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Metamath Proof Explorer

Theorem cdlemn

Description: Lemma N of Crawley p. 121 line 27. (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 cdlemn ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ) → ( 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ↔ ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 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 1 2 3 4 5 8 9 6 7 cdlemn5 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) )
11 10 3expia ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ) → ( 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) → ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) )
12 1 2 3 4 5 6 7 8 9 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 ) )
13 12 3expia ⊢ ( ( ( 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 ) ) )
14 11 13 impbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ) → ( 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ↔ ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) )