Step |
Hyp |
Ref |
Expression |
1 |
|
lkrsc.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lkrsc.d |
⊢ 𝐷 = ( Scalar ‘ 𝑊 ) |
3 |
|
lkrsc.k |
⊢ 𝐾 = ( Base ‘ 𝐷 ) |
4 |
|
lkrsc.t |
⊢ · = ( .r ‘ 𝐷 ) |
5 |
|
lkrsc.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
6 |
|
lkrsc.l |
⊢ 𝐿 = ( LKer ‘ 𝑊 ) |
7 |
|
lkrsc.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
8 |
|
lkrsc.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
9 |
|
lkrsc.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝐾 ) |
10 |
|
lkrsc.o |
⊢ 0 = ( 0g ‘ 𝐷 ) |
11 |
|
lkrsc.e |
⊢ ( 𝜑 → 𝑅 ≠ 0 ) |
12 |
1
|
fvexi |
⊢ 𝑉 ∈ V |
13 |
12
|
a1i |
⊢ ( 𝜑 → 𝑉 ∈ V ) |
14 |
2 3 1 5
|
lflf |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ) → 𝐺 : 𝑉 ⟶ 𝐾 ) |
15 |
7 8 14
|
syl2anc |
⊢ ( 𝜑 → 𝐺 : 𝑉 ⟶ 𝐾 ) |
16 |
15
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn 𝑉 ) |
17 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑣 ) = ( 𝐺 ‘ 𝑣 ) ) |
18 |
13 9 16 17
|
ofc2 |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → ( ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ‘ 𝑣 ) = ( ( 𝐺 ‘ 𝑣 ) · 𝑅 ) ) |
19 |
18
|
eqeq1d |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → ( ( ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ‘ 𝑣 ) = 0 ↔ ( ( 𝐺 ‘ 𝑣 ) · 𝑅 ) = 0 ) ) |
20 |
2
|
lvecdrng |
⊢ ( 𝑊 ∈ LVec → 𝐷 ∈ DivRing ) |
21 |
7 20
|
syl |
⊢ ( 𝜑 → 𝐷 ∈ DivRing ) |
22 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → 𝐷 ∈ DivRing ) |
23 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → 𝑊 ∈ LVec ) |
24 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → 𝐺 ∈ 𝐹 ) |
25 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → 𝑣 ∈ 𝑉 ) |
26 |
2 3 1 5
|
lflcl |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑣 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑣 ) ∈ 𝐾 ) |
27 |
23 24 25 26
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑣 ) ∈ 𝐾 ) |
28 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → 𝑅 ∈ 𝐾 ) |
29 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → 𝑅 ≠ 0 ) |
30 |
3 10 4 22 27 28 29
|
drngmuleq0 |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → ( ( ( 𝐺 ‘ 𝑣 ) · 𝑅 ) = 0 ↔ ( 𝐺 ‘ 𝑣 ) = 0 ) ) |
31 |
19 30
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → ( ( ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ‘ 𝑣 ) = 0 ↔ ( 𝐺 ‘ 𝑣 ) = 0 ) ) |
32 |
31
|
pm5.32da |
⊢ ( 𝜑 → ( ( 𝑣 ∈ 𝑉 ∧ ( ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ‘ 𝑣 ) = 0 ) ↔ ( 𝑣 ∈ 𝑉 ∧ ( 𝐺 ‘ 𝑣 ) = 0 ) ) ) |
33 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
34 |
7 33
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
35 |
1 2 3 4 5 34 8 9
|
lflvscl |
⊢ ( 𝜑 → ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ∈ 𝐹 ) |
36 |
1 2 10 5 6
|
ellkr |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ∈ 𝐹 ) → ( 𝑣 ∈ ( 𝐿 ‘ ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ) ↔ ( 𝑣 ∈ 𝑉 ∧ ( ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ‘ 𝑣 ) = 0 ) ) ) |
37 |
7 35 36
|
syl2anc |
⊢ ( 𝜑 → ( 𝑣 ∈ ( 𝐿 ‘ ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ) ↔ ( 𝑣 ∈ 𝑉 ∧ ( ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ‘ 𝑣 ) = 0 ) ) ) |
38 |
1 2 10 5 6
|
ellkr |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ) → ( 𝑣 ∈ ( 𝐿 ‘ 𝐺 ) ↔ ( 𝑣 ∈ 𝑉 ∧ ( 𝐺 ‘ 𝑣 ) = 0 ) ) ) |
39 |
7 8 38
|
syl2anc |
⊢ ( 𝜑 → ( 𝑣 ∈ ( 𝐿 ‘ 𝐺 ) ↔ ( 𝑣 ∈ 𝑉 ∧ ( 𝐺 ‘ 𝑣 ) = 0 ) ) ) |
40 |
32 37 39
|
3bitr4d |
⊢ ( 𝜑 → ( 𝑣 ∈ ( 𝐿 ‘ ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ) ↔ 𝑣 ∈ ( 𝐿 ‘ 𝐺 ) ) ) |
41 |
40
|
eqrdv |
⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐺 ∘f · ( 𝑉 × { 𝑅 } ) ) ) = ( 𝐿 ‘ 𝐺 ) ) |