Metamath Proof Explorer
Description: Deduction associated with mulgcl . (Contributed by Rohan Ridenour, 3-Aug-2023)
|
|
Ref |
Expression |
|
Hypotheses |
mulgcld.1 |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
|
|
mulgcld.2 |
⊢ · = ( .g ‘ 𝐺 ) |
|
|
mulgcld.3 |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
|
|
mulgcld.4 |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
|
|
mulgcld.5 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
|
Assertion |
mulgcld |
⊢ ( 𝜑 → ( 𝑁 · 𝑋 ) ∈ 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
mulgcld.1 |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
mulgcld.2 |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
mulgcld.3 |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
4 |
|
mulgcld.4 |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
5 |
|
mulgcld.5 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
1 2
|
mulgcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) ∈ 𝐵 ) |
7 |
3 4 5 6
|
syl3anc |
⊢ ( 𝜑 → ( 𝑁 · 𝑋 ) ∈ 𝐵 ) |