Step |
Hyp |
Ref |
Expression |
1 |
|
prjsprel.1 |
⊢ ∼ = { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ 𝐾 𝑥 = ( 𝑙 · 𝑦 ) ) } |
2 |
|
prjspertr.b |
⊢ 𝐵 = ( ( Base ‘ 𝑉 ) ∖ { ( 0g ‘ 𝑉 ) } ) |
3 |
|
prjspertr.s |
⊢ 𝑆 = ( Scalar ‘ 𝑉 ) |
4 |
|
prjspertr.x |
⊢ · = ( ·𝑠 ‘ 𝑉 ) |
5 |
|
prjspertr.k |
⊢ 𝐾 = ( Base ‘ 𝑆 ) |
6 |
|
prjsprellsp.n |
⊢ 𝑁 = ( LSpan ‘ 𝑉 ) |
7 |
|
ibar |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∃ 𝑚 ∈ ( 𝐾 ∖ { ( 0g ‘ 𝑆 ) } ) 𝑋 = ( 𝑚 · 𝑌 ) ↔ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ∃ 𝑚 ∈ ( 𝐾 ∖ { ( 0g ‘ 𝑆 ) } ) 𝑋 = ( 𝑚 · 𝑌 ) ) ) ) |
8 |
7
|
bicomd |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ∃ 𝑚 ∈ ( 𝐾 ∖ { ( 0g ‘ 𝑆 ) } ) 𝑋 = ( 𝑚 · 𝑌 ) ) ↔ ∃ 𝑚 ∈ ( 𝐾 ∖ { ( 0g ‘ 𝑆 ) } ) 𝑋 = ( 𝑚 · 𝑌 ) ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝑉 ∈ LVec ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ∃ 𝑚 ∈ ( 𝐾 ∖ { ( 0g ‘ 𝑆 ) } ) 𝑋 = ( 𝑚 · 𝑌 ) ) ↔ ∃ 𝑚 ∈ ( 𝐾 ∖ { ( 0g ‘ 𝑆 ) } ) 𝑋 = ( 𝑚 · 𝑌 ) ) ) |
10 |
|
eqid |
⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 ) |
11 |
1 2 3 4 5 10
|
prjspreln0 |
⊢ ( 𝑉 ∈ LVec → ( 𝑋 ∼ 𝑌 ↔ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ∃ 𝑚 ∈ ( 𝐾 ∖ { ( 0g ‘ 𝑆 ) } ) 𝑋 = ( 𝑚 · 𝑌 ) ) ) ) |
12 |
11
|
adantr |
⊢ ( ( 𝑉 ∈ LVec ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ∼ 𝑌 ↔ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ∃ 𝑚 ∈ ( 𝐾 ∖ { ( 0g ‘ 𝑆 ) } ) 𝑋 = ( 𝑚 · 𝑌 ) ) ) ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝑉 ) = ( Base ‘ 𝑉 ) |
14 |
|
simpl |
⊢ ( ( 𝑉 ∈ LVec ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑉 ∈ LVec ) |
15 |
|
eldifi |
⊢ ( 𝑋 ∈ ( ( Base ‘ 𝑉 ) ∖ { ( 0g ‘ 𝑉 ) } ) → 𝑋 ∈ ( Base ‘ 𝑉 ) ) |
16 |
15 2
|
eleq2s |
⊢ ( 𝑋 ∈ 𝐵 → 𝑋 ∈ ( Base ‘ 𝑉 ) ) |
17 |
16
|
ad2antrl |
⊢ ( ( 𝑉 ∈ LVec ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ ( Base ‘ 𝑉 ) ) |
18 |
|
eldifi |
⊢ ( 𝑌 ∈ ( ( Base ‘ 𝑉 ) ∖ { ( 0g ‘ 𝑉 ) } ) → 𝑌 ∈ ( Base ‘ 𝑉 ) ) |
19 |
18 2
|
eleq2s |
⊢ ( 𝑌 ∈ 𝐵 → 𝑌 ∈ ( Base ‘ 𝑉 ) ) |
20 |
19
|
ad2antll |
⊢ ( ( 𝑉 ∈ LVec ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ ( Base ‘ 𝑉 ) ) |
21 |
13 3 5 10 4 6 14 17 20
|
lspsneq |
⊢ ( ( 𝑉 ∈ LVec ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ↔ ∃ 𝑚 ∈ ( 𝐾 ∖ { ( 0g ‘ 𝑆 ) } ) 𝑋 = ( 𝑚 · 𝑌 ) ) ) |
22 |
9 12 21
|
3bitr4d |
⊢ ( ( 𝑉 ∈ LVec ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ∼ 𝑌 ↔ ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ) ) |