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Metamath Proof Explorer

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 )