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Metamath Proof Explorer

Theorem cdlemn11c

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

Ref Expression
Hypotheses cdlemn11a.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemn11a.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemn11a.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemn11a.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemn11a.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemn11a.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
cdlemn11a.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.d ⊢ + = ( +g ‘ 𝑈 )
cdlemn11a.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
cdlemn11a.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 )
cdlemn11a.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑁 )
Assertion cdlemn11c ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑄 ) ∃ 𝑧 ∈ ( 𝐼 ‘ 𝑋 ) ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) )

Proof

Step Hyp Ref Expression
1 cdlemn11a.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemn11a.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemn11a.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemn11a.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemn11a.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemn11a.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemn11a.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
8 cdlemn11a.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemn11a.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
10 cdlemn11a.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
11 cdlemn11a.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
12 cdlemn11a.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
13 cdlemn11a.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
14 cdlemn11a.d ⊢ + = ( +g ‘ 𝑈 )
15 cdlemn11a.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
16 cdlemn11a.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 )
17 cdlemn11a.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑁 )
18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 cdlemn11b ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ ∈ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) )
19 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
20 5 13 19 dvhlmod ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → 𝑈 ∈ LMod )
21 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
22 21 lsssssubg ⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
23 20 22 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
24 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
25 2 4 5 13 12 21 diclss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
26 19 24 25 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
27 23 26 sseldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) )
28 simp23l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → 𝑋 ∈ 𝐵 )
29 simp23r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → 𝑋 ≤ 𝑊 )
30 1 2 5 13 11 21 diblss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑈 ) )
31 19 28 29 30 syl12anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑈 ) )
32 23 31 sseldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝑈 ) )
33 14 15 lsmelval ⊢ ( ( ( 𝐽 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ ∈ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ↔ ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑄 ) ∃ 𝑧 ∈ ( 𝐼 ‘ 𝑋 ) ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) ) )
34 27 32 33 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ ∈ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ↔ ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑄 ) ∃ 𝑧 ∈ ( 𝐼 ‘ 𝑋 ) ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) ) )
35 18 34 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑄 ) ∃ 𝑧 ∈ ( 𝐼 ‘ 𝑋 ) ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) )