Step |
Hyp |
Ref |
Expression |
1 |
|
rnghmf1o.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
rnghmf1o.c |
⊢ 𝐶 = ( Base ‘ 𝑆 ) |
3 |
|
rnghmrcl |
⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ) |
4 |
3
|
ancomd |
⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → ( 𝑆 ∈ Rng ∧ 𝑅 ∈ Rng ) ) |
5 |
4
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → ( 𝑆 ∈ Rng ∧ 𝑅 ∈ Rng ) ) |
6 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) |
7 |
|
rnghmghm |
⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) |
8 |
7
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) |
9 |
1 2
|
ghmf1o |
⊢ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) → ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ↔ ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑅 ) ) ) |
10 |
9
|
bicomd |
⊢ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) → ( ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑅 ) ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) |
11 |
8 10
|
syl |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → ( ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑅 ) ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) |
12 |
6 11
|
mpbird |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑅 ) ) |
13 |
|
eqidd |
⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → 𝐹 = 𝐹 ) |
14 |
|
eqid |
⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) |
15 |
14 1
|
mgpbas |
⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) ) |
16 |
15
|
a1i |
⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) ) ) |
17 |
|
eqid |
⊢ ( mulGrp ‘ 𝑆 ) = ( mulGrp ‘ 𝑆 ) |
18 |
17 2
|
mgpbas |
⊢ 𝐶 = ( Base ‘ ( mulGrp ‘ 𝑆 ) ) |
19 |
18
|
a1i |
⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → 𝐶 = ( Base ‘ ( mulGrp ‘ 𝑆 ) ) ) |
20 |
13 16 19
|
f1oeq123d |
⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ↔ 𝐹 : ( Base ‘ ( mulGrp ‘ 𝑅 ) ) –1-1-onto→ ( Base ‘ ( mulGrp ‘ 𝑆 ) ) ) ) |
21 |
20
|
biimpa |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → 𝐹 : ( Base ‘ ( mulGrp ‘ 𝑅 ) ) –1-1-onto→ ( Base ‘ ( mulGrp ‘ 𝑆 ) ) ) |
22 |
14 17
|
rnghmmgmhm |
⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) |
23 |
22
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) |
24 |
|
eqid |
⊢ ( Base ‘ ( mulGrp ‘ 𝑅 ) ) = ( Base ‘ ( mulGrp ‘ 𝑅 ) ) |
25 |
|
eqid |
⊢ ( Base ‘ ( mulGrp ‘ 𝑆 ) ) = ( Base ‘ ( mulGrp ‘ 𝑆 ) ) |
26 |
24 25
|
mgmhmf1o |
⊢ ( 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) → ( 𝐹 : ( Base ‘ ( mulGrp ‘ 𝑅 ) ) –1-1-onto→ ( Base ‘ ( mulGrp ‘ 𝑆 ) ) ↔ ◡ 𝐹 ∈ ( ( mulGrp ‘ 𝑆 ) MgmHom ( mulGrp ‘ 𝑅 ) ) ) ) |
27 |
26
|
bicomd |
⊢ ( 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) → ( ◡ 𝐹 ∈ ( ( mulGrp ‘ 𝑆 ) MgmHom ( mulGrp ‘ 𝑅 ) ) ↔ 𝐹 : ( Base ‘ ( mulGrp ‘ 𝑅 ) ) –1-1-onto→ ( Base ‘ ( mulGrp ‘ 𝑆 ) ) ) ) |
28 |
23 27
|
syl |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → ( ◡ 𝐹 ∈ ( ( mulGrp ‘ 𝑆 ) MgmHom ( mulGrp ‘ 𝑅 ) ) ↔ 𝐹 : ( Base ‘ ( mulGrp ‘ 𝑅 ) ) –1-1-onto→ ( Base ‘ ( mulGrp ‘ 𝑆 ) ) ) ) |
29 |
21 28
|
mpbird |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → ◡ 𝐹 ∈ ( ( mulGrp ‘ 𝑆 ) MgmHom ( mulGrp ‘ 𝑅 ) ) ) |
30 |
12 29
|
jca |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → ( ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑅 ) ∧ ◡ 𝐹 ∈ ( ( mulGrp ‘ 𝑆 ) MgmHom ( mulGrp ‘ 𝑅 ) ) ) ) |
31 |
17 14
|
isrnghmmul |
⊢ ( ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ↔ ( ( 𝑆 ∈ Rng ∧ 𝑅 ∈ Rng ) ∧ ( ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑅 ) ∧ ◡ 𝐹 ∈ ( ( mulGrp ‘ 𝑆 ) MgmHom ( mulGrp ‘ 𝑅 ) ) ) ) ) |
32 |
5 30 31
|
sylanbrc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ) |
33 |
1 2
|
rnghmf |
⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → 𝐹 : 𝐵 ⟶ 𝐶 ) |
34 |
33
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ) → 𝐹 : 𝐵 ⟶ 𝐶 ) |
35 |
34
|
ffnd |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ) → 𝐹 Fn 𝐵 ) |
36 |
2 1
|
rnghmf |
⊢ ( ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) → ◡ 𝐹 : 𝐶 ⟶ 𝐵 ) |
37 |
36
|
adantl |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ) → ◡ 𝐹 : 𝐶 ⟶ 𝐵 ) |
38 |
37
|
ffnd |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ) → ◡ 𝐹 Fn 𝐶 ) |
39 |
|
dff1o4 |
⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ↔ ( 𝐹 Fn 𝐵 ∧ ◡ 𝐹 Fn 𝐶 ) ) |
40 |
35 38 39
|
sylanbrc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) |
41 |
32 40
|
impbida |
⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ↔ ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ) ) |