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 ) )