Metamath Proof Explorer

Theorem cdlemn9

Description: Part of proof of Lemma N of Crawley p. 121 line 36. (Contributed by NM, 27-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 cdlemn9 ( ( ( 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 ‘ 𝑄 ) = 𝑅 )

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 1 2 3 4 5 6 7 8 9 10 11 12 cdlemn8 ⊢ ( ( ( 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 = ( 𝐺 ∘ ◡ 𝐹 ) )
14 13 fveq1d ⊢ ( ( ( 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 ‘ 𝑄 ) = ( ( 𝐺 ∘ ◡ 𝐹 ) ‘ 𝑄 ) )
15 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 ) )
16 2 3 4 5 lhpocnel2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
17 16 3ad2ant1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
18 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 ) )
19 2 3 4 7 11 ltrniotacl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
20 15 17 18 19 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → 𝐹 ∈ 𝑇 )
21 1 4 7 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )
22 15 20 21 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )
23 f1ocnv ⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → ◡ 𝐹 : 𝐵 –1-1-onto→ 𝐵 )
24 f1of ⊢ ( ◡ 𝐹 : 𝐵 –1-1-onto→ 𝐵 → ◡ 𝐹 : 𝐵 ⟶ 𝐵 )
25 22 23 24 3syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ◡ 𝐹 : 𝐵 ⟶ 𝐵 )
26 simp2ll ⊢ ( ( ( 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 )
27 1 3 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
28 26 27 syl ⊢ ( ( ( 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 ∈ 𝐵 )
29 fvco3 ⊢ ( ( ◡ 𝐹 : 𝐵 ⟶ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( ( 𝐺 ∘ ◡ 𝐹 ) ‘ 𝑄 ) = ( 𝐺 ‘ ( ◡ 𝐹 ‘ 𝑄 ) ) )
30 25 28 29 syl2anc ⊢ ( ( ( 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 ∘ ◡ 𝐹 ) ‘ 𝑄 ) = ( 𝐺 ‘ ( ◡ 𝐹 ‘ 𝑄 ) ) )
31 2 3 4 7 11 ltrniotacnvval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ◡ 𝐹 ‘ 𝑄 ) = 𝑃 )
32 15 17 18 31 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( ◡ 𝐹 ‘ 𝑄 ) = 𝑃 )
33 32 fveq2d ⊢ ( ( ( 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 ‘ ( ◡ 𝐹 ‘ 𝑄 ) ) = ( 𝐺 ‘ 𝑃 ) )
34 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 ) )
35 2 3 4 7 12 ltrniotaval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝐺 ‘ 𝑃 ) = 𝑅 )
36 15 17 34 35 syl3anc ⊢ ( ( ( 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 ‘ 𝑃 ) = 𝑅 )
37 33 36 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 ‘ ( ◡ 𝐹 ‘ 𝑄 ) ) = 𝑅 )
38 14 30 37 3eqtrd ⊢ ( ( ( 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 ‘ 𝑄 ) = 𝑅 )