Step |
Hyp |
Ref |
Expression |
1 |
|
ringrng |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Rng ) |
2 |
|
ringrng |
⊢ ( 𝑆 ∈ Ring → 𝑆 ∈ Rng ) |
3 |
1 2
|
anim12i |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Ring ) → ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ) |
4 |
|
mhmismgmhm |
⊢ ( 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MndHom ( mulGrp ‘ 𝑆 ) ) → 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) |
5 |
4
|
anim2i |
⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MndHom ( mulGrp ‘ 𝑆 ) ) ) → ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) ) |
6 |
3 5
|
anim12i |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Ring ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MndHom ( mulGrp ‘ 𝑆 ) ) ) ) → ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) ) ) |
7 |
|
eqid |
⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) |
8 |
|
eqid |
⊢ ( mulGrp ‘ 𝑆 ) = ( mulGrp ‘ 𝑆 ) |
9 |
7 8
|
isrhm |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ↔ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Ring ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MndHom ( mulGrp ‘ 𝑆 ) ) ) ) ) |
10 |
7 8
|
isrnghmmul |
⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) ) ) |
11 |
6 9 10
|
3imtr4i |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ) |