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Metamath Proof Explorer

Theorem cdlemn7

Description: Part of proof of Lemma N of Crawley p. 121 line 36. (Contributed by NM, 26-Feb-2014)

Ref Expression
Hypotheses cdlemn8.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemn8.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemn8.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemn8.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemn8.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
cdlemn8.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
cdlemn8.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemn8.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
cdlemn8.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
cdlemn8.s ⊢ + = ( +g ‘ 𝑈 )
cdlemn8.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 )
cdlemn8.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑅 )
Assertion cdlemn7 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝐺 = ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) ∧ ( I ↾ 𝑇 ) = 𝑠 ) )

Proof

Step Hyp Ref Expression
1 cdlemn8.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemn8.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemn8.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 cdlemn8.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 cdlemn8.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
6 cdlemn8.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
7 cdlemn8.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemn8.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemn8.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
10 cdlemn8.s ⊢ + = ( +g ‘ 𝑈 )
11 cdlemn8.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 )
12 cdlemn8.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑅 )
13 simp33 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) )
14 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
16 simp2r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) )
17 simp31 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → 𝑠 ∈ 𝐸 )
18 simp32 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → 𝑔 ∈ 𝑇 )
19 1 2 3 4 5 6 7 8 9 10 11 cdlemn6 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) = ⟨ ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) , 𝑠 ⟩ )
20 14 15 16 17 18 19 syl122anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) = ⟨ ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) , 𝑠 ⟩ )
21 13 20 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ⟨ ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) , 𝑠 ⟩ )
22 fvex ⊢ ( 𝑠 ‘ 𝐹 ) ∈ V
23 vex ⊢ 𝑔 ∈ V
24 22 23 coex ⊢ ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) ∈ V
25 vex ⊢ 𝑠 ∈ V
26 24 25 opth2 ⊢ ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ⟨ ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) , 𝑠 ⟩ ↔ ( 𝐺 = ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) ∧ ( I ↾ 𝑇 ) = 𝑠 ) )
27 21 26 sylib ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝐺 = ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) ∧ ( I ↾ 𝑇 ) = 𝑠 ) )