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Theorem dchrmulcl 26741

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 26740	. 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 ∘_f · 𝑌))
8		mulcl 11190	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
9	8	adantl 482	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10		eqid 2732	. . . . 5 ⊢ (Base‘𝑍) = (Base‘𝑍)
11	1, 2, 3, 10, 5	dchrf 26734	. . . 4 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
12	1, 2, 3, 10, 6	dchrf 26734	. . . 4 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ)
13		fvexd 6903	. . . 4 ⊢ (𝜑 → (Base‘𝑍) ∈ V)
14		inidm 4217	. . . 4 ⊢ ((Base‘𝑍) ∩ (Base‘𝑍)) = (Base‘𝑍)
15	9, 11, 12, 13, 13, 14	off 7684	. . 3 ⊢ (𝜑 → (𝑋 ∘_f · 𝑌):(Base‘𝑍)⟶ℂ)
16		eqid 2732	. . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
17	10, 16	unitcl 20181	. . . . . . 7 ⊢ (𝑥 ∈ (Unit‘𝑍) → 𝑥 ∈ (Base‘𝑍))
18	10, 16	unitcl 20181	. . . . . . 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 26732	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
21	5, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	1, 2, 10, 16, 21, 3	dchrelbas2 26729	. . . . . . . . . . . 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 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
25		eqid 2732	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
26	25, 10	mgpbas 19987	. . . . . . . . . . . 12 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
27		eqid 2732	. . . . . . . . . . . . 13 ⊢ (._r‘𝑍) = (._r‘𝑍)
28	25, 27	mgpplusg 19985	. . . . . . . . . . . 12 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
29		eqid 2732	. . . . . . . . . . . . 13 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
30		cnfldmul 20942	. . . . . . . . . . . . 13 ⊢ · = (._r‘ℂ_fld)
31	29, 30	mgpplusg 19985	. . . . . . . . . . . 12 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
32	26, 28, 31	mhmlin 18675	. . . . . . . . . . 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 1120	. . . . . . . . . 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 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 26729	. . . . . . . . . . . 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 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
38	26, 28, 31	mhmlin 18675	. . . . . . . . . . 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 1120	. . . . . . . . . 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 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(._r‘𝑍)𝑦)) = ((𝑌‘𝑥) · (𝑌‘𝑦)))
41	34, 40	oveq12d 7423	. . . . . . . 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 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
43	42	adantrr 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘𝑥) ∈ ℂ)
44		simpr 485	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → 𝑦 ∈ (Base‘𝑍))
45		ffvelcdm 7080	. . . . . . . . . 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 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑌‘𝑥) ∈ ℂ)
48	47	adantrr 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘𝑥) ∈ ℂ)
49		ffvelcdm 7080	. . . . . . . . . 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 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 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 6715	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍))
54	53	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑋 Fn (Base‘𝑍))
55	12	ffnd 6715	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍))
56	55	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑌 Fn (Base‘𝑍))
57		fvexd 6903	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (Base‘𝑍) ∈ V)
58	21	nnnn0d 12528	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
59	2	zncrng 21091	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
60		crngring 20061	. . . . . . . . . 10 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
61	58, 59, 60	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ Ring)
62	10, 27	ringcl 20066	. . . . . . . . . 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 7683	. . . . . . . 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 837	. . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑋 Fn (Base‘𝑍))
68	55	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑌 Fn (Base‘𝑍))
69		fvexd 6903	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (Base‘𝑍) ∈ V)
70		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑥 ∈ (Base‘𝑍))
71		fnfvof 7683	. . . . . . . . . 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 837	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((𝑋 ∘_f · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
73	72	adantrr 715	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘_f · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
74		simprr 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑦 ∈ (Base‘𝑍))
75		fnfvof 7683	. . . . . . . . 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 837	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘_f · 𝑌)‘𝑦) = ((𝑋‘𝑦) · (𝑌‘𝑦)))
77	73, 76	oveq12d 7423	. . . . . . 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 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 3200	. . . 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 2732	. . . . . . . 8 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
82	10, 81	ringidcl 20076	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) ∈ (Base‘𝑍))
83	61, 82	syl 17	. . . . . 6 ⊢ (𝜑 → (1_r‘𝑍) ∈ (Base‘𝑍))
84		fnfvof 7683	. . . . . 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 837	. . . . 5 ⊢ (𝜑 → ((𝑋 ∘_f · 𝑌)‘(1_r‘𝑍)) = ((𝑋‘(1_r‘𝑍)) · (𝑌‘(1_r‘𝑍))))
86	25, 81	ringidval 20000	. . . . . . . . 9 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
87		cnfld1 20962	. . . . . . . . . 10 ⊢ 1 = (1_r‘ℂ_fld)
88	29, 87	ringidval 20000	. . . . . . . . 9 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
89	86, 88	mhm0 18676	. . . . . . . 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 18676	. . . . . . . 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 7423	. . . . . 6 ⊢ (𝜑 → ((𝑋‘(1_r‘𝑍)) · (𝑌‘(1_r‘𝑍))) = (1 · 1))
94		1t1e1 12370	. . . . . 6 ⊢ (1 · 1) = 1
95	93, 94	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → ((𝑋‘(1_r‘𝑍)) · (𝑌‘(1_r‘𝑍))) = 1)
96	85, 95	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((𝑋 ∘_f · 𝑌)‘(1_r‘𝑍)) = 1)
97	72	neeq1d 3000	. . . . . . 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 281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋 ∘_f · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋‘𝑥) ≠ 0 ∧ (𝑌‘𝑥) ≠ 0)))
100	23	simprd 496	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
101	100	r19.21bi 3248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
102	101	adantrd 492	. . . . . 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 3146	. . . 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)‘(1_r‘𝑍)) = 1 ∧ ∀𝑥 ∈ (Base‘𝑍)(((𝑋 ∘_f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
106	1, 2, 10, 16, 21, 3	dchrelbas3 26730	. . 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 711	. 2 ⊢ (𝜑 → (𝑋 ∘_f · 𝑌) ∈ 𝐷)
108	7, 107	eqeltrd 2833	1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 Vcvv 3474 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∘_f cof 7664 ℂcc 11104 0cc0 11106 1c1 11107 · cmul 11111 ℕcn 12208 ℕ₀cn0 12468 Basecbs 17140 +_gcplusg 17193 ._rcmulr 17194 MndHom cmhm 18665 mulGrpcmgp 19981 1_rcur 19998 Ringcrg 20049 CRingccrg 20050 Unitcui 20161 ℂ_fldccnfld 20936 ℤ/nℤczn 21043 DChrcdchr 26724

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 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 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-nsg 18998 df-eqg 18999 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-subrg 20353 df-lmod 20465 df-lss 20535 df-lsp 20575 df-sra 20777 df-rgmod 20778 df-lidl 20779 df-rsp 20780 df-2idl 20849 df-cnfld 20937 df-zring 21010 df-zn 21047 df-dchr 26725

This theorem is referenced by: dchrabl 26746 dchrinv 26753

W3C validator