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Metamath Proof Explorer

Theorem rngmgpf

Description: Restricted functionality of the multiplicative group on non-unital rings ( mgpf analog). (Contributed by AV, 22-Feb-2025)

Ref Expression
Assertion rngmgpf ( mulGrp ↾ Rng ) : Rng ⟶ Smgrp

Proof

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