Step |
Hyp |
Ref |
Expression |
1 |
|
isrnghmmul.m |
⊢ 𝑀 = ( mulGrp ‘ 𝑅 ) |
2 |
|
isrnghmmul.n |
⊢ 𝑁 = ( mulGrp ‘ 𝑆 ) |
3 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
4 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
5 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
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 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
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 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( ( 𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm ) ∧ 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
18 |
|
anass |
⊢ ( ( ( ( 𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm ) ∧ 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ↔ ( ( 𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm ) ∧ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
19 |
17 18
|
bitrdi |
⊢ ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( 𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm ) ∧ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) ) |
20 |
1 3
|
mgpbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑀 ) |
21 |
2 14
|
mgpbas |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑁 ) |
22 |
1 4
|
mgpplusg |
⊢ ( .r ‘ 𝑅 ) = ( +g ‘ 𝑀 ) |
23 |
2 5
|
mgpplusg |
⊢ ( .r ‘ 𝑆 ) = ( +g ‘ 𝑁 ) |
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 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
25 |
19 24
|
bitr4di |
⊢ ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ↔ 𝐹 ∈ ( 𝑀 MgmHom 𝑁 ) ) ) |
26 |
25
|
pm5.32da |
⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ↔ ( 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 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ↔ ( ( 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 ) ) ) ) |