Step |
Hyp |
Ref |
Expression |
1 |
|
0mhm.z |
⊢ 0 = ( 0g ‘ 𝑁 ) |
2 |
|
0mhm.b |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
3 |
|
id |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) → ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝑁 ) = ( Base ‘ 𝑁 ) |
5 |
4 1
|
mndidcl |
⊢ ( 𝑁 ∈ Mnd → 0 ∈ ( Base ‘ 𝑁 ) ) |
6 |
5
|
adantl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) → 0 ∈ ( Base ‘ 𝑁 ) ) |
7 |
|
fconst6g |
⊢ ( 0 ∈ ( Base ‘ 𝑁 ) → ( 𝐵 × { 0 } ) : 𝐵 ⟶ ( Base ‘ 𝑁 ) ) |
8 |
6 7
|
syl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) → ( 𝐵 × { 0 } ) : 𝐵 ⟶ ( Base ‘ 𝑁 ) ) |
9 |
|
simpr |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) → 𝑁 ∈ Mnd ) |
10 |
|
eqid |
⊢ ( +g ‘ 𝑁 ) = ( +g ‘ 𝑁 ) |
11 |
4 10 1
|
mndlid |
⊢ ( ( 𝑁 ∈ Mnd ∧ 0 ∈ ( Base ‘ 𝑁 ) ) → ( 0 ( +g ‘ 𝑁 ) 0 ) = 0 ) |
12 |
11
|
eqcomd |
⊢ ( ( 𝑁 ∈ Mnd ∧ 0 ∈ ( Base ‘ 𝑁 ) ) → 0 = ( 0 ( +g ‘ 𝑁 ) 0 ) ) |
13 |
9 5 12
|
syl2anc2 |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) → 0 = ( 0 ( +g ‘ 𝑁 ) 0 ) ) |
14 |
13
|
adantr |
⊢ ( ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 0 = ( 0 ( +g ‘ 𝑁 ) 0 ) ) |
15 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
16 |
2 15
|
mndcl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) |
17 |
16
|
3expb |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) |
18 |
17
|
adantlr |
⊢ ( ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) |
19 |
1
|
fvexi |
⊢ 0 ∈ V |
20 |
19
|
fvconst2 |
⊢ ( ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 → ( ( 𝐵 × { 0 } ) ‘ ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ) = 0 ) |
21 |
18 20
|
syl |
⊢ ( ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝐵 × { 0 } ) ‘ ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ) = 0 ) |
22 |
19
|
fvconst2 |
⊢ ( 𝑥 ∈ 𝐵 → ( ( 𝐵 × { 0 } ) ‘ 𝑥 ) = 0 ) |
23 |
19
|
fvconst2 |
⊢ ( 𝑦 ∈ 𝐵 → ( ( 𝐵 × { 0 } ) ‘ 𝑦 ) = 0 ) |
24 |
22 23
|
oveqan12d |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( 𝐵 × { 0 } ) ‘ 𝑥 ) ( +g ‘ 𝑁 ) ( ( 𝐵 × { 0 } ) ‘ 𝑦 ) ) = ( 0 ( +g ‘ 𝑁 ) 0 ) ) |
25 |
24
|
adantl |
⊢ ( ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( ( 𝐵 × { 0 } ) ‘ 𝑥 ) ( +g ‘ 𝑁 ) ( ( 𝐵 × { 0 } ) ‘ 𝑦 ) ) = ( 0 ( +g ‘ 𝑁 ) 0 ) ) |
26 |
14 21 25
|
3eqtr4d |
⊢ ( ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝐵 × { 0 } ) ‘ ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ) = ( ( ( 𝐵 × { 0 } ) ‘ 𝑥 ) ( +g ‘ 𝑁 ) ( ( 𝐵 × { 0 } ) ‘ 𝑦 ) ) ) |
27 |
26
|
ralrimivva |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝐵 × { 0 } ) ‘ ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ) = ( ( ( 𝐵 × { 0 } ) ‘ 𝑥 ) ( +g ‘ 𝑁 ) ( ( 𝐵 × { 0 } ) ‘ 𝑦 ) ) ) |
28 |
|
eqid |
⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 ) |
29 |
2 28
|
mndidcl |
⊢ ( 𝑀 ∈ Mnd → ( 0g ‘ 𝑀 ) ∈ 𝐵 ) |
30 |
29
|
adantr |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) → ( 0g ‘ 𝑀 ) ∈ 𝐵 ) |
31 |
19
|
fvconst2 |
⊢ ( ( 0g ‘ 𝑀 ) ∈ 𝐵 → ( ( 𝐵 × { 0 } ) ‘ ( 0g ‘ 𝑀 ) ) = 0 ) |
32 |
30 31
|
syl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) → ( ( 𝐵 × { 0 } ) ‘ ( 0g ‘ 𝑀 ) ) = 0 ) |
33 |
8 27 32
|
3jca |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) → ( ( 𝐵 × { 0 } ) : 𝐵 ⟶ ( Base ‘ 𝑁 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝐵 × { 0 } ) ‘ ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ) = ( ( ( 𝐵 × { 0 } ) ‘ 𝑥 ) ( +g ‘ 𝑁 ) ( ( 𝐵 × { 0 } ) ‘ 𝑦 ) ) ∧ ( ( 𝐵 × { 0 } ) ‘ ( 0g ‘ 𝑀 ) ) = 0 ) ) |
34 |
2 4 15 10 28 1
|
ismhm |
⊢ ( ( 𝐵 × { 0 } ) ∈ ( 𝑀 MndHom 𝑁 ) ↔ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) ∧ ( ( 𝐵 × { 0 } ) : 𝐵 ⟶ ( Base ‘ 𝑁 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝐵 × { 0 } ) ‘ ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ) = ( ( ( 𝐵 × { 0 } ) ‘ 𝑥 ) ( +g ‘ 𝑁 ) ( ( 𝐵 × { 0 } ) ‘ 𝑦 ) ) ∧ ( ( 𝐵 × { 0 } ) ‘ ( 0g ‘ 𝑀 ) ) = 0 ) ) ) |
35 |
3 33 34
|
sylanbrc |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) → ( 𝐵 × { 0 } ) ∈ ( 𝑀 MndHom 𝑁 ) ) |