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Metamath Proof Explorer

Theorem dihord2

Description: Part of proof after Lemma N of Crawley p. 122. Reverse ordering property. TODO: do we need -. X .<_ W and -. Y .<_ W ? (Contributed by NM, 4-Mar-2014)

Ref Expression
Hypotheses dihord2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihord2.l ⊢ ≤ = ( le ‘ 𝐾 )
dihord2.j ⊢ ∨ = ( join ‘ 𝐾 )
dihord2.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihord2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihord2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihord2.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
dihord2.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
dihord2.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihord2.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
Assertion dihord2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝑋 ≤ 𝑌 )

Proof

Step Hyp Ref Expression
1 dihord2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihord2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihord2.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dihord2.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 dihord2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 dihord2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 dihord2.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
8 dihord2.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
9 dihord2.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
10 dihord2.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
11 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
12 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
13 eqid ⊢ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) )
14 eqid ⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
15 eqid ⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
16 eqid ⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 )
17 eqid ⊢ ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑁 ) = ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑁 )
18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 dihord2pre2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) )
19 simp31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
20 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 )
21 18 19 20 3brtr3d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝑋 ≤ 𝑌 )