Metamath Proof Explorer


Theorem lspdisj2

Description: Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015)

Ref Expression
Hypotheses lspdisj2.v 𝑉 = ( Base ‘ 𝑊 )
lspdisj2.o 0 = ( 0g𝑊 )
lspdisj2.n 𝑁 = ( LSpan ‘ 𝑊 )
lspdisj2.w ( 𝜑𝑊 ∈ LVec )
lspdisj2.x ( 𝜑𝑋𝑉 )
lspdisj2.y ( 𝜑𝑌𝑉 )
lspdisj2.q ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
Assertion lspdisj2 ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } )

Proof

Step Hyp Ref Expression
1 lspdisj2.v 𝑉 = ( Base ‘ 𝑊 )
2 lspdisj2.o 0 = ( 0g𝑊 )
3 lspdisj2.n 𝑁 = ( LSpan ‘ 𝑊 )
4 lspdisj2.w ( 𝜑𝑊 ∈ LVec )
5 lspdisj2.x ( 𝜑𝑋𝑉 )
6 lspdisj2.y ( 𝜑𝑌𝑉 )
7 lspdisj2.q ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
8 sneq ( 𝑋 = 0 → { 𝑋 } = { 0 } )
9 8 fveq2d ( 𝑋 = 0 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 0 } ) )
10 lveclmod ( 𝑊 ∈ LVec → 𝑊 ∈ LMod )
11 4 10 syl ( 𝜑𝑊 ∈ LMod )
12 2 3 lspsn0 ( 𝑊 ∈ LMod → ( 𝑁 ‘ { 0 } ) = { 0 } )
13 11 12 syl ( 𝜑 → ( 𝑁 ‘ { 0 } ) = { 0 } )
14 9 13 sylan9eqr ( ( 𝜑𝑋 = 0 ) → ( 𝑁 ‘ { 𝑋 } ) = { 0 } )
15 14 ineq1d ( ( 𝜑𝑋 = 0 ) → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 } ) ) = ( { 0 } ∩ ( 𝑁 ‘ { 𝑌 } ) ) )
16 eqid ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
17 1 16 3 lspsncl ( ( 𝑊 ∈ LMod ∧ 𝑌𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) )
18 11 6 17 syl2anc ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) )
19 2 16 lss0ss ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) ) → { 0 } ⊆ ( 𝑁 ‘ { 𝑌 } ) )
20 11 18 19 syl2anc ( 𝜑 → { 0 } ⊆ ( 𝑁 ‘ { 𝑌 } ) )
21 df-ss ( { 0 } ⊆ ( 𝑁 ‘ { 𝑌 } ) ↔ ( { 0 } ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } )
22 20 21 sylib ( 𝜑 → ( { 0 } ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } )
23 22 adantr ( ( 𝜑𝑋 = 0 ) → ( { 0 } ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } )
24 15 23 eqtrd ( ( 𝜑𝑋 = 0 ) → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } )
25 4 adantr ( ( 𝜑𝑋0 ) → 𝑊 ∈ LVec )
26 18 adantr ( ( 𝜑𝑋0 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) )
27 5 adantr ( ( 𝜑𝑋0 ) → 𝑋𝑉 )
28 7 adantr ( ( 𝜑𝑋0 ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
29 25 adantr ( ( ( 𝜑𝑋0 ) ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) → 𝑊 ∈ LVec )
30 6 adantr ( ( 𝜑𝑋0 ) → 𝑌𝑉 )
31 30 adantr ( ( ( 𝜑𝑋0 ) ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) → 𝑌𝑉 )
32 simpr ( ( ( 𝜑𝑋0 ) ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) )
33 simplr ( ( ( 𝜑𝑋0 ) ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) → 𝑋0 )
34 1 2 3 29 31 32 33 lspsneleq ( ( ( 𝜑𝑋0 ) ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) )
35 34 ex ( ( 𝜑𝑋0 ) → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ) )
36 35 necon3ad ( ( 𝜑𝑋0 ) → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) )
37 28 36 mpd ( ( 𝜑𝑋0 ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) )
38 1 2 3 16 25 26 27 37 lspdisj ( ( 𝜑𝑋0 ) → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } )
39 24 38 pm2.61dane ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } )