This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dih1dimb

Description: Two expressions for a 1-dimensional subspace of vector space H (when F is a nonzero vector i.e. non-identity translation). (Contributed by NM, 27-Apr-2014)

Ref Expression
Hypotheses dih1dimb.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dih1dimb.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dih1dimb.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dih1dimb.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
dih1dimb.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
dih1dimb.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dih1dimb.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dih1dimb.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
Assertion dih1dimb ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑁 ‘ { ⟨ 𝐹 , 𝑂 ⟩ } ) )

Proof

Step Hyp Ref Expression
1 dih1dimb.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dih1dimb.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 dih1dimb.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 dih1dimb.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
5 dih1dimb.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
6 dih1dimb.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
7 dih1dimb.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
8 dih1dimb.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
9 simpl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 1 2 3 4 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐵 )
11 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
12 11 2 3 4 trlle ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ( le ‘ 𝐾 ) 𝑊 )
13 eqid ⊢ ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
14 1 11 2 7 13 dihvalb ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑅 ‘ 𝐹 ) ∈ 𝐵 ∧ ( 𝑅 ‘ 𝐹 ) ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) )
15 9 10 12 14 syl12anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) )
16 1 2 3 4 5 6 13 8 dib1dim2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑁 ‘ { ⟨ 𝐹 , 𝑂 ⟩ } ) )
17 15 16 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑁 ‘ { ⟨ 𝐹 , 𝑂 ⟩ } ) )