Step |
Hyp |
Ref |
Expression |
1 |
|
fldextfld1 |
⊢ ( 𝐸 /FldExt 𝐹 → 𝐸 ∈ Field ) |
2 |
|
isfld |
⊢ ( 𝐸 ∈ Field ↔ ( 𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing ) ) |
3 |
2
|
simplbi |
⊢ ( 𝐸 ∈ Field → 𝐸 ∈ DivRing ) |
4 |
1 3
|
syl |
⊢ ( 𝐸 /FldExt 𝐹 → 𝐸 ∈ DivRing ) |
5 |
|
fldextress |
⊢ ( 𝐸 /FldExt 𝐹 → 𝐹 = ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) ) |
6 |
|
fldextfld2 |
⊢ ( 𝐸 /FldExt 𝐹 → 𝐹 ∈ Field ) |
7 |
|
isfld |
⊢ ( 𝐹 ∈ Field ↔ ( 𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing ) ) |
8 |
7
|
simplbi |
⊢ ( 𝐹 ∈ Field → 𝐹 ∈ DivRing ) |
9 |
6 8
|
syl |
⊢ ( 𝐸 /FldExt 𝐹 → 𝐹 ∈ DivRing ) |
10 |
5 9
|
eqeltrrd |
⊢ ( 𝐸 /FldExt 𝐹 → ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) ∈ DivRing ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 ) |
12 |
11
|
fldextsubrg |
⊢ ( 𝐸 /FldExt 𝐹 → ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ 𝐸 ) ) |
13 |
|
eqid |
⊢ ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) = ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) |
14 |
|
eqid |
⊢ ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) = ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) |
15 |
13 14
|
sralvec |
⊢ ( ( 𝐸 ∈ DivRing ∧ ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) ∈ DivRing ∧ ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ 𝐸 ) ) → ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ∈ LVec ) |
16 |
4 10 12 15
|
syl3anc |
⊢ ( 𝐸 /FldExt 𝐹 → ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ∈ LVec ) |
17 |
|
eqid |
⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 ) |
18 |
17
|
subrgss |
⊢ ( ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ 𝐸 ) → ( Base ‘ 𝐹 ) ⊆ ( Base ‘ 𝐸 ) ) |
19 |
12 18
|
syl |
⊢ ( 𝐸 /FldExt 𝐹 → ( Base ‘ 𝐹 ) ⊆ ( Base ‘ 𝐸 ) ) |
20 |
13 17
|
sradrng |
⊢ ( ( 𝐸 ∈ DivRing ∧ ( Base ‘ 𝐹 ) ⊆ ( Base ‘ 𝐸 ) ) → ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ∈ DivRing ) |
21 |
4 19 20
|
syl2anc |
⊢ ( 𝐸 /FldExt 𝐹 → ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ∈ DivRing ) |
22 |
|
drngdimgt0 |
⊢ ( ( ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ∈ LVec ∧ ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ∈ DivRing ) → 0 < ( dim ‘ ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ) ) |
23 |
16 21 22
|
syl2anc |
⊢ ( 𝐸 /FldExt 𝐹 → 0 < ( dim ‘ ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ) ) |
24 |
|
extdgval |
⊢ ( 𝐸 /FldExt 𝐹 → ( 𝐸 [:] 𝐹 ) = ( dim ‘ ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ) ) |
25 |
23 24
|
breqtrrd |
⊢ ( 𝐸 /FldExt 𝐹 → 0 < ( 𝐸 [:] 𝐹 ) ) |