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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 · = (+g𝐺)
dchrmul.x (𝜑𝑋𝐷)
dchrmul.y (𝜑𝑌𝐷)
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.
StepHypRef 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 · = (+g𝐺)
5 dchrmul.x . . 3 (𝜑𝑋𝐷)
6 dchrmul.y . . 3 (𝜑𝑌𝐷)
71, 2, 3, 4, 5, 6dchrmul 24773 . 2 (𝜑 → (𝑋 · 𝑌) = (𝑋𝑓 · 𝑌))
8 mulcl 9899 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
98adantl 481 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10 eqid 2610 . . . . 5 (Base‘𝑍) = (Base‘𝑍)
111, 2, 3, 10, 5dchrf 24767 . . . 4 (𝜑𝑋:(Base‘𝑍)⟶ℂ)
121, 2, 3, 10, 6dchrf 24767 . . . 4 (𝜑𝑌:(Base‘𝑍)⟶ℂ)
13 fvex 6113 . . . . 5 (Base‘𝑍) ∈ V
1413a1i 11 . . . 4 (𝜑 → (Base‘𝑍) ∈ V)
15 inidm 3784 . . . 4 ((Base‘𝑍) ∩ (Base‘𝑍)) = (Base‘𝑍)
169, 11, 12, 14, 14, 15off 6810 . . 3 (𝜑 → (𝑋𝑓 · 𝑌):(Base‘𝑍)⟶ℂ)
17 eqid 2610 . . . . . . . 8 (Unit‘𝑍) = (Unit‘𝑍)
1810, 17unitcl 18482 . . . . . . 7 (𝑥 ∈ (Unit‘𝑍) → 𝑥 ∈ (Base‘𝑍))
1910, 17unitcl 18482 . . . . . . 7 (𝑦 ∈ (Unit‘𝑍) → 𝑦 ∈ (Base‘𝑍))
2018, 19anim12i 588 . . . . . 6 ((𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍)) → (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)))
211, 3dchrrcl 24765 . . . . . . . . . . . . . 14 (𝑋𝐷𝑁 ∈ ℕ)
225, 21syl 17 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℕ)
231, 2, 10, 17, 22, 3dchrelbas2 24762 . . . . . . . . . . . 12 (𝜑 → (𝑋𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
245, 23mpbid 221 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
2524simpld 474 . . . . . . . . . 10 (𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
26 eqid 2610 . . . . . . . . . . . . 13 (mulGrp‘𝑍) = (mulGrp‘𝑍)
2726, 10mgpbas 18318 . . . . . . . . . . . 12 (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
28 eqid 2610 . . . . . . . . . . . . 13 (.r𝑍) = (.r𝑍)
2926, 28mgpplusg 18316 . . . . . . . . . . . 12 (.r𝑍) = (+g‘(mulGrp‘𝑍))
30 eqid 2610 . . . . . . . . . . . . 13 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
31 cnfldmul 19573 . . . . . . . . . . . . 13 · = (.r‘ℂfld)
3230, 31mgpplusg 18316 . . . . . . . . . . . 12 · = (+g‘(mulGrp‘ℂfld))
3327, 29, 32mhmlin 17165 . . . . . . . . . . 11 ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
34333expb 1258 . . . . . . . . . 10 ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
3525, 34sylan 487 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
361, 2, 10, 17, 22, 3dchrelbas2 24762 . . . . . . . . . . . 12 (𝜑 → (𝑌𝐷 ↔ (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
376, 36mpbid 221 . . . . . . . . . . 11 (𝜑 → (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
3837simpld 474 . . . . . . . . . 10 (𝜑𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
3927, 29, 32mhmlin 17165 . . . . . . . . . . 11 ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
40393expb 1258 . . . . . . . . . 10 ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
4138, 40sylan 487 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
4235, 41oveq12d 6567 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))) = (((𝑋𝑥) · (𝑋𝑦)) · ((𝑌𝑥) · (𝑌𝑦))))
4311ffvelrnda 6267 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (𝑋𝑥) ∈ ℂ)
4443adantrr 749 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋𝑥) ∈ 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)) → (𝑋𝑦) ∈ ℂ)
4711, 45, 46syl2an 493 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋𝑦) ∈ ℂ)
4812ffvelrnda 6267 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (𝑌𝑥) ∈ ℂ)
4948adantrr 749 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌𝑥) ∈ ℂ)
50 ffvelrn 6265 . . . . . . . . . 10 ((𝑌:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌𝑦) ∈ ℂ)
5112, 45, 50syl2an 493 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌𝑦) ∈ ℂ)
5244, 47, 49, 51mul4d 10127 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋𝑥) · (𝑋𝑦)) · ((𝑌𝑥) · (𝑌𝑦))) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
5342, 52eqtrd 2644 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
54 ffn 5958 . . . . . . . . . 10 (𝑋:(Base‘𝑍)⟶ℂ → 𝑋 Fn (Base‘𝑍))
5511, 54syl 17 . . . . . . . . 9 (𝜑𝑋 Fn (Base‘𝑍))
5655adantr 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))
5812, 57syl 17 . . . . . . . . 9 (𝜑𝑌 Fn (Base‘𝑍))
5958adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑌 Fn (Base‘𝑍))
6013a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (Base‘𝑍) ∈ V)
6122nnnn0d 11228 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
622zncrng 19712 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑍 ∈ CRing)
63 crngring 18381 . . . . . . . . . 10 (𝑍 ∈ CRing → 𝑍 ∈ Ring)
6461, 62, 633syl 18 . . . . . . . . 9 (𝜑𝑍 ∈ Ring)
6510, 28ringcl 18384 . . . . . . . . . 10 ((𝑍 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
66653expb 1258 . . . . . . . . 9 ((𝑍 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
6764, 66sylan 487 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
68 fnfvof 6809 . . . . . . . 8 (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))))
6956, 59, 60, 67, 68syl22anc 1319 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))))
7055adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑋 Fn (Base‘𝑍))
7158adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑌 Fn (Base‘𝑍))
7213a1i 11 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (Base‘𝑍) ∈ V)
73 simpr 476 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑥 ∈ (Base‘𝑍))
74 fnfvof 6809 . . . . . . . . . 10 (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ 𝑥 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
7570, 71, 72, 73, 74syl22anc 1319 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑍)) → ((𝑋𝑓 · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
7675adantrr 749 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
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))) → ((𝑋𝑓 · 𝑌)‘𝑦) = ((𝑋𝑦) · (𝑌𝑦)))
7956, 59, 60, 77, 78syl22anc 1319 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘𝑦) = ((𝑋𝑦) · (𝑌𝑦)))
8076, 79oveq12d 6567 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
8153, 69, 803eqtr4d 2654 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)))
8220, 81sylan2 490 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)))
8382ralrimivva 2954 . . . 4 (𝜑 → ∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)))
84 eqid 2610 . . . . . . . 8 (1r𝑍) = (1r𝑍)
8510, 84ringidcl 18391 . . . . . . 7 (𝑍 ∈ Ring → (1r𝑍) ∈ (Base‘𝑍))
8664, 85syl 17 . . . . . 6 (𝜑 → (1r𝑍) ∈ (Base‘𝑍))
87 fnfvof 6809 . . . . . 6 (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ (1r𝑍) ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(1r𝑍)) = ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))))
8855, 58, 14, 86, 87syl22anc 1319 . . . . 5 (𝜑 → ((𝑋𝑓 · 𝑌)‘(1r𝑍)) = ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))))
8926, 84ringidval 18326 . . . . . . . . 9 (1r𝑍) = (0g‘(mulGrp‘𝑍))
90 cnfld1 19590 . . . . . . . . . 10 1 = (1r‘ℂfld)
9130, 90ringidval 18326 . . . . . . . . 9 1 = (0g‘(mulGrp‘ℂfld))
9289, 91mhm0 17166 . . . . . . . 8 (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑋‘(1r𝑍)) = 1)
9325, 92syl 17 . . . . . . 7 (𝜑 → (𝑋‘(1r𝑍)) = 1)
9489, 91mhm0 17166 . . . . . . . 8 (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑌‘(1r𝑍)) = 1)
9538, 94syl 17 . . . . . . 7 (𝜑 → (𝑌‘(1r𝑍)) = 1)
9693, 95oveq12d 6567 . . . . . 6 (𝜑 → ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))) = (1 · 1))
97 1t1e1 11052 . . . . . 6 (1 · 1) = 1
9896, 97syl6eq 2660 . . . . 5 (𝜑 → ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))) = 1)
9988, 98eqtrd 2644 . . . 4 (𝜑 → ((𝑋𝑓 · 𝑌)‘(1r𝑍)) = 1)
10075neeq1d 2841 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋𝑥) · (𝑌𝑥)) ≠ 0))
10143, 48mulne0bd 10557 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0) ↔ ((𝑋𝑥) · (𝑌𝑥)) ≠ 0))
102100, 101bitr4d 270 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0)))
10324simprd 478 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
104103r19.21bi 2916 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → ((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
105104adantrd 483 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0) → 𝑥 ∈ (Unit‘𝑍)))
106102, 105sylbid 229 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
107106ralrimiva 2949 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
10883, 99, 1073jca 1235 . . 3 (𝜑 → (∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)) ∧ ((𝑋𝑓 · 𝑌)‘(1r𝑍)) = 1 ∧ ∀𝑥 ∈ (Base‘𝑍)(((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
1091, 2, 10, 17, 22, 3dchrelbas3 24763 . . 3 (𝜑 → ((𝑋𝑓 · 𝑌) ∈ 𝐷 ↔ ((𝑋𝑓 · 𝑌):(Base‘𝑍)⟶ℂ ∧ (∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)) ∧ ((𝑋𝑓 · 𝑌)‘(1r𝑍)) = 1 ∧ ∀𝑥 ∈ (Base‘𝑍)(((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))))
11016, 108, 109mpbir2and 959 . 2 (𝜑 → (𝑋𝑓 · 𝑌) ∈ 𝐷)
1117, 110eqeltrd 2688 1 (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)
 Colors of variables: wff setvar class 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  1rcur 18324  Ringcrg 18370  CRingccrg 18371  Unitcui 18462  ℂfldccnfld 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
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