Step |
Hyp |
Ref |
Expression |
1 |
|
mulgmhm.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
mulgmhm.m |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
cmnmnd |
⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd ) |
4 |
3
|
adantr |
⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) → 𝐺 ∈ Mnd ) |
5 |
1 2
|
mulgnn0cl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑀 ∈ ℕ0 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑀 · 𝑥 ) ∈ 𝐵 ) |
6 |
3 5
|
syl3an1 |
⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑀 · 𝑥 ) ∈ 𝐵 ) |
7 |
6
|
3expa |
⊢ ( ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝑀 · 𝑥 ) ∈ 𝐵 ) |
8 |
7
|
fmpttd |
⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) : 𝐵 ⟶ 𝐵 ) |
9 |
|
3anass |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ↔ ( 𝑀 ∈ ℕ0 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ) |
10 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
11 |
1 2 10
|
mulgnn0di |
⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( 𝑀 · 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑀 · 𝑧 ) ) ) |
12 |
9 11
|
sylan2br |
⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑀 ∈ ℕ0 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ) → ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( 𝑀 · 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑀 · 𝑧 ) ) ) |
13 |
12
|
anassrs |
⊢ ( ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( 𝑀 · 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑀 · 𝑧 ) ) ) |
14 |
1 10
|
mndcl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝐵 ) |
15 |
14
|
3expb |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝐵 ) |
16 |
4 15
|
sylan |
⊢ ( ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝐵 ) |
17 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) → ( 𝑀 · 𝑥 ) = ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
18 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) = ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) |
19 |
|
ovex |
⊢ ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ V |
20 |
17 18 19
|
fvmpt |
⊢ ( ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝐵 → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
21 |
16 20
|
syl |
⊢ ( ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
22 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑀 · 𝑥 ) = ( 𝑀 · 𝑦 ) ) |
23 |
|
ovex |
⊢ ( 𝑀 · 𝑦 ) ∈ V |
24 |
22 18 23
|
fvmpt |
⊢ ( 𝑦 ∈ 𝐵 → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑦 ) = ( 𝑀 · 𝑦 ) ) |
25 |
|
oveq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝑀 · 𝑥 ) = ( 𝑀 · 𝑧 ) ) |
26 |
|
ovex |
⊢ ( 𝑀 · 𝑧 ) ∈ V |
27 |
25 18 26
|
fvmpt |
⊢ ( 𝑧 ∈ 𝐵 → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑧 ) = ( 𝑀 · 𝑧 ) ) |
28 |
24 27
|
oveqan12d |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑧 ) ) = ( ( 𝑀 · 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑀 · 𝑧 ) ) ) |
29 |
28
|
adantl |
⊢ ( ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑧 ) ) = ( ( 𝑀 · 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑀 · 𝑧 ) ) ) |
30 |
13 21 29
|
3eqtr4d |
⊢ ( ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑧 ) ) ) |
31 |
30
|
ralrimivva |
⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) → ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑧 ) ) ) |
32 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
33 |
1 32
|
mndidcl |
⊢ ( 𝐺 ∈ Mnd → ( 0g ‘ 𝐺 ) ∈ 𝐵 ) |
34 |
|
oveq2 |
⊢ ( 𝑥 = ( 0g ‘ 𝐺 ) → ( 𝑀 · 𝑥 ) = ( 𝑀 · ( 0g ‘ 𝐺 ) ) ) |
35 |
|
ovex |
⊢ ( 𝑀 · ( 0g ‘ 𝐺 ) ) ∈ V |
36 |
34 18 35
|
fvmpt |
⊢ ( ( 0g ‘ 𝐺 ) ∈ 𝐵 → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 0g ‘ 𝐺 ) ) = ( 𝑀 · ( 0g ‘ 𝐺 ) ) ) |
37 |
4 33 36
|
3syl |
⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 0g ‘ 𝐺 ) ) = ( 𝑀 · ( 0g ‘ 𝐺 ) ) ) |
38 |
1 2 32
|
mulgnn0z |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑀 ∈ ℕ0 ) → ( 𝑀 · ( 0g ‘ 𝐺 ) ) = ( 0g ‘ 𝐺 ) ) |
39 |
3 38
|
sylan |
⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) → ( 𝑀 · ( 0g ‘ 𝐺 ) ) = ( 0g ‘ 𝐺 ) ) |
40 |
37 39
|
eqtrd |
⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 0g ‘ 𝐺 ) ) = ( 0g ‘ 𝐺 ) ) |
41 |
8 31 40
|
3jca |
⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) : 𝐵 ⟶ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑧 ) ) ∧ ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 0g ‘ 𝐺 ) ) = ( 0g ‘ 𝐺 ) ) ) |
42 |
1 1 10 10 32 32
|
ismhm |
⊢ ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ∈ ( 𝐺 MndHom 𝐺 ) ↔ ( ( 𝐺 ∈ Mnd ∧ 𝐺 ∈ Mnd ) ∧ ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) : 𝐵 ⟶ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑧 ) ) ∧ ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 0g ‘ 𝐺 ) ) = ( 0g ‘ 𝐺 ) ) ) ) |
43 |
4 4 41 42
|
syl21anbrc |
⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0 ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ∈ ( 𝐺 MndHom 𝐺 ) ) |