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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 = ( 0g ‘ 𝑈 )
dih0vb.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
dih0vb.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
dih0vb.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
Assertion dih0vbN ( 𝜑 → ( 𝑋 = 𝑍 ↔ ( 𝑁 ‘ { 𝑋 } ) = ( 𝐼 ‘ 0 ) ) )

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 = ( 0g ‘ 𝑈 )
7 dih0vb.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8 dih0vb.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 dih0vb.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
10 2 1 3 4 6 dih0 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ 0 ) = { 𝑍 } )
11 8 10 syl ⊢ ( 𝜑 → ( 𝐼 ‘ 0 ) = { 𝑍 } )
12 11 eqeq2d ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) = ( 𝐼 ‘ 0 ) ↔ ( 𝑁 ‘ { 𝑋 } ) = { 𝑍 } ) )
13 1 4 8 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
14 5 6 7 lspsneq0 ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 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 } ) = ( 𝐼 ‘ 0 ) ) )