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 )