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Theorem dchrmulcl 25387
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 25386 . 2 (𝜑 → (𝑋 · 𝑌) = (𝑋𝑓 · 𝑌))
8 mulcl 10336 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
98adantl 475 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10 eqid 2825 . . . . 5 (Base‘𝑍) = (Base‘𝑍)
111, 2, 3, 10, 5dchrf 25380 . . . 4 (𝜑𝑋:(Base‘𝑍)⟶ℂ)
121, 2, 3, 10, 6dchrf 25380 . . . 4 (𝜑𝑌:(Base‘𝑍)⟶ℂ)
13 fvexd 6448 . . . 4 (𝜑 → (Base‘𝑍) ∈ V)
14 inidm 4047 . . . 4 ((Base‘𝑍) ∩ (Base‘𝑍)) = (Base‘𝑍)
159, 11, 12, 13, 13, 14off 7172 . . 3 (𝜑 → (𝑋𝑓 · 𝑌):(Base‘𝑍)⟶ℂ)
16 eqid 2825 . . . . . . . 8 (Unit‘𝑍) = (Unit‘𝑍)
1710, 16unitcl 19013 . . . . . . 7 (𝑥 ∈ (Unit‘𝑍) → 𝑥 ∈ (Base‘𝑍))
1810, 16unitcl 19013 . . . . . . 7 (𝑦 ∈ (Unit‘𝑍) → 𝑦 ∈ (Base‘𝑍))
1917, 18anim12i 606 . . . . . 6 ((𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍)) → (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)))
201, 3dchrrcl 25378 . . . . . . . . . . . . . 14 (𝑋𝐷𝑁 ∈ ℕ)
215, 20syl 17 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℕ)
221, 2, 10, 16, 21, 3dchrelbas2 25375 . . . . . . . . . . . 12 (𝜑 → (𝑋𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
235, 22mpbid 224 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
2423simpld 490 . . . . . . . . . 10 (𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
25 eqid 2825 . . . . . . . . . . . . 13 (mulGrp‘𝑍) = (mulGrp‘𝑍)
2625, 10mgpbas 18849 . . . . . . . . . . . 12 (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
27 eqid 2825 . . . . . . . . . . . . 13 (.r𝑍) = (.r𝑍)
2825, 27mgpplusg 18847 . . . . . . . . . . . 12 (.r𝑍) = (+g‘(mulGrp‘𝑍))
29 eqid 2825 . . . . . . . . . . . . 13 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
30 cnfldmul 20112 . . . . . . . . . . . . 13 · = (.r‘ℂfld)
3129, 30mgpplusg 18847 . . . . . . . . . . . 12 · = (+g‘(mulGrp‘ℂfld))
3226, 28, 31mhmlin 17695 . . . . . . . . . . 11 ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
33323expb 1153 . . . . . . . . . 10 ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
3424, 33sylan 575 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
351, 2, 10, 16, 21, 3dchrelbas2 25375 . . . . . . . . . . . 12 (𝜑 → (𝑌𝐷 ↔ (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
366, 35mpbid 224 . . . . . . . . . . 11 (𝜑 → (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
3736simpld 490 . . . . . . . . . 10 (𝜑𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
3826, 28, 31mhmlin 17695 . . . . . . . . . . 11 ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
39383expb 1153 . . . . . . . . . 10 ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
4037, 39sylan 575 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
4134, 40oveq12d 6923 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))) = (((𝑋𝑥) · (𝑋𝑦)) · ((𝑌𝑥) · (𝑌𝑦))))
4211ffvelrnda 6608 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (𝑋𝑥) ∈ ℂ)
4342adantrr 708 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋𝑥) ∈ ℂ)
44 simpr 479 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → 𝑦 ∈ (Base‘𝑍))
45 ffvelrn 6606 . . . . . . . . . 10 ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋𝑦) ∈ ℂ)
4611, 44, 45syl2an 589 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋𝑦) ∈ ℂ)
4712ffvelrnda 6608 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (𝑌𝑥) ∈ ℂ)
4847adantrr 708 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌𝑥) ∈ ℂ)
49 ffvelrn 6606 . . . . . . . . . 10 ((𝑌:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌𝑦) ∈ ℂ)
5012, 44, 49syl2an 589 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌𝑦) ∈ ℂ)
5143, 46, 48, 50mul4d 10567 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋𝑥) · (𝑋𝑦)) · ((𝑌𝑥) · (𝑌𝑦))) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
5241, 51eqtrd 2861 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
5311ffnd 6279 . . . . . . . . 9 (𝜑𝑋 Fn (Base‘𝑍))
5453adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑋 Fn (Base‘𝑍))
5512ffnd 6279 . . . . . . . . 9 (𝜑𝑌 Fn (Base‘𝑍))
5655adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑌 Fn (Base‘𝑍))
57 fvexd 6448 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (Base‘𝑍) ∈ V)
5821nnnn0d 11678 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
592zncrng 20252 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑍 ∈ CRing)
60 crngring 18912 . . . . . . . . . 10 (𝑍 ∈ CRing → 𝑍 ∈ Ring)
6158, 59, 603syl 18 . . . . . . . . 9 (𝜑𝑍 ∈ Ring)
6210, 27ringcl 18915 . . . . . . . . . 10 ((𝑍 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
63623expb 1153 . . . . . . . . 9 ((𝑍 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
6461, 63sylan 575 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
65 fnfvof 7171 . . . . . . . 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))))
6654, 56, 57, 64, 65syl22anc 872 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))))
6753adantr 474 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑋 Fn (Base‘𝑍))
6855adantr 474 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑌 Fn (Base‘𝑍))
69 fvexd 6448 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (Base‘𝑍) ∈ V)
70 simpr 479 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑥 ∈ (Base‘𝑍))
71 fnfvof 7171 . . . . . . . . . 10 (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ 𝑥 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
7267, 68, 69, 70, 71syl22anc 872 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑍)) → ((𝑋𝑓 · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
7372adantrr 708 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
74 simprr 789 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑦 ∈ (Base‘𝑍))
75 fnfvof 7171 . . . . . . . . 9 (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘𝑦) = ((𝑋𝑦) · (𝑌𝑦)))
7654, 56, 57, 74, 75syl22anc 872 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘𝑦) = ((𝑋𝑦) · (𝑌𝑦)))
7773, 76oveq12d 6923 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
7852, 66, 773eqtr4d 2871 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)))
7919, 78sylan2 586 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)))
8079ralrimivva 3180 . . . 4 (𝜑 → ∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)((𝑋𝑓 · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋𝑓 · 𝑌)‘𝑥) · ((𝑋𝑓 · 𝑌)‘𝑦)))
81 eqid 2825 . . . . . . . 8 (1r𝑍) = (1r𝑍)
8210, 81ringidcl 18922 . . . . . . 7 (𝑍 ∈ Ring → (1r𝑍) ∈ (Base‘𝑍))
8361, 82syl 17 . . . . . 6 (𝜑 → (1r𝑍) ∈ (Base‘𝑍))
84 fnfvof 7171 . . . . . 6 (((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) ∧ ((Base‘𝑍) ∈ V ∧ (1r𝑍) ∈ (Base‘𝑍))) → ((𝑋𝑓 · 𝑌)‘(1r𝑍)) = ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))))
8553, 55, 13, 83, 84syl22anc 872 . . . . 5 (𝜑 → ((𝑋𝑓 · 𝑌)‘(1r𝑍)) = ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))))
8625, 81ringidval 18857 . . . . . . . . 9 (1r𝑍) = (0g‘(mulGrp‘𝑍))
87 cnfld1 20131 . . . . . . . . . 10 1 = (1r‘ℂfld)
8829, 87ringidval 18857 . . . . . . . . 9 1 = (0g‘(mulGrp‘ℂfld))
8986, 88mhm0 17696 . . . . . . . 8 (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑋‘(1r𝑍)) = 1)
9024, 89syl 17 . . . . . . 7 (𝜑 → (𝑋‘(1r𝑍)) = 1)
9186, 88mhm0 17696 . . . . . . . 8 (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑌‘(1r𝑍)) = 1)
9237, 91syl 17 . . . . . . 7 (𝜑 → (𝑌‘(1r𝑍)) = 1)
9390, 92oveq12d 6923 . . . . . 6 (𝜑 → ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))) = (1 · 1))
94 1t1e1 11520 . . . . . 6 (1 · 1) = 1
9593, 94syl6eq 2877 . . . . 5 (𝜑 → ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))) = 1)
9685, 95eqtrd 2861 . . . 4 (𝜑 → ((𝑋𝑓 · 𝑌)‘(1r𝑍)) = 1)
9772neeq1d 3058 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋𝑥) · (𝑌𝑥)) ≠ 0))
9842, 47mulne0bd 11003 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0) ↔ ((𝑋𝑥) · (𝑌𝑥)) ≠ 0))
9997, 98bitr4d 274 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0)))
10023simprd 491 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
101100r19.21bi 3141 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → ((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
102101adantrd 487 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0) → 𝑥 ∈ (Unit‘𝑍)))
10399, 102sylbid 232 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
104103ralrimiva 3175 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(((𝑋𝑓 · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
10580, 96, 1043jca 1162 . . 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))))
1061, 2, 10, 16, 21, 3dchrelbas3 25376 . . 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))))))
10715, 105, 106mpbir2and 704 . 2 (𝜑 → (𝑋𝑓 · 𝑌) ∈ 𝐷)
1087, 107eqeltrd 2906 1 (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999  wral 3117  Vcvv 3414   Fn wfn 6118  wf 6119  cfv 6123  (class class class)co 6905  𝑓 cof 7155  cc 10250  0cc0 10252  1c1 10253   · cmul 10257  cn 11350  0cn0 11618  Basecbs 16222  +gcplusg 16305  .rcmulr 16306   MndHom cmhm 17686  mulGrpcmgp 18843  1rcur 18855  Ringcrg 18901  CRingccrg 18902  Unitcui 18993  fldccnfld 20106  ℤ/nczn 20211  DChrcdchr 25370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-addf 10331  ax-mulf 10332
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-of 7157  df-om 7327  df-1st 7428  df-2nd 7429  df-tpos 7617  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-ec 8011  df-qs 8015  df-map 8124  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-sup 8617  df-inf 8618  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-z 11705  df-dec 11822  df-uz 11969  df-fz 12620  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-mulr 16319  df-starv 16320  df-sca 16321  df-vsca 16322  df-ip 16323  df-tset 16324  df-ple 16325  df-ds 16327  df-unif 16328  df-0g 16455  df-imas 16521  df-qus 16522  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-mhm 17688  df-grp 17779  df-minusg 17780  df-sbg 17781  df-subg 17942  df-nsg 17943  df-eqg 17944  df-cmn 18548  df-abl 18549  df-mgp 18844  df-ur 18856  df-ring 18903  df-cring 18904  df-oppr 18977  df-dvdsr 18995  df-unit 18996  df-subrg 19134  df-lmod 19221  df-lss 19289  df-lsp 19331  df-sra 19533  df-rgmod 19534  df-lidl 19535  df-rsp 19536  df-2idl 19593  df-cnfld 20107  df-zring 20179  df-zn 20215  df-dchr 25371
This theorem is referenced by:  dchrabl  25392  dchrinv  25399
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