Theorem cdlemn11

Description: Part of proof of Lemma N of Crawley p. 121 line 37. (Contributed by NM, 27-Feb-2014)

	Ref	Expression
Hypotheses	cdlemn11.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdlemn11.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemn11.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemn11.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemn11.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemn11.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
	cdlemn11.J	⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
	cdlemn11.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	cdlemn11.s	⊢ ⊕ = ( LSSum ‘ 𝑈 )
Assertion	cdlemn11	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemn11.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdlemn11.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdlemn11.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdlemn11.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdlemn11.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdlemn11.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
7		cdlemn11.J	⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
8		cdlemn11.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
9		cdlemn11.s	⊢ ⊕ = ( LSSum ‘ 𝑈 )
10		eqid	⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
11		eqid	⊢ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) )
12		eqid	⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
13		eqid	⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
14		eqid	⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
15		eqid	⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 )
16		eqid	⊢ ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) = ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 )
17		eqid	⊢ ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑅 ) = ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑅 )
18	1 2 3 4 5 10 11 12 13 14 6 7 8 15 9 16 17	cdlemn11pre	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) )