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Metamath Proof Explorer

Theorem idrhm

Description: The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020)

Ref Expression
Hypothesis idrhm.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
Assertion idrhm ( 𝑅 ∈ Ring → ( I ↾ 𝐵 ) ∈ ( 𝑅 RingHom 𝑅 ) )

Proof

Step Hyp Ref Expression
1 idrhm.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 id ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Ring )
3 ringgrp ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )
4 1 idghm ⊢ ( 𝑅 ∈ Grp → ( I ↾ 𝐵 ) ∈ ( 𝑅 GrpHom 𝑅 ) )
5 3 4 syl ⊢ ( 𝑅 ∈ Ring → ( I ↾ 𝐵 ) ∈ ( 𝑅 GrpHom 𝑅 ) )
6 eqid ⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
7 6 ringmgp ⊢ ( 𝑅 ∈ Ring → ( mulGrp ‘ 𝑅 ) ∈ Mnd )
8 6 1 mgpbas ⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
9 8 idmhm ⊢ ( ( mulGrp ‘ 𝑅 ) ∈ Mnd → ( I ↾ 𝐵 ) ∈ ( ( mulGrp ‘ 𝑅 ) MndHom ( mulGrp ‘ 𝑅 ) ) )
10 7 9 syl ⊢ ( 𝑅 ∈ Ring → ( I ↾ 𝐵 ) ∈ ( ( mulGrp ‘ 𝑅 ) MndHom ( mulGrp ‘ 𝑅 ) ) )
11 5 10 jca ⊢ ( 𝑅 ∈ Ring → ( ( I ↾ 𝐵 ) ∈ ( 𝑅 GrpHom 𝑅 ) ∧ ( I ↾ 𝐵 ) ∈ ( ( mulGrp ‘ 𝑅 ) MndHom ( mulGrp ‘ 𝑅 ) ) ) )
12 6 6 isrhm ⊢ ( ( I ↾ 𝐵 ) ∈ ( 𝑅 RingHom 𝑅 ) ↔ ( ( 𝑅 ∈ Ring ∧ 𝑅 ∈ Ring ) ∧ ( ( I ↾ 𝐵 ) ∈ ( 𝑅 GrpHom 𝑅 ) ∧ ( I ↾ 𝐵 ) ∈ ( ( mulGrp ‘ 𝑅 ) MndHom ( mulGrp ‘ 𝑅 ) ) ) ) )
13 2 2 11 12 syl21anbrc ⊢ ( 𝑅 ∈ Ring → ( I ↾ 𝐵 ) ∈ ( 𝑅 RingHom 𝑅 ) )