Step |
Hyp |
Ref |
Expression |
1 |
|
rnghomsub.1 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
2 |
|
rnghomsub.2 |
⊢ 𝑋 = ran 𝐺 |
3 |
|
rnghomsub.3 |
⊢ 𝐻 = ( /𝑔 ‘ 𝐺 ) |
4 |
|
rnghomsub.4 |
⊢ 𝐽 = ( 1st ‘ 𝑆 ) |
5 |
|
rnghomsub.5 |
⊢ 𝐾 = ( /𝑔 ‘ 𝐽 ) |
6 |
1
|
rngogrpo |
⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp ) |
7 |
6
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → 𝐺 ∈ GrpOp ) |
8 |
4
|
rngogrpo |
⊢ ( 𝑆 ∈ RingOps → 𝐽 ∈ GrpOp ) |
9 |
8
|
3ad2ant2 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → 𝐽 ∈ GrpOp ) |
10 |
1 4
|
rngogrphom |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → 𝐹 ∈ ( 𝐺 GrpOpHom 𝐽 ) ) |
11 |
7 9 10
|
3jca |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ ( 𝐺 GrpOpHom 𝐽 ) ) ) |
12 |
2 3 5
|
ghomdiv |
⊢ ( ( ( 𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ ( 𝐺 GrpOpHom 𝐽 ) ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐹 ‘ ( 𝐴 𝐻 𝐵 ) ) = ( ( 𝐹 ‘ 𝐴 ) 𝐾 ( 𝐹 ‘ 𝐵 ) ) ) |
13 |
11 12
|
sylan |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐹 ‘ ( 𝐴 𝐻 𝐵 ) ) = ( ( 𝐹 ‘ 𝐴 ) 𝐾 ( 𝐹 ‘ 𝐵 ) ) ) |