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Theorem dchrmulcl 27095
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 27094 . 2 (𝜑 → (𝑋 · 𝑌) = (𝑋f · 𝑌))
8 mulcl 11200 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
98adantl 481 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10 eqid 2731 . . . . 5 (Base‘𝑍) = (Base‘𝑍)
111, 2, 3, 10, 5dchrf 27088 . . . 4 (𝜑𝑋:(Base‘𝑍)⟶ℂ)
121, 2, 3, 10, 6dchrf 27088 . . . 4 (𝜑𝑌:(Base‘𝑍)⟶ℂ)
13 fvexd 6906 . . . 4 (𝜑 → (Base‘𝑍) ∈ V)
14 inidm 4218 . . . 4 ((Base‘𝑍) ∩ (Base‘𝑍)) = (Base‘𝑍)
159, 11, 12, 13, 13, 14off 7692 . . 3 (𝜑 → (𝑋f · 𝑌):(Base‘𝑍)⟶ℂ)
16 eqid 2731 . . . . . . . 8 (Unit‘𝑍) = (Unit‘𝑍)
1710, 16unitcl 20273 . . . . . . 7 (𝑥 ∈ (Unit‘𝑍) → 𝑥 ∈ (Base‘𝑍))
1810, 16unitcl 20273 . . . . . . 7 (𝑦 ∈ (Unit‘𝑍) → 𝑦 ∈ (Base‘𝑍))
1917, 18anim12i 612 . . . . . 6 ((𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍)) → (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)))
201, 3dchrrcl 27086 . . . . . . . . . . . . . 14 (𝑋𝐷𝑁 ∈ ℕ)
215, 20syl 17 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℕ)
221, 2, 10, 16, 21, 3dchrelbas2 27083 . . . . . . . . . . . 12 (𝜑 → (𝑋𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
235, 22mpbid 231 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
2423simpld 494 . . . . . . . . . 10 (𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
25 eqid 2731 . . . . . . . . . . . . 13 (mulGrp‘𝑍) = (mulGrp‘𝑍)
2625, 10mgpbas 20041 . . . . . . . . . . . 12 (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
27 eqid 2731 . . . . . . . . . . . . 13 (.r𝑍) = (.r𝑍)
2825, 27mgpplusg 20039 . . . . . . . . . . . 12 (.r𝑍) = (+g‘(mulGrp‘𝑍))
29 eqid 2731 . . . . . . . . . . . . 13 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
30 cnfldmul 21239 . . . . . . . . . . . . 13 · = (.r‘ℂfld)
3129, 30mgpplusg 20039 . . . . . . . . . . . 12 · = (+g‘(mulGrp‘ℂfld))
3226, 28, 31mhmlin 18721 . . . . . . . . . . 11 ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
33323expb 1119 . . . . . . . . . 10 ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
3424, 33sylan 579 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
351, 2, 10, 16, 21, 3dchrelbas2 27083 . . . . . . . . . . . 12 (𝜑 → (𝑌𝐷 ↔ (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))
366, 35mpbid 231 . . . . . . . . . . 11 (𝜑 → (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ (Base‘𝑍)((𝑌𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))
3736simpld 494 . . . . . . . . . 10 (𝜑𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
3826, 28, 31mhmlin 18721 . . . . . . . . . . 11 ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
39383expb 1119 . . . . . . . . . 10 ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
4037, 39sylan 579 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
4134, 40oveq12d 7430 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))) = (((𝑋𝑥) · (𝑋𝑦)) · ((𝑌𝑥) · (𝑌𝑦))))
4211ffvelcdmda 7086 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (𝑋𝑥) ∈ ℂ)
4342adantrr 714 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋𝑥) ∈ ℂ)
44 simpr 484 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → 𝑦 ∈ (Base‘𝑍))
45 ffvelcdm 7083 . . . . . . . . . 10 ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋𝑦) ∈ ℂ)
4611, 44, 45syl2an 595 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋𝑦) ∈ ℂ)
4712ffvelcdmda 7086 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (𝑌𝑥) ∈ ℂ)
4847adantrr 714 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌𝑥) ∈ ℂ)
49 ffvelcdm 7083 . . . . . . . . . 10 ((𝑌:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌𝑦) ∈ ℂ)
5012, 44, 49syl2an 595 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌𝑦) ∈ ℂ)
5143, 46, 48, 50mul4d 11433 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋𝑥) · (𝑋𝑦)) · ((𝑌𝑥) · (𝑌𝑦))) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
5241, 51eqtrd 2771 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
5311ffnd 6718 . . . . . . . . 9 (𝜑𝑋 Fn (Base‘𝑍))
5453adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑋 Fn (Base‘𝑍))
5512ffnd 6718 . . . . . . . . 9 (𝜑𝑌 Fn (Base‘𝑍))
5655adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑌 Fn (Base‘𝑍))
57 fvexd 6906 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (Base‘𝑍) ∈ V)
5821nnnn0d 12539 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
592zncrng 21410 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑍 ∈ CRing)
60 crngring 20146 . . . . . . . . . 10 (𝑍 ∈ CRing → 𝑍 ∈ Ring)
6158, 59, 603syl 18 . . . . . . . . 9 (𝜑𝑍 ∈ Ring)
6210, 27ringcl 20151 . . . . . . . . . 10 ((𝑍 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
63623expb 1119 . . . . . . . . 9 ((𝑍 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
6461, 63sylan 579 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
65 fnfvof 7691 . . . . . . . 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 836 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋f · 𝑌)‘(𝑥(.r𝑍)𝑦)) = ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))))
6753adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑋 Fn (Base‘𝑍))
6855adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑌 Fn (Base‘𝑍))
69 fvexd 6906 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (Base‘𝑍) ∈ V)
70 simpr 484 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑥 ∈ (Base‘𝑍))
71 fnfvof 7691 . . . . . . . . . 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 836 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑍)) → ((𝑋f · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
7372adantrr 714 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋f · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
74 simprr 770 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑦 ∈ (Base‘𝑍))
75 fnfvof 7691 . . . . . . . . 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 836 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋f · 𝑌)‘𝑦) = ((𝑋𝑦) · (𝑌𝑦)))
7773, 76oveq12d 7430 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋f · 𝑌)‘𝑥) · ((𝑋f · 𝑌)‘𝑦)) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
7852, 66, 773eqtr4d 2781 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋f · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋f · 𝑌)‘𝑥) · ((𝑋f · 𝑌)‘𝑦)))
7919, 78sylan2 592 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍))) → ((𝑋f · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋f · 𝑌)‘𝑥) · ((𝑋f · 𝑌)‘𝑦)))
8079ralrimivva 3199 . . . 4 (𝜑 → ∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)((𝑋f · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋f · 𝑌)‘𝑥) · ((𝑋f · 𝑌)‘𝑦)))
81 eqid 2731 . . . . . . . 8 (1r𝑍) = (1r𝑍)
8210, 81ringidcl 20161 . . . . . . 7 (𝑍 ∈ Ring → (1r𝑍) ∈ (Base‘𝑍))
8361, 82syl 17 . . . . . 6 (𝜑 → (1r𝑍) ∈ (Base‘𝑍))
84 fnfvof 7691 . . . . . 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 836 . . . . 5 (𝜑 → ((𝑋f · 𝑌)‘(1r𝑍)) = ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))))
8625, 81ringidval 20084 . . . . . . . . 9 (1r𝑍) = (0g‘(mulGrp‘𝑍))
87 cnfld1 21259 . . . . . . . . . 10 1 = (1r‘ℂfld)
8829, 87ringidval 20084 . . . . . . . . 9 1 = (0g‘(mulGrp‘ℂfld))
8986, 88mhm0 18722 . . . . . . . 8 (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑋‘(1r𝑍)) = 1)
9024, 89syl 17 . . . . . . 7 (𝜑 → (𝑋‘(1r𝑍)) = 1)
9186, 88mhm0 18722 . . . . . . . 8 (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑌‘(1r𝑍)) = 1)
9237, 91syl 17 . . . . . . 7 (𝜑 → (𝑌‘(1r𝑍)) = 1)
9390, 92oveq12d 7430 . . . . . 6 (𝜑 → ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))) = (1 · 1))
94 1t1e1 12381 . . . . . 6 (1 · 1) = 1
9593, 94eqtrdi 2787 . . . . 5 (𝜑 → ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))) = 1)
9685, 95eqtrd 2771 . . . 4 (𝜑 → ((𝑋f · 𝑌)‘(1r𝑍)) = 1)
9772neeq1d 2999 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋f · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋𝑥) · (𝑌𝑥)) ≠ 0))
9842, 47mulne0bd 11872 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0) ↔ ((𝑋𝑥) · (𝑌𝑥)) ≠ 0))
9997, 98bitr4d 282 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋f · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0)))
10023simprd 495 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
101100r19.21bi 3247 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → ((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
102101adantrd 491 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0) → 𝑥 ∈ (Unit‘𝑍)))
10399, 102sylbid 239 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
104103ralrimiva 3145 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(((𝑋f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
10580, 96, 1043jca 1127 . . 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 27084 . . 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 710 . 2 (𝜑 → (𝑋f · 𝑌) ∈ 𝐷)
1087, 107eqeltrd 2832 1 (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wral 3060  Vcvv 3473   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  f cof 7672  cc 11114  0cc0 11116  1c1 11117   · cmul 11121  cn 12219  0cn0 12479  Basecbs 17151  +gcplusg 17204  .rcmulr 17205   MndHom cmhm 18709  mulGrpcmgp 20035  1rcur 20082  Ringcrg 20134  CRingccrg 20135  Unitcui 20253  fldccnfld 21233  ℤ/nczn 21362  DChrcdchr 27078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-addf 11195  ax-mulf 11196
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-tpos 8217  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-ec 8711  df-qs 8715  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-starv 17219  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-unif 17227  df-0g 17394  df-imas 17461  df-qus 17462  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-mhm 18711  df-grp 18864  df-minusg 18865  df-sbg 18866  df-subg 19046  df-nsg 19047  df-eqg 19048  df-cmn 19698  df-abl 19699  df-mgp 20036  df-rng 20054  df-ur 20083  df-ring 20136  df-cring 20137  df-oppr 20232  df-dvdsr 20255  df-unit 20256  df-subrng 20442  df-subrg 20467  df-lmod 20704  df-lss 20775  df-lsp 20815  df-sra 21019  df-rgmod 21020  df-lidl 21021  df-rsp 21022  df-2idl 21095  df-cnfld 21234  df-zring 21307  df-zn 21366  df-dchr 27079
This theorem is referenced by:  dchrabl  27100  dchrinv  27107
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