Step |
Hyp |
Ref |
Expression |
1 |
|
lkrshp4.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lkrshp4.h |
⊢ 𝐻 = ( LSHyp ‘ 𝑊 ) |
3 |
|
lkrshp4.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
4 |
|
lkrshp4.k |
⊢ 𝐾 = ( LKer ‘ 𝑊 ) |
5 |
|
lkrshp4.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
6 |
|
lkrshp4.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
7 |
1 2 3 4 5 6
|
lkrshpor |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ∨ ( 𝐾 ‘ 𝐺 ) = 𝑉 ) ) |
8 |
7
|
orcomd |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) = 𝑉 ∨ ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ) ) |
9 |
|
neor |
⊢ ( ( ( 𝐾 ‘ 𝐺 ) = 𝑉 ∨ ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ) ↔ ( ( 𝐾 ‘ 𝐺 ) ≠ 𝑉 → ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ) ) |
10 |
8 9
|
sylib |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ≠ 𝑉 → ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ) ) |
11 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
12 |
5 11
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
13 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ) → 𝑊 ∈ LMod ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ) → ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ) |
15 |
1 2 13 14
|
lshpne |
⊢ ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ) → ( 𝐾 ‘ 𝐺 ) ≠ 𝑉 ) |
16 |
15
|
ex |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 → ( 𝐾 ‘ 𝐺 ) ≠ 𝑉 ) ) |
17 |
10 16
|
impbid |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ≠ 𝑉 ↔ ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ) ) |