Metamath Proof Explorer


Theorem prjsprellsp

Description: Two vectors are equivalent iff their spans are equal. (Contributed by Steven Nguyen, 31-May-2023)

Ref Expression
Hypotheses prjsprel.1 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐵𝑦𝐵 ) ∧ ∃ 𝑙𝐾 𝑥 = ( 𝑙 · 𝑦 ) ) }
prjspertr.b 𝐵 = ( ( Base ‘ 𝑉 ) ∖ { ( 0g𝑉 ) } )
prjspertr.s 𝑆 = ( Scalar ‘ 𝑉 )
prjspertr.x · = ( ·𝑠𝑉 )
prjspertr.k 𝐾 = ( Base ‘ 𝑆 )
prjsprellsp.n 𝑁 = ( LSpan ‘ 𝑉 )
Assertion prjsprellsp ( ( 𝑉 ∈ LVec ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 𝑌 ↔ ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ) )

Proof

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