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Theorem dchrmulcl 26597
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 26596 . 2 (𝜑 → (𝑋 · 𝑌) = (𝑋f · 𝑌))
8 mulcl 11135 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
98adantl 482 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10 eqid 2736 . . . . 5 (Base‘𝑍) = (Base‘𝑍)
111, 2, 3, 10, 5dchrf 26590 . . . 4 (𝜑𝑋:(Base‘𝑍)⟶ℂ)
121, 2, 3, 10, 6dchrf 26590 . . . 4 (𝜑𝑌:(Base‘𝑍)⟶ℂ)
13 fvexd 6857 . . . 4 (𝜑 → (Base‘𝑍) ∈ V)
14 inidm 4178 . . . 4 ((Base‘𝑍) ∩ (Base‘𝑍)) = (Base‘𝑍)
159, 11, 12, 13, 13, 14off 7635 . . 3 (𝜑 → (𝑋f · 𝑌):(Base‘𝑍)⟶ℂ)
16 eqid 2736 . . . . . . . 8 (Unit‘𝑍) = (Unit‘𝑍)
1710, 16unitcl 20088 . . . . . . 7 (𝑥 ∈ (Unit‘𝑍) → 𝑥 ∈ (Base‘𝑍))
1810, 16unitcl 20088 . . . . . . 7 (𝑦 ∈ (Unit‘𝑍) → 𝑦 ∈ (Base‘𝑍))
1917, 18anim12i 613 . . . . . 6 ((𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍)) → (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)))
201, 3dchrrcl 26588 . . . . . . . . . . . . . 14 (𝑋𝐷𝑁 ∈ ℕ)
215, 20syl 17 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℕ)
221, 2, 10, 16, 21, 3dchrelbas2 26585 . . . . . . . . . . . 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 495 . . . . . . . . . 10 (𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
25 eqid 2736 . . . . . . . . . . . . 13 (mulGrp‘𝑍) = (mulGrp‘𝑍)
2625, 10mgpbas 19902 . . . . . . . . . . . 12 (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
27 eqid 2736 . . . . . . . . . . . . 13 (.r𝑍) = (.r𝑍)
2825, 27mgpplusg 19900 . . . . . . . . . . . 12 (.r𝑍) = (+g‘(mulGrp‘𝑍))
29 eqid 2736 . . . . . . . . . . . . 13 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
30 cnfldmul 20802 . . . . . . . . . . . . 13 · = (.r‘ℂfld)
3129, 30mgpplusg 19900 . . . . . . . . . . . 12 · = (+g‘(mulGrp‘ℂfld))
3226, 28, 31mhmlin 18609 . . . . . . . . . . 11 ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
33323expb 1120 . . . . . . . . . 10 ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
3424, 33sylan 580 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)))
351, 2, 10, 16, 21, 3dchrelbas2 26585 . . . . . . . . . . . 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 495 . . . . . . . . . 10 (𝜑𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
3826, 28, 31mhmlin 18609 . . . . . . . . . . 11 ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
39383expb 1120 . . . . . . . . . 10 ((𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
4037, 39sylan 580 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌‘(𝑥(.r𝑍)𝑦)) = ((𝑌𝑥) · (𝑌𝑦)))
4134, 40oveq12d 7375 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))) = (((𝑋𝑥) · (𝑋𝑦)) · ((𝑌𝑥) · (𝑌𝑦))))
4211ffvelcdmda 7035 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (𝑋𝑥) ∈ ℂ)
4342adantrr 715 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋𝑥) ∈ ℂ)
44 simpr 485 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → 𝑦 ∈ (Base‘𝑍))
45 ffvelcdm 7032 . . . . . . . . . 10 ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑋𝑦) ∈ ℂ)
4611, 44, 45syl2an 596 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑋𝑦) ∈ ℂ)
4712ffvelcdmda 7035 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (𝑌𝑥) ∈ ℂ)
4847adantrr 715 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌𝑥) ∈ ℂ)
49 ffvelcdm 7032 . . . . . . . . . 10 ((𝑌:(Base‘𝑍)⟶ℂ ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑌𝑦) ∈ ℂ)
5012, 44, 49syl2an 596 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑌𝑦) ∈ ℂ)
5143, 46, 48, 50mul4d 11367 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋𝑥) · (𝑋𝑦)) · ((𝑌𝑥) · (𝑌𝑦))) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
5241, 51eqtrd 2776 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
5311ffnd 6669 . . . . . . . . 9 (𝜑𝑋 Fn (Base‘𝑍))
5453adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑋 Fn (Base‘𝑍))
5512ffnd 6669 . . . . . . . . 9 (𝜑𝑌 Fn (Base‘𝑍))
5655adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑌 Fn (Base‘𝑍))
57 fvexd 6857 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (Base‘𝑍) ∈ V)
5821nnnn0d 12473 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
592zncrng 20951 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑍 ∈ CRing)
60 crngring 19976 . . . . . . . . . 10 (𝑍 ∈ CRing → 𝑍 ∈ Ring)
6158, 59, 603syl 18 . . . . . . . . 9 (𝜑𝑍 ∈ Ring)
6210, 27ringcl 19981 . . . . . . . . . 10 ((𝑍 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍)) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
63623expb 1120 . . . . . . . . 9 ((𝑍 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
6461, 63sylan 580 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (𝑥(.r𝑍)𝑦) ∈ (Base‘𝑍))
65 fnfvof 7634 . . . . . . . 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 837 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋f · 𝑌)‘(𝑥(.r𝑍)𝑦)) = ((𝑋‘(𝑥(.r𝑍)𝑦)) · (𝑌‘(𝑥(.r𝑍)𝑦))))
6753adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑋 Fn (Base‘𝑍))
6855adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑌 Fn (Base‘𝑍))
69 fvexd 6857 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → (Base‘𝑍) ∈ V)
70 simpr 485 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑍)) → 𝑥 ∈ (Base‘𝑍))
71 fnfvof 7634 . . . . . . . . . 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 837 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑍)) → ((𝑋f · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
7372adantrr 715 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋f · 𝑌)‘𝑥) = ((𝑋𝑥) · (𝑌𝑥)))
74 simprr 771 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → 𝑦 ∈ (Base‘𝑍))
75 fnfvof 7634 . . . . . . . . 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 837 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋f · 𝑌)‘𝑦) = ((𝑋𝑦) · (𝑌𝑦)))
7773, 76oveq12d 7375 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → (((𝑋f · 𝑌)‘𝑥) · ((𝑋f · 𝑌)‘𝑦)) = (((𝑋𝑥) · (𝑌𝑥)) · ((𝑋𝑦) · (𝑌𝑦))))
7852, 66, 773eqtr4d 2786 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑍) ∧ 𝑦 ∈ (Base‘𝑍))) → ((𝑋f · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋f · 𝑌)‘𝑥) · ((𝑋f · 𝑌)‘𝑦)))
7919, 78sylan2 593 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Unit‘𝑍) ∧ 𝑦 ∈ (Unit‘𝑍))) → ((𝑋f · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋f · 𝑌)‘𝑥) · ((𝑋f · 𝑌)‘𝑦)))
8079ralrimivva 3197 . . . 4 (𝜑 → ∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)((𝑋f · 𝑌)‘(𝑥(.r𝑍)𝑦)) = (((𝑋f · 𝑌)‘𝑥) · ((𝑋f · 𝑌)‘𝑦)))
81 eqid 2736 . . . . . . . 8 (1r𝑍) = (1r𝑍)
8210, 81ringidcl 19989 . . . . . . 7 (𝑍 ∈ Ring → (1r𝑍) ∈ (Base‘𝑍))
8361, 82syl 17 . . . . . 6 (𝜑 → (1r𝑍) ∈ (Base‘𝑍))
84 fnfvof 7634 . . . . . 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 837 . . . . 5 (𝜑 → ((𝑋f · 𝑌)‘(1r𝑍)) = ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))))
8625, 81ringidval 19915 . . . . . . . . 9 (1r𝑍) = (0g‘(mulGrp‘𝑍))
87 cnfld1 20822 . . . . . . . . . 10 1 = (1r‘ℂfld)
8829, 87ringidval 19915 . . . . . . . . 9 1 = (0g‘(mulGrp‘ℂfld))
8986, 88mhm0 18610 . . . . . . . 8 (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑋‘(1r𝑍)) = 1)
9024, 89syl 17 . . . . . . 7 (𝜑 → (𝑋‘(1r𝑍)) = 1)
9186, 88mhm0 18610 . . . . . . . 8 (𝑌 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑌‘(1r𝑍)) = 1)
9237, 91syl 17 . . . . . . 7 (𝜑 → (𝑌‘(1r𝑍)) = 1)
9390, 92oveq12d 7375 . . . . . 6 (𝜑 → ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))) = (1 · 1))
94 1t1e1 12315 . . . . . 6 (1 · 1) = 1
9593, 94eqtrdi 2792 . . . . 5 (𝜑 → ((𝑋‘(1r𝑍)) · (𝑌‘(1r𝑍))) = 1)
9685, 95eqtrd 2776 . . . 4 (𝜑 → ((𝑋f · 𝑌)‘(1r𝑍)) = 1)
9772neeq1d 3003 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋f · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋𝑥) · (𝑌𝑥)) ≠ 0))
9842, 47mulne0bd 11806 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0) ↔ ((𝑋𝑥) · (𝑌𝑥)) ≠ 0))
9997, 98bitr4d 281 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑍)) → (((𝑋f · 𝑌)‘𝑥) ≠ 0 ↔ ((𝑋𝑥) ≠ 0 ∧ (𝑌𝑥) ≠ 0)))
10023simprd 496 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ (Base‘𝑍)((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
101100r19.21bi 3234 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑍)) → ((𝑋𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
102101adantrd 492 . . . . . 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 3143 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(((𝑋f · 𝑌)‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))
10580, 96, 1043jca 1128 . . 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 26586 . . 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 711 . 2 (𝜑 → (𝑋f · 𝑌) ∈ 𝐷)
1087, 107eqeltrd 2838 1 (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  cc 11049  0cc0 11051  1c1 11052   · cmul 11056  cn 12153  0cn0 12413  Basecbs 17083  +gcplusg 17133  .rcmulr 17134   MndHom cmhm 18599  mulGrpcmgp 19896  1rcur 19913  Ringcrg 19964  CRingccrg 19965  Unitcui 20068  fldccnfld 20796  ℤ/nczn 20903  DChrcdchr 26580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-ec 8650  df-qs 8654  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-0g 17323  df-imas 17390  df-qus 17391  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-nsg 18926  df-eqg 18927  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-subrg 20220  df-lmod 20324  df-lss 20393  df-lsp 20433  df-sra 20633  df-rgmod 20634  df-lidl 20635  df-rsp 20636  df-2idl 20702  df-cnfld 20797  df-zring 20870  df-zn 20907  df-dchr 26581
This theorem is referenced by:  dchrabl  26602  dchrinv  26609
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