Step |
Hyp |
Ref |
Expression |
1 |
|
subgmulgcld.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
subgmulgcld.x |
⊢ · = ( .g ‘ 𝑅 ) |
3 |
|
subgmulgcld.r |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
4 |
|
subgmulgcld.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑆 ) |
5 |
|
subgmulgcld.s |
⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝑅 ) ) |
6 |
|
subgmulgcld.z |
⊢ ( 𝜑 → 𝑍 ∈ ℤ ) |
7 |
|
eqid |
⊢ ( Base ‘ ( 𝑅 ↾s 𝑆 ) ) = ( Base ‘ ( 𝑅 ↾s 𝑆 ) ) |
8 |
|
eqid |
⊢ ( .g ‘ ( 𝑅 ↾s 𝑆 ) ) = ( .g ‘ ( 𝑅 ↾s 𝑆 ) ) |
9 |
|
eqid |
⊢ ( 𝑅 ↾s 𝑆 ) = ( 𝑅 ↾s 𝑆 ) |
10 |
9
|
subggrp |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝑅 ) → ( 𝑅 ↾s 𝑆 ) ∈ Grp ) |
11 |
5 10
|
syl |
⊢ ( 𝜑 → ( 𝑅 ↾s 𝑆 ) ∈ Grp ) |
12 |
1
|
subgss |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝑅 ) → 𝑆 ⊆ 𝐵 ) |
13 |
9 1
|
ressbas2 |
⊢ ( 𝑆 ⊆ 𝐵 → 𝑆 = ( Base ‘ ( 𝑅 ↾s 𝑆 ) ) ) |
14 |
5 12 13
|
3syl |
⊢ ( 𝜑 → 𝑆 = ( Base ‘ ( 𝑅 ↾s 𝑆 ) ) ) |
15 |
4 14
|
eleqtrd |
⊢ ( 𝜑 → 𝐴 ∈ ( Base ‘ ( 𝑅 ↾s 𝑆 ) ) ) |
16 |
7 8 11 6 15
|
mulgcld |
⊢ ( 𝜑 → ( 𝑍 ( .g ‘ ( 𝑅 ↾s 𝑆 ) ) 𝐴 ) ∈ ( Base ‘ ( 𝑅 ↾s 𝑆 ) ) ) |
17 |
2 9 8
|
subgmulg |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝑅 ) ∧ 𝑍 ∈ ℤ ∧ 𝐴 ∈ 𝑆 ) → ( 𝑍 · 𝐴 ) = ( 𝑍 ( .g ‘ ( 𝑅 ↾s 𝑆 ) ) 𝐴 ) ) |
18 |
5 6 4 17
|
syl3anc |
⊢ ( 𝜑 → ( 𝑍 · 𝐴 ) = ( 𝑍 ( .g ‘ ( 𝑅 ↾s 𝑆 ) ) 𝐴 ) ) |
19 |
16 18 14
|
3eltr4d |
⊢ ( 𝜑 → ( 𝑍 · 𝐴 ) ∈ 𝑆 ) |