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Metamath Proof Explorer

Theorem lssvnegcl

Description: Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014)

Ref Expression
Hypotheses lssvnegcl.s ⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
lssvnegcl.n ⊢ 𝑁 = ( invg ‘ 𝑊 )
Assertion lssvnegcl ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝑈 )

Proof

Step Hyp Ref Expression
1 lssvnegcl.s ⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
2 lssvnegcl.n ⊢ 𝑁 = ( invg ‘ 𝑊 )
3 eqid ⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
4 3 1 lssel ⊢ ( ( 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → 𝑋 ∈ ( Base ‘ 𝑊 ) )
5 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
6 eqid ⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 )
7 eqid ⊢ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) = ( 1r ‘ ( Scalar ‘ 𝑊 ) )
8 eqid ⊢ ( invg ‘ ( Scalar ‘ 𝑊 ) ) = ( invg ‘ ( Scalar ‘ 𝑊 ) )
9 3 2 5 6 7 8 lmodvneg1 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) = ( 𝑁 ‘ 𝑋 ) )
10 4 9 sylan2 ⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) ) → ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) = ( 𝑁 ‘ 𝑋 ) )
11 10 3impb ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) = ( 𝑁 ‘ 𝑋 ) )
12 simp1 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → 𝑊 ∈ LMod )
13 simp2 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → 𝑈 ∈ 𝑆 )
14 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
15 5 lmodring ⊢ ( 𝑊 ∈ LMod → ( Scalar ‘ 𝑊 ) ∈ Ring )
16 15 3ad2ant1 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → ( Scalar ‘ 𝑊 ) ∈ Ring )
17 16 ringgrpd ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → ( Scalar ‘ 𝑊 ) ∈ Grp )
18 14 7 ringidcl ⊢ ( ( Scalar ‘ 𝑊 ) ∈ Ring → ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
19 16 18 syl ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
20 14 8 17 19 grpinvcld ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
21 simp3 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → 𝑋 ∈ 𝑈 )
22 5 6 14 1 lssvscl ⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ) ∧ ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑋 ∈ 𝑈 ) ) → ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ 𝑈 )
23 12 13 20 21 22 syl22anc ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ 𝑈 )
24 11 23 eqeltrrd ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝑈 )