 17431  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mhm 18709  df-grp 18870  df-minusg 18871  df-sbg 18872  df-subg 19057  df-nsg 19058  df-eqg 19059  df-cmn 19715  df-abl 19716  df-mgp 20080  df-rng 20092  df-ur 20121  df-ring 20174  df-cring 20175  df-oppr 20275  df-dvdsr 20295  df-unit 20296  df-subrng 20481  df-subrg 20505  df-lmod 20815  df-lss 20885  df-lsp 20925  df-sra 21127  df-rgmod 21128  df-lidl 21165  df-rsp 21166  df-2idl 21207  df-cnfld 21312  df-zring 21404  df-zn 21463  df-dchr 27184

This theorem is referenced by:  dchrabl  27205  dchrinv  27212