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Metamath Proof Explorer

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 ) )