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Metamath Proof Explorer

Theorem lspsn

Description: Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014) (Proof shortened by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses lspsn.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
lspsn.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
lspsn.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
lspsn.t ⊢ · = ( ·𝑠 ‘ 𝑊 )
lspsn.n ⊢ 𝑁 = ( LSpan ‘ 𝑊 )
Assertion lspsn ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) = { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } )

Proof

Step Hyp Ref Expression
1 lspsn.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2 lspsn.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
3 lspsn.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
4 lspsn.t ⊢ · = ( ·𝑠 ‘ 𝑊 )
5 lspsn.n ⊢ 𝑁 = ( LSpan ‘ 𝑊 )
6 eqid ⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
7 simpl ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑊 ∈ LMod )
8 3 1 4 2 6 lss1d ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ∈ ( LSubSp ‘ 𝑊 ) )
9 eqid ⊢ ( 1r ‘ 𝐹 ) = ( 1r ‘ 𝐹 )
10 1 2 9 lmod1cl ⊢ ( 𝑊 ∈ LMod → ( 1r ‘ 𝐹 ) ∈ 𝐾 )
11 3 1 4 9 lmodvs1 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 1r ‘ 𝐹 ) · 𝑋 ) = 𝑋 )
12 11 eqcomd ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 = ( ( 1r ‘ 𝐹 ) · 𝑋 ) )
13 oveq1 ⊢ ( 𝑘 = ( 1r ‘ 𝐹 ) → ( 𝑘 · 𝑋 ) = ( ( 1r ‘ 𝐹 ) · 𝑋 ) )
14 13 rspceeqv ⊢ ( ( ( 1r ‘ 𝐹 ) ∈ 𝐾 ∧ 𝑋 = ( ( 1r ‘ 𝐹 ) · 𝑋 ) ) → ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) )
15 10 12 14 syl2an2r ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) )
16 eqeq1 ⊢ ( 𝑣 = 𝑋 → ( 𝑣 = ( 𝑘 · 𝑋 ) ↔ 𝑋 = ( 𝑘 · 𝑋 ) ) )
17 16 rexbidv ⊢ ( 𝑣 = 𝑋 → ( ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) ) )
18 17 elabg ⊢ ( 𝑋 ∈ 𝑉 → ( 𝑋 ∈ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) ) )
19 18 adantl ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 ∈ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) ) )
20 15 19 mpbird ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } )
21 6 5 7 8 20 ellspsn5 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ⊆ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } )
22 7 adantr ⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → 𝑊 ∈ LMod )
23 3 6 5 lspsncl ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) )
24 23 adantr ⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) )
25 simpr ⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → 𝑘 ∈ 𝐾 )
26 3 5 lspsnid ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) )
27 26 adantr ⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) )
28 1 4 2 6 lssvscl ⊢ ( ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) ∧ ( 𝑘 ∈ 𝐾 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) ) → ( 𝑘 · 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) )
29 22 24 25 27 28 syl22anc ⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → ( 𝑘 · 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) )
30 eleq1a ⊢ ( ( 𝑘 · 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) → ( 𝑣 = ( 𝑘 · 𝑋 ) → 𝑣 ∈ ( 𝑁 ‘ { 𝑋 } ) ) )
31 29 30 syl ⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → ( 𝑣 = ( 𝑘 · 𝑋 ) → 𝑣 ∈ ( 𝑁 ‘ { 𝑋 } ) ) )
32 31 rexlimdva ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) → 𝑣 ∈ ( 𝑁 ‘ { 𝑋 } ) ) )
33 32 abssdv ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ⊆ ( 𝑁 ‘ { 𝑋 } ) )
34 21 33 eqssd ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) = { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } )