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Metamath Proof Explorer

Theorem mgpf

Description: Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015)

Ref Expression
Assertion mgpf ( mulGrp ↾ Ring ) : Ring ⟶ Mnd

Proof

Step Hyp Ref Expression
1 fnmgp ⊢ mulGrp Fn V
2 ssv ⊢ Ring ⊆ V
3 fnssres ⊢ ( ( mulGrp Fn V ∧ Ring ⊆ V ) → ( mulGrp ↾ Ring ) Fn Ring )
4 1 2 3 mp2an ⊢ ( mulGrp ↾ Ring ) Fn Ring
5 fvres ⊢ ( 𝑎 ∈ Ring → ( ( mulGrp ↾ Ring ) ‘ 𝑎 ) = ( mulGrp ‘ 𝑎 ) )
6 eqid ⊢ ( mulGrp ‘ 𝑎 ) = ( mulGrp ‘ 𝑎 )
7 6 ringmgp ⊢ ( 𝑎 ∈ Ring → ( mulGrp ‘ 𝑎 ) ∈ Mnd )
8 5 7 eqeltrd ⊢ ( 𝑎 ∈ Ring → ( ( mulGrp ↾ Ring ) ‘ 𝑎 ) ∈ Mnd )
9 8 rgen ⊢ ∀ 𝑎 ∈ Ring ( ( mulGrp ↾ Ring ) ‘ 𝑎 ) ∈ Mnd
10 ffnfv ⊢ ( ( mulGrp ↾ Ring ) : Ring ⟶ Mnd ↔ ( ( mulGrp ↾ Ring ) Fn Ring ∧ ∀ 𝑎 ∈ Ring ( ( mulGrp ↾ Ring ) ‘ 𝑎 ) ∈ Mnd ) )
11 4 9 10 mpbir2an ⊢ ( mulGrp ↾ Ring ) : Ring ⟶ Mnd