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Metamath Proof Explorer

Theorem cdlemn11pre

Description: Part of proof of Lemma N of Crawley p. 121 line 37. TODO: combine cdlemn11a , cdlemn11b , cdlemn11c , cdlemn11pre into one? (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 cdlemn11pre ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) )

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 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 ) )
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 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 ) )
21 2 4 5 6 8 10 12 16 dicelval3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑦 ∈ ( 𝐽 ‘ 𝑄 ) ↔ ∃ 𝑠 ∈ 𝐸 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ) )
22 19 20 21 syl2anc ⊢ ( ( ( 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 ) ↔ ∃ 𝑠 ∈ 𝐸 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ) )
23 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) )
24 1 2 5 8 9 7 11 dibelval3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝑧 ∈ ( 𝐼 ‘ 𝑋 ) ↔ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) )
25 19 23 24 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( 𝑧 ∈ ( 𝐼 ‘ 𝑋 ) ↔ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) )
26 22 25 anbi12d ⊢ ( ( ( 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 ) ) ↔ ( ∃ 𝑠 ∈ 𝐸 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) ) )
27 reeanv ⊢ ( ∃ 𝑠 ∈ 𝐸 ∃ 𝑔 ∈ 𝑇 ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) ↔ ( ∃ 𝑠 ∈ 𝐸 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) )
28 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
29 simpl21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
30 simpl22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) )
31 simpl23 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) )
32 simpr1r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → 𝑔 ∈ 𝑇 )
33 simpr1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → 𝑠 ∈ 𝐸 )
34 simpr3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) )
35 1 2 4 5 6 7 8 10 13 14 16 17 cdlemn9 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑔 ‘ 𝑄 ) = 𝑁 )
36 28 29 30 33 32 34 35 syl123anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑔 ‘ 𝑄 ) = 𝑁 )
37 simpr2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 )
38 1 2 3 4 5 8 9 cdlemn10 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑁 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) )
39 28 29 30 31 32 36 37 38 syl133anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) )
40 39 3exp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) → ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) ) )
41 oveq12 ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( 𝑦 + 𝑧 ) = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) )
42 41 eqeq2d ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) ↔ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) )
43 42 imbi1d ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ↔ ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) )
44 43 imbi2d ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) ↔ ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) ) )
45 44 biimprd ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) → ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) ) )
46 45 com23 ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) ) )
47 46 impr ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) )
48 47 com12 ⊢ ( ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) → ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) )
49 40 48 syl6 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) → ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) ) )
50 49 rexlimdvv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( ∃ 𝑠 ∈ 𝐸 ∃ 𝑔 ∈ 𝑇 ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) )
51 27 50 biimtrrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( ( ∃ 𝑠 ∈ 𝐸 𝑦 = ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ ∧ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( 𝑦 + 𝑧 ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) )
52 26 51 sylbid ⊢ ( ( ( 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 ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) ) )
53 52 rexlimdvv ⊢ ( ( ( 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 ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) ) )
54 18 53 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → 𝑁 ≤ ( 𝑄 ∨ 𝑋 ) )