Theorem dihord5a

Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014)

	Ref	Expression
Hypotheses	dihord.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihord.l	⊢ ≤ = ( le ‘ 𝐾 )
	dihord.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihord.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dihord5a	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → 𝑋 ≤ 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		dihord.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihord.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dihord.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dihord.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
5		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
6		eqid	⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
7		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
8		eqid	⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
9		eqid	⊢ ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
10	1 2 3 5 6 7 8 9 4	dihord5apre	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → 𝑋 ≤ 𝑌 )