Theorem dchrmulcl 24774

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 24773	. 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 ∘𝑓 · 𝑌))
8		mulcl 9899	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
9	8	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10		eqid 2610	. . . . 5 ⊢ (Base‘𝑍) = (Base‘𝑍)
11	1, 2, 3, 10, 5	dchrf 24767	. . . 4 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
12	1, 2, 3, 10, 6	dchrf 24767	. . . 4 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ)
13		fvex 6113	. . . . 5 ⊢ (Base‘𝑍) ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → (Base‘𝑍) ∈ V)
15		inidm 3784	. . . 4 ⊢ ((Base‘𝑍) ∩ (Base‘𝑍)) = (Base‘𝑍)
16	9, 11, 12, 14, 14, 15	off 6810	. . 3 ⊢ (𝜑 → (𝑋 ∘𝑓 · 𝑌):(Base‘𝑍)⟶ℂ)
17		eqid 2610	. . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
18	10, 17	unitcl 18482	. . . . . . 7 ⊢ (𝑥 ∈ (Unit‘𝑍) → 𝑥 ∈ (Base‘𝑍))
19	10, 17	unitcl 18482	. . . . . . 7 ⊢ (𝑦 ∈ (Unit‘𝑍) → 𝑦 ∈ (Base‘𝑍))
20	18, 19	anim12i 588	. . . . . 6 ⊢ ((𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍)) → (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)))
21	1, 3	dchrrcl 24765	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
22	5, 21	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
23	1, 2, 10, 17, 22, 3	dchrelbas2 24762	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
24	5, 23	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
25	24	simpld 474	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
26		eqid 2610	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
27	26, 10	mgpbas 18318	. . . . . . . . . . . 12 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
28		eqid 2610	. . . . . . . . . . . . 13 ⊢ (.r‘𝑍) = (.r‘𝑍)
29	26, 28	mgpplusg 18316	. . . . . . . . . . . 12 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍))
30		eqid 2610	. . . . . . . . . . . . 13 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
31		cnfldmul 19573	. . . . . . . . . . . . 13 ⊢ · = (.r‘ℂfld)
32	30, 31	mgpplusg 18316	. . . . . . . . . . . 12 ⊢ · = (+g‘(mulGrp‘ℂfld))
33	27, 29, 32	mhmlin 17165	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘(𝑥(.r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
34	33	3expb 1258	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
35	25, 34	sylan 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
36	1, 2, 10, 17, 22, 3	dchrelbas2 24762	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌 ∈ 𝐷 ↔ (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
37	6, 36	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
38	37	simpld 474	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
39	27, 29, 32	mhmlin 17165	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘(𝑥(.r‘𝑍)𝑦)) = ((𝑌‘𝑥) · (𝑌‘𝑦)))
40	39	3expb 1258	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r‘𝑍)𝑦)) = ((𝑌‘𝑥) · (𝑌‘𝑦)))
41	38, 40	sylan 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r‘𝑍)𝑦)) = ((𝑌‘𝑥) · (𝑌‘𝑦)))
42	35, 41	oveq12d 6567	. . . . . . . 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))))
43	11	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
44	43	adantrr 749	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘𝑥) ∈ ℂ)
45		simpr 476	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → 𝑦 ∈ (Base‘𝑍))
46		ffvelrn 6265	. . . . . . . . . 10 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘𝑦) ∈ ℂ)
47	11, 45, 46	syl2an 493	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘𝑦) ∈ ℂ)
48	12	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑌‘𝑥) ∈ ℂ)
49	48	adantrr 749	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘𝑥) ∈ ℂ)
50		ffvelrn 6265	. . . . . . . . . 10 ⊢ ((𝑌:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘𝑦) ∈ ℂ)
51	12, 45, 50	syl2an 493	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘𝑦) ∈ ℂ)
52	44, 47, 49, 51	mul4d 10127	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋‘𝑥) · (𝑋‘𝑦)) · ((𝑌‘𝑥) · (𝑌‘𝑦))) = (((𝑋‘𝑥) · (𝑌‘𝑥)) · ((𝑋‘𝑦) · (𝑌‘𝑦))))
53	42, 52	eqtrd 2644	. . . . . . 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))))
54		ffn 5958	. . . . . . . . . 10 ⊢ (𝑋:(Base‘𝑍)⟶ℂ → 𝑋 Fn (Base‘𝑍))
55	11, 54	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍))
56	55	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑋 Fn (Base‘𝑍))
57		ffn 5958	. . . . . . . . . 10 ⊢ (𝑌:(Base‘𝑍)⟶ℂ → 𝑌 Fn (Base‘𝑍))
58	12, 57	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍))
59	58	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑌 Fn (Base‘𝑍))
60	13	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (Base‘𝑍) ∈ V)
61	22	nnnn0d 11228	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
62	2	zncrng 19712	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing)
63		crngring 18381	. . . . . . . . . 10 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
64	61, 62, 63	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ Ring)
65	10, 28	ringcl 18384	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑥(.r‘𝑍)𝑦) ∈ (Base‘𝑍))
66	65	3expb 1258	. . . . . . . . 9 ⊢ ((𝑍 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r‘𝑍)𝑦) ∈ (Base‘𝑍))
67	64, 66	sylan 487	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r‘𝑍)𝑦) ∈ (Base‘𝑍))
68		fnfvof 6809	. . . . . . . 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))))
69	56, 59, 60, 67, 68	syl22anc 1319	. . . . . . 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))))
70	55	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑋 Fn (Base‘𝑍))
71	58	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑌 Fn (Base‘𝑍))
72	13	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (Base‘𝑍) ∈ V)
73		simpr 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → 𝑥 ∈ (Base‘𝑍))
74		fnfvof 6809	. . . . . . . . . 10 ⊢ (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ 𝑥 ∈ (Base‘𝑍))) → ((𝑋 ∘𝑓 · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
75	70, 71, 72, 73, 74	syl22anc 1319	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((𝑋 ∘𝑓 · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
76	75	adantrr 749	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘𝑓 · 𝑌)‘𝑥) = ((𝑋‘𝑥) · (𝑌‘𝑥)))
77		simprr 792	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑦 ∈ (Base‘𝑍))
78		fnfvof 6809	. . . . . . . . 9 ⊢ (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘𝑓 · 𝑌)‘𝑦) = ((𝑋‘𝑦) · (𝑌‘𝑦)))
79	56, 59, 60, 77, 78	syl22anc 1319	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘𝑓 · 𝑌)‘𝑦) = ((𝑋‘𝑦) · (𝑌‘𝑦)))
80	76, 79	oveq12d 6567	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋 ∘𝑓 · 𝑌)‘𝑥) · ((𝑋 ∘𝑓 · 𝑌)‘𝑦)) = (((𝑋‘𝑥) · (𝑌‘𝑥)) · ((𝑋‘𝑦) · (𝑌‘𝑦))))
81	53, 69, 80	3eqtr4d 2654	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋 ∘𝑓 · 𝑌)‘(𝑥(.r‘𝑍)𝑦)) = (((𝑋 ∘𝑓 · 𝑌)‘𝑥) · ((𝑋 ∘𝑓 · 𝑌)‘𝑦)))
82	20, 81	sylan2 490	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍))) → ((𝑋 ∘𝑓 · 𝑌)‘(𝑥(.r‘𝑍)𝑦)) = (((𝑋 ∘𝑓 · 𝑌)‘𝑥) · ((𝑋 ∘𝑓 · 𝑌)‘𝑦)))
83	82	ralrimivva 2954	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)((𝑋 ∘𝑓 · 𝑌)‘(𝑥(.r‘𝑍)𝑦)) = (((𝑋 ∘𝑓 · 𝑌)‘𝑥) · ((𝑋 ∘𝑓 · 𝑌)‘𝑦)))
84		eqid 2610	. . . . . . . 8 ⊢ (1r‘𝑍) = (1r‘𝑍)
85	10, 84	ringidcl 18391	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (1r‘𝑍) ∈ (Base‘𝑍))
86	64, 85	syl 17	. . . . . 6 ⊢ (𝜑 → (1r‘𝑍) ∈ (Base‘𝑍))
87		fnfvof 6809	. . . . . 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))))
88	55, 58, 14, 86, 87	syl22anc 1319	. . . . 5 ⊢ (𝜑 → ((𝑋 ∘𝑓 · 𝑌)‘(1r‘𝑍)) = ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))))
89	26, 84	ringidval 18326	. . . . . . . . 9 ⊢ (1r‘𝑍) = (0g‘(mulGrp‘𝑍))
90		cnfld1 19590	. . . . . . . . . 10 ⊢ 1 = (1r‘ℂfld)
91	30, 90	ringidval 18326	. . . . . . . . 9 ⊢ 1 = (0g‘(mulGrp‘ℂfld))
92	89, 91	mhm0 17166	. . . . . . . 8 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑋‘(1r‘𝑍)) = 1)
93	25, 92	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑋‘(1r‘𝑍)) = 1)
94	89, 91	mhm0 17166	. . . . . . . 8 ⊢ (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑌‘(1r‘𝑍)) = 1)
95	38, 94	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑌‘(1r‘𝑍)) = 1)
96	93, 95	oveq12d 6567	. . . . . 6 ⊢ (𝜑 → ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))) = (1 · 1))
97		1t1e1 11052	. . . . . 6 ⊢ (1 · 1) = 1
98	96, 97	syl6eq 2660	. . . . 5 ⊢ (𝜑 → ((𝑋‘(1r‘𝑍)) · (𝑌‘(1r‘𝑍))) = 1)
99	88, 98	eqtrd 2644	. . . 4 ⊢ (𝜑 → ((𝑋 ∘𝑓 · 𝑌)‘(1r‘𝑍)) = 1)
100	75	neeq1d 2841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋 ∘𝑓 · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋‘𝑥) · (𝑌‘𝑥)) ≠ 0))
101	43, 48	mulne0bd 10557	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋‘𝑥) ≠ 0 ∧ (𝑌‘𝑥) ≠ 0) ↔ ((𝑋‘𝑥) · (𝑌‘𝑥)) ≠ 0))
102	100, 101	bitr4d 270	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋 ∘𝑓 · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋‘𝑥) ≠ 0 ∧ (𝑌‘𝑥) ≠ 0)))
103	24	simprd 478	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
104	103	r19.21bi 2916	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
105	104	adantrd 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋‘𝑥) ≠ 0 ∧ (𝑌‘𝑥) ≠ 0) → 𝑥 ∈ (Unit‘𝑍)))
106	102, 105	sylbid 229	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (((𝑋 ∘𝑓 · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
107	106	ralrimiva 2949	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(((𝑋 ∘𝑓 · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
108	83, 99, 107	3jca 1235	. . 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))))
109	1, 2, 10, 17, 22, 3	dchrelbas3 24763	. . 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))))))
110	16, 108, 109	mpbir2and 959	. 2 ⊢ (𝜑 → (𝑋 ∘𝑓 · 𝑌) ∈ 𝐷)
111	7, 110	eqeltrd 2688	1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  ℂcc 9813  0cc0 9815  1c1 9816   · cmul 9820  ℕcn 10897  ℕ₀cn0 11169  Basecbs 15695  +_gcplusg 15768  ._rcmulr 15769   MndHom cmhm 17156  mulGrpcmgp 18312  1_rcur 18324  Ringcrg 18370  CRingccrg 18371  Unitcui 18462  ℂ_fldccnfld 19567  ℤ/nℤczn 19670  DChrcdchr 24757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-0g 15925  df-imas 15991  df-qus 15992  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-nsg 17415  df-eqg 17416  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-subrg 18601  df-lmod 18688  df-lss 18754  df-lsp 18793  df-sra 18993  df-rgmod 18994  df-lidl 18995  df-rsp 18996  df-2idl 19053  df-cnfld 19568  df-zring 19638  df-zn 19674  df-dchr 24758

This theorem is referenced by:  dchrabl  24779  dchrinv  24786