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Metamath Proof Explorer

Theorem dihsmsprn

Description: Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015)

Ref Expression
Hypotheses dihsmsprn.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihsmsprn.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihsmsprn.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
dihsmsprn.p ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihsmsprn.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
dihsmsprn.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihsmsprn.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
dihsmsprn.x ⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
dihsmsprn.t ⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
Assertion dihsmsprn ( 𝜑 → ( 𝑋 ⊕ ( 𝑁 ‘ { 𝑇 } ) ) ∈ ran 𝐼 )

Proof

Step Hyp Ref Expression
1 dihsmsprn.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dihsmsprn.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 dihsmsprn.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
4 dihsmsprn.p ⊢ ⊕ = ( LSSum ‘ 𝑈 )
5 dihsmsprn.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
6 dihsmsprn.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
7 dihsmsprn.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8 dihsmsprn.x ⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
9 dihsmsprn.t ⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
10 eqid ⊢ ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( joinH ‘ 𝐾 ) ‘ 𝑊 )
11 1 2 3 4 5 6 10 7 8 9 dihjat1 ⊢ ( 𝜑 → ( 𝑋 ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑁 ‘ { 𝑇 } ) ) = ( 𝑋 ⊕ ( 𝑁 ‘ { 𝑇 } ) ) )
12 1 2 6 3 dihrnss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → 𝑋 ⊆ 𝑉 )
13 7 8 12 syl2anc ⊢ ( 𝜑 → 𝑋 ⊆ 𝑉 )
14 1 2 7 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
15 9 snssd ⊢ ( 𝜑 → { 𝑇 } ⊆ 𝑉 )
16 3 5 lspssv ⊢ ( ( 𝑈 ∈ LMod ∧ { 𝑇 } ⊆ 𝑉 ) → ( 𝑁 ‘ { 𝑇 } ) ⊆ 𝑉 )
17 14 15 16 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑇 } ) ⊆ 𝑉 )
18 1 6 2 3 10 djhcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ⊆ 𝑉 ∧ ( 𝑁 ‘ { 𝑇 } ) ⊆ 𝑉 ) ) → ( 𝑋 ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑁 ‘ { 𝑇 } ) ) ∈ ran 𝐼 )
19 7 13 17 18 syl12anc ⊢ ( 𝜑 → ( 𝑋 ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑁 ‘ { 𝑇 } ) ) ∈ ran 𝐼 )
20 11 19 eqeltrrd ⊢ ( 𝜑 → ( 𝑋 ⊕ ( 𝑁 ‘ { 𝑇 } ) ) ∈ ran 𝐼 )