Metamath Proof Explorer


Theorem isrnghmmul

Description: A function is a non-unital ring homomorphism iff it preserves both addition and multiplication. (Contributed by AV, 27-Feb-2020)

Ref Expression
Hypotheses isrnghmmul.m 𝑀 = ( mulGrp ‘ 𝑅 )
isrnghmmul.n 𝑁 = ( mulGrp ‘ 𝑆 )
Assertion isrnghmmul ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( 𝑀 MgmHom 𝑁 ) ) ) )

Proof

Step Hyp Ref Expression
1 isrnghmmul.m 𝑀 = ( mulGrp ‘ 𝑅 )
2 isrnghmmul.n 𝑁 = ( mulGrp ‘ 𝑆 )
3 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
4 eqid ( .r𝑅 ) = ( .r𝑅 )
5 eqid ( .r𝑆 ) = ( .r𝑆 )
6 3 4 5 isrnghm ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( .r𝑆 ) ( 𝐹𝑦 ) ) ) ) )
7 1 rngmgp ( 𝑅 ∈ Rng → 𝑀 ∈ Smgrp )
8 sgrpmgm ( 𝑀 ∈ Smgrp → 𝑀 ∈ Mgm )
9 7 8 syl ( 𝑅 ∈ Rng → 𝑀 ∈ Mgm )
10 2 rngmgp ( 𝑆 ∈ Rng → 𝑁 ∈ Smgrp )
11 sgrpmgm ( 𝑁 ∈ Smgrp → 𝑁 ∈ Mgm )
12 10 11 syl ( 𝑆 ∈ Rng → 𝑁 ∈ Mgm )
13 9 12 anim12i ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( 𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm ) )
14 eqid ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
15 3 14 ghmf ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) → 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) )
16 13 15 anim12i ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) → ( ( 𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm ) ∧ 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ) )
17 16 biantrurd ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( .r𝑆 ) ( 𝐹𝑦 ) ) ↔ ( ( ( 𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm ) ∧ 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( .r𝑆 ) ( 𝐹𝑦 ) ) ) ) )
18 anass ( ( ( ( 𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm ) ∧ 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( .r𝑆 ) ( 𝐹𝑦 ) ) ) ↔ ( ( 𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm ) ∧ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( .r𝑆 ) ( 𝐹𝑦 ) ) ) ) )
19 17 18 bitrdi ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( .r𝑆 ) ( 𝐹𝑦 ) ) ↔ ( ( 𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm ) ∧ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( .r𝑆 ) ( 𝐹𝑦 ) ) ) ) ) )
20 1 3 mgpbas ( Base ‘ 𝑅 ) = ( Base ‘ 𝑀 )
21 2 14 mgpbas ( Base ‘ 𝑆 ) = ( Base ‘ 𝑁 )
22 1 4 mgpplusg ( .r𝑅 ) = ( +g𝑀 )
23 2 5 mgpplusg ( .r𝑆 ) = ( +g𝑁 )
24 20 21 22 23 ismgmhm ( 𝐹 ∈ ( 𝑀 MgmHom 𝑁 ) ↔ ( ( 𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm ) ∧ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( .r𝑆 ) ( 𝐹𝑦 ) ) ) ) )
25 19 24 bitr4di ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( .r𝑆 ) ( 𝐹𝑦 ) ) ↔ 𝐹 ∈ ( 𝑀 MgmHom 𝑁 ) ) )
26 25 pm5.32da ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( .r𝑆 ) ( 𝐹𝑦 ) ) ) ↔ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( 𝑀 MgmHom 𝑁 ) ) ) )
27 26 pm5.32i ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( .r𝑆 ) ( 𝐹𝑦 ) ) ) ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( 𝑀 MgmHom 𝑁 ) ) ) )
28 6 27 bitri ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( 𝑀 MgmHom 𝑁 ) ) ) )