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Metamath Proof Explorer

Theorem dihord6a

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

Ref Expression
Hypotheses dihord3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihord3.l ⊢ ≤ = ( le ‘ 𝐾 )
dihord3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihord3.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihord6a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → 𝑋 ≤ 𝑌 )

Proof

Step Hyp Ref Expression
1 dihord3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihord3.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihord3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihord3.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
5 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
6 eqid ⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
7 eqid ⊢ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) )
8 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
9 eqid ⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
10 eqid ⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
11 eqid ⊢ ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
12 eqid ⊢ ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 ) = ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 )
13 1 2 5 3 6 7 8 9 4 10 11 12 dihord6apre ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → 𝑋 ≤ 𝑌 )