Metamath Proof Explorer

Theorem cdlemn8

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 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 ) )

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 coass ⊢ ( ( ◡ 𝐹 ∘ 𝐹 ) ∘ 𝑔 ) = ( ◡ 𝐹 ∘ ( 𝐹 ∘ 𝑔 ) )
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 2 3 4 5 lhpocnel2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
16 15 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 ) )
17 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 ) )
18 2 3 4 7 11 ltrniotacl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
19 14 16 17 18 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 )
20 1 4 7 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )
21 14 19 20 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 )
22 f1ococnv1 ⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ 𝐵 ) )
23 21 22 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 ⟩ ) ) ) → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ 𝐵 ) )
24 23 coeq1d ⊢ ( ( ( 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 ∘ 𝐹 ) ∘ 𝑔 ) = ( ( I ↾ 𝐵 ) ∘ 𝑔 ) )
25 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 )
26 1 4 7 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ) → 𝑔 : 𝐵 –1-1-onto→ 𝐵 )
27 14 25 26 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 : 𝐵 –1-1-onto→ 𝐵 )
28 f1of ⊢ ( 𝑔 : 𝐵 –1-1-onto→ 𝐵 → 𝑔 : 𝐵 ⟶ 𝐵 )
29 fcoi2 ⊢ ( 𝑔 : 𝐵 ⟶ 𝐵 → ( ( I ↾ 𝐵 ) ∘ 𝑔 ) = 𝑔 )
30 27 28 29 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 ⟩ ) ) ) → ( ( I ↾ 𝐵 ) ∘ 𝑔 ) = 𝑔 )
31 24 30 eqtr2d ⊢ ( ( ( 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 ∘ 𝐹 ) ∘ 𝑔 ) )
32 1 2 3 4 5 6 7 8 9 10 11 12 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 ) )
33 32 simpld ⊢ ( ( ( 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 ) )
34 32 simprd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ = ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( I ↾ 𝑇 ) = 𝑠 )
35 34 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 ⟩ ) ) ) → ( ( I ↾ 𝑇 ) ‘ 𝐹 ) = ( 𝑠 ‘ 𝐹 ) )
36 fvresi ⊢ ( 𝐹 ∈ 𝑇 → ( ( I ↾ 𝑇 ) ‘ 𝐹 ) = 𝐹 )
37 19 36 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 ⟩ ) ) ) → ( ( I ↾ 𝑇 ) ‘ 𝐹 ) = 𝐹 )
38 35 37 eqtr3d ⊢ ( ( ( 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 ) = 𝐹 )
39 38 coeq1d ⊢ ( ( ( 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 ) ∘ 𝑔 ) = ( 𝐹 ∘ 𝑔 ) )
40 33 39 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 ∘ 𝑔 ) )
41 40 coeq2d ⊢ ( ( ( 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 ∘ 𝐺 ) = ( ◡ 𝐹 ∘ ( 𝐹 ∘ 𝑔 ) ) )
42 13 31 41 3eqtr4a ⊢ ( ( ( 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 ∘ 𝐺 ) )
43 4 7 ltrncnv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ◡ 𝐹 ∈ 𝑇 )
44 14 19 43 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 ∈ 𝑇 )
45 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 ) )
46 2 3 4 7 12 ltrniotacl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → 𝐺 ∈ 𝑇 )
47 14 16 45 46 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 ∈ 𝑇 )
48 4 7 ltrncom ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ◡ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ◡ 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ ◡ 𝐹 ) )
49 14 44 47 48 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 ∘ 𝐺 ) = ( 𝐺 ∘ ◡ 𝐹 ) )
50 42 49 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 = ( 𝐺 ∘ ◡ 𝐹 ) )