Metamath Proof Explorer

Theorem cdlemn5pre

Description: Part of proof of Lemma N of Crawley p. 121 line 32. (Contributed by NM, 25-Feb-2014)

Ref Expression
Hypotheses cdlemn5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemn5.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemn5.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemn5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemn5.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemn5.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
cdlemn5.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
cdlemn5.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
cdlemn5.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
cdlemn5.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
cdlemn5.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
cdlemn5.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemn5.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
cdlemn5.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
cdlemn5.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 )
cdlemn5.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑅 )
cdlemn5.m ⊢ 𝑀 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑄 ) = 𝑅 )
Assertion cdlemn5pre ( ( ( 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 cdlemn5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemn5.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemn5.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemn5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemn5.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemn5.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemn5.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
8 cdlemn5.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemn5.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
10 cdlemn5.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
11 cdlemn5.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
12 cdlemn5.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
13 cdlemn5.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
14 cdlemn5.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
15 cdlemn5.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 )
16 cdlemn5.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑅 )
17 cdlemn5.m ⊢ 𝑀 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑄 ) = 𝑅 )
18 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
19 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) )
20 2 4 5 10 12 9 6 14 16 diclspsn ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑅 ) = ( 𝑁 ‘ { ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ } ) )
21 18 19 20 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐽 ‘ 𝑅 ) = ( 𝑁 ‘ { ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ } ) )
22 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
23 1 2 4 10 5 12 11 6 15 16 17 14 7 cdlemn4a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝑁 ‘ { ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ } ) ⊆ ( ( 𝑁 ‘ { ⟨ 𝐹 , ( I ↾ 𝑇 ) ⟩ } ) ⊕ ( 𝑁 ‘ { ⟨ 𝑀 , 𝑂 ⟩ } ) ) )
24 18 22 19 23 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑁 ‘ { ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ } ) ⊆ ( ( 𝑁 ‘ { ⟨ 𝐹 , ( I ↾ 𝑇 ) ⟩ } ) ⊕ ( 𝑁 ‘ { ⟨ 𝑀 , 𝑂 ⟩ } ) ) )
25 2 4 5 10 12 9 6 14 15 diclspsn ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑄 ) = ( 𝑁 ‘ { ⟨ 𝐹 , ( I ↾ 𝑇 ) ⟩ } ) )
26 18 22 25 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐽 ‘ 𝑄 ) = ( 𝑁 ‘ { ⟨ 𝐹 , ( I ↾ 𝑇 ) ⟩ } ) )
27 26 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝑁 ‘ { ⟨ 𝑀 , 𝑂 ⟩ } ) ) = ( ( 𝑁 ‘ { ⟨ 𝐹 , ( I ↾ 𝑇 ) ⟩ } ) ⊕ ( 𝑁 ‘ { ⟨ 𝑀 , 𝑂 ⟩ } ) ) )
28 24 27 sseqtrrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑁 ‘ { ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ } ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝑁 ‘ { ⟨ 𝑀 , 𝑂 ⟩ } ) ) )
29 5 6 18 dvhlmod ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝑈 ∈ LMod )
30 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
31 30 lsssssubg ⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
32 29 31 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
33 2 4 5 6 9 30 diclss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
34 18 22 33 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
35 32 34 sseldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) )
36 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) )
37 1 2 5 6 8 30 diblss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑈 ) )
38 18 36 37 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑈 ) )
39 32 38 sseldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝑈 ) )
40 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
41 1 2 3 4 5 12 40 11 8 6 14 17 cdlemn2a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑁 ‘ { ⟨ 𝑀 , 𝑂 ⟩ } ) ⊆ ( 𝐼 ‘ 𝑋 ) )
42 7 lsmless2 ⊢ ( ( ( 𝐽 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝑁 ‘ { ⟨ 𝑀 , 𝑂 ⟩ } ) ⊆ ( 𝐼 ‘ 𝑋 ) ) → ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝑁 ‘ { ⟨ 𝑀 , 𝑂 ⟩ } ) ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) )
43 35 39 41 42 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝑁 ‘ { ⟨ 𝑀 , 𝑂 ⟩ } ) ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) )
44 28 43 sstrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑁 ‘ { ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ } ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) )
45 21 44 eqsstrd ⊢ ( ( ( 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 ) ) )