Theorem idrnghm

Description: The identity homomorphism on a non-unital ring. (Contributed by AV, 27-Feb-2020)

	Ref	Expression
Hypothesis	idrnghm.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
Assertion	idrnghm	⊢ ( 𝑅 ∈ Rng → ( I ↾ 𝐵 ) ∈ ( 𝑅 RngHom 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		idrnghm.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		id	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Rng )
3	2 2	jca	⊢ ( 𝑅 ∈ Rng → ( 𝑅 ∈ Rng ∧ 𝑅 ∈ Rng ) )
4		rngabl	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Abel )
5		ablgrp	⊢ ( 𝑅 ∈ Abel → 𝑅 ∈ Grp )
6	1	idghm	⊢ ( 𝑅 ∈ Grp → ( I ↾ 𝐵 ) ∈ ( 𝑅 GrpHom 𝑅 ) )
7	4 5 6	3syl	⊢ ( 𝑅 ∈ Rng → ( I ↾ 𝐵 ) ∈ ( 𝑅 GrpHom 𝑅 ) )
8		eqid	⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
9	8	rngmgp	⊢ ( 𝑅 ∈ Rng → ( mulGrp ‘ 𝑅 ) ∈ Smgrp )
10		sgrpmgm	⊢ ( ( mulGrp ‘ 𝑅 ) ∈ Smgrp → ( mulGrp ‘ 𝑅 ) ∈ Mgm )
11	8 1	mgpbas	⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
12	11	idmgmhm	⊢ ( ( mulGrp ‘ 𝑅 ) ∈ Mgm → ( I ↾ 𝐵 ) ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑅 ) ) )
13	9 10 12	3syl	⊢ ( 𝑅 ∈ Rng → ( I ↾ 𝐵 ) ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑅 ) ) )
14	7 13	jca	⊢ ( 𝑅 ∈ Rng → ( ( I ↾ 𝐵 ) ∈ ( 𝑅 GrpHom 𝑅 ) ∧ ( I ↾ 𝐵 ) ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑅 ) ) ) )
15	8 8	isrnghmmul	⊢ ( ( I ↾ 𝐵 ) ∈ ( 𝑅 RngHom 𝑅 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑅 ∈ Rng ) ∧ ( ( I ↾ 𝐵 ) ∈ ( 𝑅 GrpHom 𝑅 ) ∧ ( I ↾ 𝐵 ) ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑅 ) ) ) ) )
16	3 14 15	sylanbrc	⊢ ( 𝑅 ∈ Rng → ( I ↾ 𝐵 ) ∈ ( 𝑅 RngHom 𝑅 ) )