Step |
Hyp |
Ref |
Expression |
1 |
|
lshpset.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lshpset.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
3 |
|
lshpset.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
4 |
|
lshpset.h |
⊢ 𝐻 = ( LSHyp ‘ 𝑊 ) |
5 |
1 2 3 4
|
lshpset |
⊢ ( 𝑊 ∈ 𝑋 → 𝐻 = { 𝑠 ∈ 𝑆 ∣ ( 𝑠 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑠 ∪ { 𝑣 } ) ) = 𝑉 ) } ) |
6 |
5
|
eleq2d |
⊢ ( 𝑊 ∈ 𝑋 → ( 𝑈 ∈ 𝐻 ↔ 𝑈 ∈ { 𝑠 ∈ 𝑆 ∣ ( 𝑠 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑠 ∪ { 𝑣 } ) ) = 𝑉 ) } ) ) |
7 |
|
neeq1 |
⊢ ( 𝑠 = 𝑈 → ( 𝑠 ≠ 𝑉 ↔ 𝑈 ≠ 𝑉 ) ) |
8 |
|
uneq1 |
⊢ ( 𝑠 = 𝑈 → ( 𝑠 ∪ { 𝑣 } ) = ( 𝑈 ∪ { 𝑣 } ) ) |
9 |
8
|
fveqeq2d |
⊢ ( 𝑠 = 𝑈 → ( ( 𝑁 ‘ ( 𝑠 ∪ { 𝑣 } ) ) = 𝑉 ↔ ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) ) |
10 |
9
|
rexbidv |
⊢ ( 𝑠 = 𝑈 → ( ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑠 ∪ { 𝑣 } ) ) = 𝑉 ↔ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) ) |
11 |
7 10
|
anbi12d |
⊢ ( 𝑠 = 𝑈 → ( ( 𝑠 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑠 ∪ { 𝑣 } ) ) = 𝑉 ) ↔ ( 𝑈 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) ) ) |
12 |
11
|
elrab |
⊢ ( 𝑈 ∈ { 𝑠 ∈ 𝑆 ∣ ( 𝑠 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑠 ∪ { 𝑣 } ) ) = 𝑉 ) } ↔ ( 𝑈 ∈ 𝑆 ∧ ( 𝑈 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) ) ) |
13 |
|
3anass |
⊢ ( ( 𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) ↔ ( 𝑈 ∈ 𝑆 ∧ ( 𝑈 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) ) ) |
14 |
12 13
|
bitr4i |
⊢ ( 𝑈 ∈ { 𝑠 ∈ 𝑆 ∣ ( 𝑠 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑠 ∪ { 𝑣 } ) ) = 𝑉 ) } ↔ ( 𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) ) |
15 |
6 14
|
bitrdi |
⊢ ( 𝑊 ∈ 𝑋 → ( 𝑈 ∈ 𝐻 ↔ ( 𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) ) ) |