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Theorem dchrmulcl 24874
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 24873 . 2 (𝜑 → (𝑋 · 𝑌) = (𝑋𝑓 · 𝑌))
8 mulcl 9964 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
98adantl 482 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10 eqid 2621 . . . . 5 (Base‘𝑍) = (Base‘𝑍)
111, 2, 3, 10, 5dchrf 24867 . . . 4 (𝜑𝑋:(Base‘𝑍)⟶ℂ)
121, 2, 3, 10, 6dchrf 24867 . . . 4 (𝜑𝑌:(Base‘𝑍)⟶ℂ)
13 fvex 6158 . . . . 5 (Base‘𝑍) ∈ V
1413a1i 11 . . . 4 (𝜑 → (Base‘𝑍) ∈ V)
15 inidm 3800 . . . 4 ((Base‘𝑍) ∩ (Base‘𝑍)) = (Base‘𝑍)
169, 11, 12, 14, 14, 15off 6865 . . 3 (𝜑 → (𝑋𝑓 · 𝑌):(Base‘𝑍)⟶ℂ)
17 eqid 2621 . . . . . . . 8 (Unit‘𝑍) = (Unit‘𝑍)
1810, 17unitcl 18580 . . . . . . 7 (𝑥 ∈ (Unit‘𝑍) → 𝑥 ∈ (Base‘𝑍))
1910, 17unitcl 18580 . . . . . . 7 (𝑦 ∈ (Unit‘𝑍) → 𝑦 ∈ (Base‘𝑍))
2018, 19anim12i 589 . . . . . 6 ((𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍)) → (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)))
211, 3dchrrcl 24865 . . . . . . . . . . . . . 14 (𝑋𝐷𝑁 ∈ ℕ)
225, 21syl 17 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℕ)
231, 2, 10, 17, 22, 3dchrelbas2 24862 . . . . . . . . . . . 12 (𝜑 → (𝑋𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
245, 23mpbid 222 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
2524simpld 475 . . . . . . . . . 10 (𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
26 eqid 2621 . . . . . . . . . . . . 13 (mulGrp‘𝑍) = (mulGrp‘𝑍)
2726, 10mgpbas 18416 . . . . . . . . . . . 12 (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
28 eqid 2621 . . . . . . . . . . . . 13 (.r𝑍) = (.r𝑍)
2926, 28mgpplusg 18414 . . . . . . . . . . . 12 (.r𝑍) = (+g‘(mulGrp‘𝑍))
30 eqid 2621 . . . . . . . . . . . . 13 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
31 cnfldmul 19671 . . . . . . . . . . . . 13 · = (.r‘ℂfld)
3230, 31mgpplusg 18414 . . . . . . . . . . . 12 · = (+g‘(mulGrp‘ℂfld))
3327, 29, 32mhmlin 17263 . . . . . . . . . . 11 ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
34333expb 1263 . . . . . . . . . 10 ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
3525, 34sylan 488 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
361, 2, 10, 17, 22, 3dchrelbas2 24862 . . . . . . . . . . . 12 (𝜑 → (𝑌𝐷 ↔ (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
376, 36mpbid 222 . . . . . . . . . . 11 (𝜑 → (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
3837simpld 475 . . . . . . . . . 10 (𝜑𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
3927, 29, 32mhmlin 17263 . . . . . . . . . . 11 ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
40393expb 1263 . . . . . . . . . 10 ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
4138, 40sylan 488 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
4235, 41oveq12d 6622 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))) = (((𝑋𝑥) · (𝑋𝑦)) · ((𝑌𝑥) · (𝑌𝑦))))
4311ffvelrnda 6315 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (𝑋𝑥) ∈ ℂ)
4443adantrr 752 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋𝑥) ∈ ℂ)
45 simpr 477 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → 𝑦 ∈ (Base‘𝑍))
46 ffvelrn 6313 . . . . . . . . . 10 ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋𝑦) ∈ ℂ)
4711, 45, 46syl2an 494 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋𝑦) ∈ ℂ)
4812ffvelrnda 6315 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (𝑌𝑥) ∈ ℂ)
4948adantrr 752 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌𝑥) ∈ ℂ)
50 ffvelrn 6313 . . . . . . . . . 10 ((𝑌:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌𝑦) ∈ ℂ)
5112, 45, 50syl2an 494 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌𝑦) ∈ ℂ)
5244, 47, 49, 51mul4d 10192 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋𝑥) · (𝑋𝑦)) · ((𝑌𝑥) · (𝑌𝑦))) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
5342, 52eqtrd 2655 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
54 ffn 6002 . . . . . . . . . 10 (𝑋:(Base‘𝑍)⟶ℂ → 𝑋 Fn (Base‘𝑍))
5511, 54syl 17 . . . . . . . . 9 (𝜑𝑋 Fn (Base‘𝑍))
5655adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑋 Fn (Base‘𝑍))
57 ffn 6002 . . . . . . . . . 10 (𝑌:(Base‘𝑍)⟶ℂ → 𝑌 Fn (Base‘𝑍))
5812, 57syl 17 . . . . . . . . 9 (𝜑𝑌 Fn (Base‘𝑍))
5958adantr 481 . . . . . . . 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 11295 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
622zncrng 19812 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑍 ∈ CRing)
63 crngring 18479 . . . . . . . . . 10 (𝑍 ∈ CRing → 𝑍 ∈ Ring)
6461, 62, 633syl 18 . . . . . . . . 9 (𝜑𝑍 ∈ Ring)
6510, 28ringcl 18482 . . . . . . . . . 10 ((𝑍 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
66653expb 1263 . . . . . . . . 9 ((𝑍 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
6764, 66sylan 488 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
68 fnfvof 6864 . . . . . . . 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 1324 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))))
7055adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑋 Fn (Base‘𝑍))
7158adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑌 Fn (Base‘𝑍))
7213a1i 11 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (Base‘𝑍) ∈ V)
73 simpr 477 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑥 ∈ (Base‘𝑍))
74 fnfvof 6864 . . . . . . . . . 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 1324 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑍)) → ((𝑋𝑓 · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
7675adantrr 752 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
77 simprr 795 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑦 ∈ (Base‘𝑍))
78 fnfvof 6864 . . . . . . . . 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 1324 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘𝑦) = ((𝑋𝑦) · (𝑌𝑦)))
8076, 79oveq12d 6622 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
8153, 69, 803eqtr4d 2665 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)))
8220, 81sylan2 491 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)))
8382ralrimivva 2965 . . . 4 (𝜑 → ∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)))
84 eqid 2621 . . . . . . . 8 (1r𝑍) = (1r𝑍)
8510, 84ringidcl 18489 . . . . . . 7 (𝑍 ∈ Ring → (1r𝑍) ∈ (Base‘𝑍))
8664, 85syl 17 . . . . . 6 (𝜑 → (1r𝑍) ∈ (Base‘𝑍))
87 fnfvof 6864 . . . . . 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 1324 . . . . 5 (𝜑 → ((𝑋𝑓 · 𝑌)‘(1r𝑍)) = ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))))
8926, 84ringidval 18424 . . . . . . . . 9 (1r𝑍) = (0g‘(mulGrp‘𝑍))
90 cnfld1 19690 . . . . . . . . . 10 1 = (1r‘ℂfld)
9130, 90ringidval 18424 . . . . . . . . 9 1 = (0g‘(mulGrp‘ℂfld))
9289, 91mhm0 17264 . . . . . . . 8 (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑋‘(1r𝑍)) = 1)
9325, 92syl 17 . . . . . . 7 (𝜑 → (𝑋‘(1r𝑍)) = 1)
9489, 91mhm0 17264 . . . . . . . 8 (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑌‘(1r𝑍)) = 1)
9538, 94syl 17 . . . . . . 7 (𝜑 → (𝑌‘(1r𝑍)) = 1)
9693, 95oveq12d 6622 . . . . . 6 (𝜑 → ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))) = (1 · 1))
97 1t1e1 11119 . . . . . 6 (1 · 1) = 1
9896, 97syl6eq 2671 . . . . 5 (𝜑 → ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))) = 1)
9988, 98eqtrd 2655 . . . 4 (𝜑 → ((𝑋𝑓 · 𝑌)‘(1r𝑍)) = 1)
10075neeq1d 2849 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋𝑥) · (𝑌𝑥)) ≠ 0))
10143, 48mulne0bd 10622 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0) ↔ ((𝑋𝑥) · (𝑌𝑥)) ≠ 0))
102100, 101bitr4d 271 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0)))
10324simprd 479 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
104103r19.21bi 2927 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → ((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
105104adantrd 484 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0) → 𝑥 ∈ (Unit‘𝑍)))
106102, 105sylbid 230 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
107106ralrimiva 2960 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
10883, 99, 1073jca 1240 . . 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 24863 . . 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 956 . 2 (𝜑 → (𝑋𝑓 · 𝑌) ∈ 𝐷)
1117, 110eqeltrd 2698 1 (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  𝑓 cof 6848  cc 9878  0cc0 9880  1c1 9881   · cmul 9885  cn 10964  0cn0 11236  Basecbs 15781  +gcplusg 15862  .rcmulr 15863   MndHom cmhm 17254  mulGrpcmgp 18410  1rcur 18422  Ringcrg 18468  CRingccrg 18469  Unitcui 18560  fldccnfld 19665  ℤ/nczn 19770  DChrcdchr 24857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-tpos 7297  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-ec 7689  df-qs 7693  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-0g 16023  df-imas 16089  df-qus 16090  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-grp 17346  df-minusg 17347  df-sbg 17348  df-subg 17512  df-nsg 17513  df-eqg 17514  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-ring 18470  df-cring 18471  df-oppr 18544  df-dvdsr 18562  df-unit 18563  df-subrg 18699  df-lmod 18786  df-lss 18852  df-lsp 18891  df-sra 19091  df-rgmod 19092  df-lidl 19093  df-rsp 19094  df-2idl 19151  df-cnfld 19666  df-zring 19738  df-zn 19774  df-dchr 24858
This theorem is referenced by:  dchrabl  24879  dchrinv  24886
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