Step |
Hyp |
Ref |
Expression |
1 |
|
lindfind2.k |
⊢ 𝐾 = ( LSpan ‘ 𝑊 ) |
2 |
|
lindfind2.l |
⊢ 𝐿 = ( Scalar ‘ 𝑊 ) |
3 |
|
simp1 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ) ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) → ( 𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ) ) |
4 |
|
linds2 |
⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) → ( I ↾ 𝐹 ) LIndF 𝑊 ) |
5 |
4
|
3ad2ant2 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ) ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) → ( I ↾ 𝐹 ) LIndF 𝑊 ) |
6 |
|
dmresi |
⊢ dom ( I ↾ 𝐹 ) = 𝐹 |
7 |
6
|
eleq2i |
⊢ ( 𝐸 ∈ dom ( I ↾ 𝐹 ) ↔ 𝐸 ∈ 𝐹 ) |
8 |
7
|
biimpri |
⊢ ( 𝐸 ∈ 𝐹 → 𝐸 ∈ dom ( I ↾ 𝐹 ) ) |
9 |
8
|
3ad2ant3 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ) ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) → 𝐸 ∈ dom ( I ↾ 𝐹 ) ) |
10 |
1 2
|
lindfind2 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ) ∧ ( I ↾ 𝐹 ) LIndF 𝑊 ∧ 𝐸 ∈ dom ( I ↾ 𝐹 ) ) → ¬ ( ( I ↾ 𝐹 ) ‘ 𝐸 ) ∈ ( 𝐾 ‘ ( ( I ↾ 𝐹 ) “ ( dom ( I ↾ 𝐹 ) ∖ { 𝐸 } ) ) ) ) |
11 |
3 5 9 10
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ) ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) → ¬ ( ( I ↾ 𝐹 ) ‘ 𝐸 ) ∈ ( 𝐾 ‘ ( ( I ↾ 𝐹 ) “ ( dom ( I ↾ 𝐹 ) ∖ { 𝐸 } ) ) ) ) |
12 |
|
fvresi |
⊢ ( 𝐸 ∈ 𝐹 → ( ( I ↾ 𝐹 ) ‘ 𝐸 ) = 𝐸 ) |
13 |
6
|
difeq1i |
⊢ ( dom ( I ↾ 𝐹 ) ∖ { 𝐸 } ) = ( 𝐹 ∖ { 𝐸 } ) |
14 |
13
|
imaeq2i |
⊢ ( ( I ↾ 𝐹 ) “ ( dom ( I ↾ 𝐹 ) ∖ { 𝐸 } ) ) = ( ( I ↾ 𝐹 ) “ ( 𝐹 ∖ { 𝐸 } ) ) |
15 |
|
difss |
⊢ ( 𝐹 ∖ { 𝐸 } ) ⊆ 𝐹 |
16 |
|
resiima |
⊢ ( ( 𝐹 ∖ { 𝐸 } ) ⊆ 𝐹 → ( ( I ↾ 𝐹 ) “ ( 𝐹 ∖ { 𝐸 } ) ) = ( 𝐹 ∖ { 𝐸 } ) ) |
17 |
15 16
|
ax-mp |
⊢ ( ( I ↾ 𝐹 ) “ ( 𝐹 ∖ { 𝐸 } ) ) = ( 𝐹 ∖ { 𝐸 } ) |
18 |
14 17
|
eqtri |
⊢ ( ( I ↾ 𝐹 ) “ ( dom ( I ↾ 𝐹 ) ∖ { 𝐸 } ) ) = ( 𝐹 ∖ { 𝐸 } ) |
19 |
18
|
fveq2i |
⊢ ( 𝐾 ‘ ( ( I ↾ 𝐹 ) “ ( dom ( I ↾ 𝐹 ) ∖ { 𝐸 } ) ) ) = ( 𝐾 ‘ ( 𝐹 ∖ { 𝐸 } ) ) |
20 |
19
|
a1i |
⊢ ( 𝐸 ∈ 𝐹 → ( 𝐾 ‘ ( ( I ↾ 𝐹 ) “ ( dom ( I ↾ 𝐹 ) ∖ { 𝐸 } ) ) ) = ( 𝐾 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) |
21 |
12 20
|
eleq12d |
⊢ ( 𝐸 ∈ 𝐹 → ( ( ( I ↾ 𝐹 ) ‘ 𝐸 ) ∈ ( 𝐾 ‘ ( ( I ↾ 𝐹 ) “ ( dom ( I ↾ 𝐹 ) ∖ { 𝐸 } ) ) ) ↔ 𝐸 ∈ ( 𝐾 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) ) |
22 |
21
|
3ad2ant3 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ) ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) → ( ( ( I ↾ 𝐹 ) ‘ 𝐸 ) ∈ ( 𝐾 ‘ ( ( I ↾ 𝐹 ) “ ( dom ( I ↾ 𝐹 ) ∖ { 𝐸 } ) ) ) ↔ 𝐸 ∈ ( 𝐾 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) ) |
23 |
11 22
|
mtbid |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ) ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) → ¬ 𝐸 ∈ ( 𝐾 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) |