Theorem dih0bN

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

	Ref	Expression
Hypotheses	dih0b.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dih0b.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dih0b.o	⊢ 0 = ( 0. ‘ 𝐾 )
	dih0b.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dih0b.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dih0b.z	⊢ 𝑍 = ( 0g ‘ 𝑈 )
	dih0b.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	dih0b.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	dih0bN	⊢ ( 𝜑 → ( 𝑋 = 0 ↔ ( 𝐼 ‘ 𝑋 ) = { 𝑍 } ) )

Proof

Step	Hyp	Ref	Expression
1		dih0b.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dih0b.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		dih0b.o	⊢ 0 = ( 0. ‘ 𝐾 )
4		dih0b.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
5		dih0b.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6		dih0b.z	⊢ 𝑍 = ( 0g ‘ 𝑈 )
7		dih0b.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8		dih0b.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
9	7	simpld	⊢ ( 𝜑 → 𝐾 ∈ HL )
10		hlop	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
11	1 3	op0cl	⊢ ( 𝐾 ∈ OP → 0 ∈ 𝐵 )
12	9 10 11	3syl	⊢ ( 𝜑 → 0 ∈ 𝐵 )
13	1 2 4	dih11	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) = ( 𝐼 ‘ 0 ) ↔ 𝑋 = 0 ) )
14	7 8 12 13	syl3anc	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) = ( 𝐼 ‘ 0 ) ↔ 𝑋 = 0 ) )
15	3 2 4 5 6	dih0	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ 0 ) = { 𝑍 } )
16	7 15	syl	⊢ ( 𝜑 → ( 𝐼 ‘ 0 ) = { 𝑍 } )
17	16	eqeq2d	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) = ( 𝐼 ‘ 0 ) ↔ ( 𝐼 ‘ 𝑋 ) = { 𝑍 } ) )
18	14 17	bitr3d	⊢ ( 𝜑 → ( 𝑋 = 0 ↔ ( 𝐼 ‘ 𝑋 ) = { 𝑍 } ) )