Step |
Hyp |
Ref |
Expression |
1 |
|
ghmpropd.a |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐽 ) ) |
2 |
|
ghmpropd.b |
⊢ ( 𝜑 → 𝐶 = ( Base ‘ 𝐾 ) ) |
3 |
|
ghmpropd.c |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) ) |
4 |
|
ghmpropd.d |
⊢ ( 𝜑 → 𝐶 = ( Base ‘ 𝑀 ) ) |
5 |
|
ghmpropd.e |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐽 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ) |
6 |
|
ghmpropd.f |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ) |
7 |
1 3 5
|
grppropd |
⊢ ( 𝜑 → ( 𝐽 ∈ Grp ↔ 𝐿 ∈ Grp ) ) |
8 |
2 4 6
|
grppropd |
⊢ ( 𝜑 → ( 𝐾 ∈ Grp ↔ 𝑀 ∈ Grp ) ) |
9 |
7 8
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝐽 ∈ Grp ∧ 𝐾 ∈ Grp ) ↔ ( 𝐿 ∈ Grp ∧ 𝑀 ∈ Grp ) ) ) |
10 |
1 2 3 4 5 6
|
mhmpropd |
⊢ ( 𝜑 → ( 𝐽 MndHom 𝐾 ) = ( 𝐿 MndHom 𝑀 ) ) |
11 |
10
|
eleq2d |
⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝐽 MndHom 𝐾 ) ↔ 𝑓 ∈ ( 𝐿 MndHom 𝑀 ) ) ) |
12 |
9 11
|
anbi12d |
⊢ ( 𝜑 → ( ( ( 𝐽 ∈ Grp ∧ 𝐾 ∈ Grp ) ∧ 𝑓 ∈ ( 𝐽 MndHom 𝐾 ) ) ↔ ( ( 𝐿 ∈ Grp ∧ 𝑀 ∈ Grp ) ∧ 𝑓 ∈ ( 𝐿 MndHom 𝑀 ) ) ) ) |
13 |
|
ghmgrp1 |
⊢ ( 𝑓 ∈ ( 𝐽 GrpHom 𝐾 ) → 𝐽 ∈ Grp ) |
14 |
|
ghmgrp2 |
⊢ ( 𝑓 ∈ ( 𝐽 GrpHom 𝐾 ) → 𝐾 ∈ Grp ) |
15 |
13 14
|
jca |
⊢ ( 𝑓 ∈ ( 𝐽 GrpHom 𝐾 ) → ( 𝐽 ∈ Grp ∧ 𝐾 ∈ Grp ) ) |
16 |
|
ghmmhmb |
⊢ ( ( 𝐽 ∈ Grp ∧ 𝐾 ∈ Grp ) → ( 𝐽 GrpHom 𝐾 ) = ( 𝐽 MndHom 𝐾 ) ) |
17 |
16
|
eleq2d |
⊢ ( ( 𝐽 ∈ Grp ∧ 𝐾 ∈ Grp ) → ( 𝑓 ∈ ( 𝐽 GrpHom 𝐾 ) ↔ 𝑓 ∈ ( 𝐽 MndHom 𝐾 ) ) ) |
18 |
15 17
|
biadanii |
⊢ ( 𝑓 ∈ ( 𝐽 GrpHom 𝐾 ) ↔ ( ( 𝐽 ∈ Grp ∧ 𝐾 ∈ Grp ) ∧ 𝑓 ∈ ( 𝐽 MndHom 𝐾 ) ) ) |
19 |
|
ghmgrp1 |
⊢ ( 𝑓 ∈ ( 𝐿 GrpHom 𝑀 ) → 𝐿 ∈ Grp ) |
20 |
|
ghmgrp2 |
⊢ ( 𝑓 ∈ ( 𝐿 GrpHom 𝑀 ) → 𝑀 ∈ Grp ) |
21 |
19 20
|
jca |
⊢ ( 𝑓 ∈ ( 𝐿 GrpHom 𝑀 ) → ( 𝐿 ∈ Grp ∧ 𝑀 ∈ Grp ) ) |
22 |
|
ghmmhmb |
⊢ ( ( 𝐿 ∈ Grp ∧ 𝑀 ∈ Grp ) → ( 𝐿 GrpHom 𝑀 ) = ( 𝐿 MndHom 𝑀 ) ) |
23 |
22
|
eleq2d |
⊢ ( ( 𝐿 ∈ Grp ∧ 𝑀 ∈ Grp ) → ( 𝑓 ∈ ( 𝐿 GrpHom 𝑀 ) ↔ 𝑓 ∈ ( 𝐿 MndHom 𝑀 ) ) ) |
24 |
21 23
|
biadanii |
⊢ ( 𝑓 ∈ ( 𝐿 GrpHom 𝑀 ) ↔ ( ( 𝐿 ∈ Grp ∧ 𝑀 ∈ Grp ) ∧ 𝑓 ∈ ( 𝐿 MndHom 𝑀 ) ) ) |
25 |
12 18 24
|
3bitr4g |
⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝐽 GrpHom 𝐾 ) ↔ 𝑓 ∈ ( 𝐿 GrpHom 𝑀 ) ) ) |
26 |
25
|
eqrdv |
⊢ ( 𝜑 → ( 𝐽 GrpHom 𝐾 ) = ( 𝐿 GrpHom 𝑀 ) ) |