Step |
Hyp |
Ref |
Expression |
1 |
|
dih0vb.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dih0vb.o |
⊢ 0 = ( 0. ‘ 𝐾 ) |
3 |
|
dih0vb.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dih0vb.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dih0vb.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
6 |
|
dih0vb.z |
⊢ 𝑍 = ( 0g ‘ 𝑈 ) |
7 |
|
dih0vb.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
8 |
|
dih0vb.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
dih0vb.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
10 |
2 1 3 4 6
|
dih0 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ 0 ) = { 𝑍 } ) |
11 |
8 10
|
syl |
⊢ ( 𝜑 → ( 𝐼 ‘ 0 ) = { 𝑍 } ) |
12 |
11
|
eqeq2d |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) = ( 𝐼 ‘ 0 ) ↔ ( 𝑁 ‘ { 𝑋 } ) = { 𝑍 } ) ) |
13 |
1 4 8
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
14 |
5 6 7
|
lspsneq0 |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑁 ‘ { 𝑋 } ) = { 𝑍 } ↔ 𝑋 = 𝑍 ) ) |
15 |
13 9 14
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) = { 𝑍 } ↔ 𝑋 = 𝑍 ) ) |
16 |
12 15
|
bitr2d |
⊢ ( 𝜑 → ( 𝑋 = 𝑍 ↔ ( 𝑁 ‘ { 𝑋 } ) = ( 𝐼 ‘ 0 ) ) ) |