Step |
Hyp |
Ref |
Expression |
1 |
|
rngkerinj.1 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
2 |
|
rngkerinj.2 |
⊢ 𝑋 = ran 𝐺 |
3 |
|
rngkerinj.3 |
⊢ 𝑊 = ( GId ‘ 𝐺 ) |
4 |
|
rngkerinj.4 |
⊢ 𝐽 = ( 1st ‘ 𝑆 ) |
5 |
|
rngkerinj.5 |
⊢ 𝑌 = ran 𝐽 |
6 |
|
rngkerinj.6 |
⊢ 𝑍 = ( GId ‘ 𝐽 ) |
7 |
|
eqid |
⊢ ( 1st ‘ 𝑅 ) = ( 1st ‘ 𝑅 ) |
8 |
7
|
rngogrpo |
⊢ ( 𝑅 ∈ RingOps → ( 1st ‘ 𝑅 ) ∈ GrpOp ) |
9 |
8
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 1st ‘ 𝑅 ) ∈ GrpOp ) |
10 |
|
eqid |
⊢ ( 1st ‘ 𝑆 ) = ( 1st ‘ 𝑆 ) |
11 |
10
|
rngogrpo |
⊢ ( 𝑆 ∈ RingOps → ( 1st ‘ 𝑆 ) ∈ GrpOp ) |
12 |
11
|
3ad2ant2 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 1st ‘ 𝑆 ) ∈ GrpOp ) |
13 |
7 10
|
rngogrphom |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → 𝐹 ∈ ( ( 1st ‘ 𝑅 ) GrpOpHom ( 1st ‘ 𝑆 ) ) ) |
14 |
1
|
rneqi |
⊢ ran 𝐺 = ran ( 1st ‘ 𝑅 ) |
15 |
2 14
|
eqtri |
⊢ 𝑋 = ran ( 1st ‘ 𝑅 ) |
16 |
1
|
fveq2i |
⊢ ( GId ‘ 𝐺 ) = ( GId ‘ ( 1st ‘ 𝑅 ) ) |
17 |
3 16
|
eqtri |
⊢ 𝑊 = ( GId ‘ ( 1st ‘ 𝑅 ) ) |
18 |
4
|
rneqi |
⊢ ran 𝐽 = ran ( 1st ‘ 𝑆 ) |
19 |
5 18
|
eqtri |
⊢ 𝑌 = ran ( 1st ‘ 𝑆 ) |
20 |
4
|
fveq2i |
⊢ ( GId ‘ 𝐽 ) = ( GId ‘ ( 1st ‘ 𝑆 ) ) |
21 |
6 20
|
eqtri |
⊢ 𝑍 = ( GId ‘ ( 1st ‘ 𝑆 ) ) |
22 |
15 17 19 21
|
grpokerinj |
⊢ ( ( ( 1st ‘ 𝑅 ) ∈ GrpOp ∧ ( 1st ‘ 𝑆 ) ∈ GrpOp ∧ 𝐹 ∈ ( ( 1st ‘ 𝑅 ) GrpOpHom ( 1st ‘ 𝑆 ) ) ) → ( 𝐹 : 𝑋 –1-1→ 𝑌 ↔ ( ◡ 𝐹 “ { 𝑍 } ) = { 𝑊 } ) ) |
23 |
9 12 13 22
|
syl3anc |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐹 : 𝑋 –1-1→ 𝑌 ↔ ( ◡ 𝐹 “ { 𝑍 } ) = { 𝑊 } ) ) |