Metamath Proof Explorer

Theorem cdlemn2a

Description: Part of proof of Lemma N of Crawley p. 121. (Contributed by NM, 24-Feb-2014)

Ref Expression
Hypotheses cdlemn2a.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemn2a.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemn2a.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemn2a.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemn2a.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemn2a.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemn2a.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemn2a.o ⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
cdlemn2a.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
cdlemn2a.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
cdlemn2a.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
cdlemn2a.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑄 ) = 𝑆 )
Assertion cdlemn2a ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑁 ‘ { ⟨ 𝐹 , 𝑂 ⟩ } ) ⊆ ( 𝐼 ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 cdlemn2a.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemn2a.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemn2a.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemn2a.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemn2a.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemn2a.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemn2a.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemn2a.o ⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
9 cdlemn2a.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
10 cdlemn2a.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
11 cdlemn2a.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
12 cdlemn2a.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑄 ) = 𝑆 )
13 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
15 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) )
16 2 4 5 6 12 ltrniotacl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
17 13 14 15 16 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝐹 ∈ 𝑇 )
18 1 5 6 7 8 10 9 11 dib1dim2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑁 ‘ { ⟨ 𝐹 , 𝑂 ⟩ } ) )
19 13 17 18 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑁 ‘ { ⟨ 𝐹 , 𝑂 ⟩ } ) )
20 1 2 3 4 5 6 7 12 cdlemn2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 )
21 1 5 6 7 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐵 )
22 13 17 21 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐵 )
23 2 5 6 7 trlle ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 )
24 13 17 23 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 )
25 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) )
26 1 2 5 9 dibord ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑅 ‘ 𝐹 ) ∈ 𝐵 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) ⊆ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) )
27 13 22 24 25 26 syl121anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) ⊆ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) )
28 20 27 mpbird ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) ⊆ ( 𝐼 ‘ 𝑋 ) )
29 19 28 eqsstrrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑁 ‘ { ⟨ 𝐹 , 𝑂 ⟩ } ) ⊆ ( 𝐼 ‘ 𝑋 ) )