Metamath Proof Explorer


Theorem lspun0

Description: The span of a union with the zero subspace. (Contributed by NM, 22-May-2015)

Ref Expression
Hypotheses lspun0.v 𝑉 = ( Base ‘ 𝑊 )
lspun0.o 0 = ( 0g𝑊 )
lspun0.n 𝑁 = ( LSpan ‘ 𝑊 )
lspun0.w ( 𝜑𝑊 ∈ LMod )
lspun0.x ( 𝜑𝑋𝑉 )
Assertion lspun0 ( 𝜑 → ( 𝑁 ‘ ( 𝑋 ∪ { 0 } ) ) = ( 𝑁𝑋 ) )

Proof

Step Hyp Ref Expression
1 lspun0.v 𝑉 = ( Base ‘ 𝑊 )
2 lspun0.o 0 = ( 0g𝑊 )
3 lspun0.n 𝑁 = ( LSpan ‘ 𝑊 )
4 lspun0.w ( 𝜑𝑊 ∈ LMod )
5 lspun0.x ( 𝜑𝑋𝑉 )
6 1 2 lmod0vcl ( 𝑊 ∈ LMod → 0𝑉 )
7 4 6 syl ( 𝜑0𝑉 )
8 7 snssd ( 𝜑 → { 0 } ⊆ 𝑉 )
9 1 3 lspun ( ( 𝑊 ∈ LMod ∧ 𝑋𝑉 ∧ { 0 } ⊆ 𝑉 ) → ( 𝑁 ‘ ( 𝑋 ∪ { 0 } ) ) = ( 𝑁 ‘ ( ( 𝑁𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) ) )
10 4 5 8 9 syl3anc ( 𝜑 → ( 𝑁 ‘ ( 𝑋 ∪ { 0 } ) ) = ( 𝑁 ‘ ( ( 𝑁𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) ) )
11 2 3 lspsn0 ( 𝑊 ∈ LMod → ( 𝑁 ‘ { 0 } ) = { 0 } )
12 4 11 syl ( 𝜑 → ( 𝑁 ‘ { 0 } ) = { 0 } )
13 12 uneq2d ( 𝜑 → ( ( 𝑁𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) = ( ( 𝑁𝑋 ) ∪ { 0 } ) )
14 eqid ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
15 1 14 3 lspcl ( ( 𝑊 ∈ LMod ∧ 𝑋𝑉 ) → ( 𝑁𝑋 ) ∈ ( LSubSp ‘ 𝑊 ) )
16 4 5 15 syl2anc ( 𝜑 → ( 𝑁𝑋 ) ∈ ( LSubSp ‘ 𝑊 ) )
17 2 14 lss0ss ( ( 𝑊 ∈ LMod ∧ ( 𝑁𝑋 ) ∈ ( LSubSp ‘ 𝑊 ) ) → { 0 } ⊆ ( 𝑁𝑋 ) )
18 4 16 17 syl2anc ( 𝜑 → { 0 } ⊆ ( 𝑁𝑋 ) )
19 ssequn2 ( { 0 } ⊆ ( 𝑁𝑋 ) ↔ ( ( 𝑁𝑋 ) ∪ { 0 } ) = ( 𝑁𝑋 ) )
20 18 19 sylib ( 𝜑 → ( ( 𝑁𝑋 ) ∪ { 0 } ) = ( 𝑁𝑋 ) )
21 13 20 eqtrd ( 𝜑 → ( ( 𝑁𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) = ( 𝑁𝑋 ) )
22 21 fveq2d ( 𝜑 → ( 𝑁 ‘ ( ( 𝑁𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) ) = ( 𝑁 ‘ ( 𝑁𝑋 ) ) )
23 1 3 lspidm ( ( 𝑊 ∈ LMod ∧ 𝑋𝑉 ) → ( 𝑁 ‘ ( 𝑁𝑋 ) ) = ( 𝑁𝑋 ) )
24 4 5 23 syl2anc ( 𝜑 → ( 𝑁 ‘ ( 𝑁𝑋 ) ) = ( 𝑁𝑋 ) )
25 22 24 eqtrd ( 𝜑 → ( 𝑁 ‘ ( ( 𝑁𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) ) = ( 𝑁𝑋 ) )
26 10 25 eqtrd ( 𝜑 → ( 𝑁 ‘ ( 𝑋 ∪ { 0 } ) ) = ( 𝑁𝑋 ) )