Step |
Hyp |
Ref |
Expression |
1 |
|
qqhval2.0 |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
qqhval2.1 |
⊢ / = ( /r ‘ 𝑅 ) |
3 |
|
qqhval2.2 |
⊢ 𝐿 = ( ℤRHom ‘ 𝑅 ) |
4 |
|
zssq |
⊢ ℤ ⊆ ℚ |
5 |
|
simpr1 |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝑋 ∈ ℤ ) |
6 |
4 5
|
sselid |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝑋 ∈ ℚ ) |
7 |
|
simpr2 |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝑌 ∈ ℤ ) |
8 |
4 7
|
sselid |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝑌 ∈ ℚ ) |
9 |
|
simpr3 |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝑌 ≠ 0 ) |
10 |
|
qdivcl |
⊢ ( ( 𝑋 ∈ ℚ ∧ 𝑌 ∈ ℚ ∧ 𝑌 ≠ 0 ) → ( 𝑋 / 𝑌 ) ∈ ℚ ) |
11 |
6 8 9 10
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝑋 / 𝑌 ) ∈ ℚ ) |
12 |
1 2 3
|
qqhvval |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 / 𝑌 ) ∈ ℚ ) → ( ( ℚHom ‘ 𝑅 ) ‘ ( 𝑋 / 𝑌 ) ) = ( ( 𝐿 ‘ ( numer ‘ ( 𝑋 / 𝑌 ) ) ) / ( 𝐿 ‘ ( denom ‘ ( 𝑋 / 𝑌 ) ) ) ) ) |
13 |
11 12
|
syldan |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( ℚHom ‘ 𝑅 ) ‘ ( 𝑋 / 𝑌 ) ) = ( ( 𝐿 ‘ ( numer ‘ ( 𝑋 / 𝑌 ) ) ) / ( 𝐿 ‘ ( denom ‘ ( 𝑋 / 𝑌 ) ) ) ) ) |
14 |
1 2 3
|
qqhval2lem |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( 𝐿 ‘ ( numer ‘ ( 𝑋 / 𝑌 ) ) ) / ( 𝐿 ‘ ( denom ‘ ( 𝑋 / 𝑌 ) ) ) ) = ( ( 𝐿 ‘ 𝑋 ) / ( 𝐿 ‘ 𝑌 ) ) ) |
15 |
13 14
|
eqtrd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( ℚHom ‘ 𝑅 ) ‘ ( 𝑋 / 𝑌 ) ) = ( ( 𝐿 ‘ 𝑋 ) / ( 𝐿 ‘ 𝑌 ) ) ) |