Step |
Hyp |
Ref |
Expression |
1 |
|
dicelval2nd.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dicelval2nd.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
dicelval2nd.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dicelval2nd.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dicelval2nd.i |
⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
eqid |
⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
eqid |
⊢ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
8 |
1 2 3 5 6 7
|
dicssdvh |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ⊆ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
9 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
3 9 4 6 7
|
dvhvbase |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) ) |
12 |
8 11
|
sseqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ⊆ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) ) |
13 |
12
|
sseld |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) → 𝑌 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) ) ) |
14 |
13
|
3impia |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) → 𝑌 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) ) |
15 |
|
xp2nd |
⊢ ( 𝑌 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) → ( 2nd ‘ 𝑌 ) ∈ 𝐸 ) |
16 |
14 15
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) → ( 2nd ‘ 𝑌 ) ∈ 𝐸 ) |