Metamath Proof Explorer


Theorem dih0vbN

Description: A vector is zero iff its span is the isomorphism of lattice zero. (Contributed by NM, 16-Aug-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dih0vb.h 𝐻 = ( LHyp ‘ 𝐾 )
dih0vb.o 0 = ( 0. ‘ 𝐾 )
dih0vb.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dih0vb.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dih0vb.v 𝑉 = ( Base ‘ 𝑈 )
dih0vb.z 𝑍 = ( 0g𝑈 )
dih0vb.n 𝑁 = ( LSpan ‘ 𝑈 )
dih0vb.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
dih0vb.x ( 𝜑𝑋𝑉 )
Assertion dih0vbN ( 𝜑 → ( 𝑋 = 𝑍 ↔ ( 𝑁 ‘ { 𝑋 } ) = ( 𝐼0 ) ) )

Proof

Step Hyp Ref Expression
1 dih0vb.h 𝐻 = ( LHyp ‘ 𝐾 )
2 dih0vb.o 0 = ( 0. ‘ 𝐾 )
3 dih0vb.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4 dih0vb.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5 dih0vb.v 𝑉 = ( Base ‘ 𝑈 )
6 dih0vb.z 𝑍 = ( 0g𝑈 )
7 dih0vb.n 𝑁 = ( LSpan ‘ 𝑈 )
8 dih0vb.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 dih0vb.x ( 𝜑𝑋𝑉 )
10 2 1 3 4 6 dih0 ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( 𝐼0 ) = { 𝑍 } )
11 8 10 syl ( 𝜑 → ( 𝐼0 ) = { 𝑍 } )
12 11 eqeq2d ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) = ( 𝐼0 ) ↔ ( 𝑁 ‘ { 𝑋 } ) = { 𝑍 } ) )
13 1 4 8 dvhlmod ( 𝜑𝑈 ∈ LMod )
14 5 6 7 lspsneq0 ( ( 𝑈 ∈ LMod ∧ 𝑋𝑉 ) → ( ( 𝑁 ‘ { 𝑋 } ) = { 𝑍 } ↔ 𝑋 = 𝑍 ) )
15 13 9 14 syl2anc ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) = { 𝑍 } ↔ 𝑋 = 𝑍 ) )
16 12 15 bitr2d ( 𝜑 → ( 𝑋 = 𝑍 ↔ ( 𝑁 ‘ { 𝑋 } ) = ( 𝐼0 ) ) )