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Metamath Proof Explorer

Theorem cdlemg47a

Description: TODO: fix comment. TODO: Use this above in place of ( FP ) = P antecedents? (Contributed by NM, 5-Jun-2013)

Ref Expression
Hypotheses cdlemg46.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemg46.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg46.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg47a ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 cdlemg46.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemg46.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 cdlemg46.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
5 simp2r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → 𝐺 ∈ 𝑇 )
6 1 2 3 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → 𝐺 : 𝐵 –1-1-onto→ 𝐵 )
7 4 5 6 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → 𝐺 : 𝐵 –1-1-onto→ 𝐵 )
8 f1of ⊢ ( 𝐺 : 𝐵 –1-1-onto→ 𝐵 → 𝐺 : 𝐵 ⟶ 𝐵 )
9 7 8 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → 𝐺 : 𝐵 ⟶ 𝐵 )
10 fcoi1 ⊢ ( 𝐺 : 𝐵 ⟶ 𝐵 → ( 𝐺 ∘ ( I ↾ 𝐵 ) ) = 𝐺 )
11 9 10 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝐺 ∘ ( I ↾ 𝐵 ) ) = 𝐺 )
12 simp3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → 𝐹 = ( I ↾ 𝐵 ) )
13 12 coeq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝐺 ∘ 𝐹 ) = ( 𝐺 ∘ ( I ↾ 𝐵 ) ) )
14 12 coeq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝐹 ∘ 𝐺 ) = ( ( I ↾ 𝐵 ) ∘ 𝐺 ) )
15 fcoi2 ⊢ ( 𝐺 : 𝐵 ⟶ 𝐵 → ( ( I ↾ 𝐵 ) ∘ 𝐺 ) = 𝐺 )
16 9 15 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( ( I ↾ 𝐵 ) ∘ 𝐺 ) = 𝐺 )
17 14 16 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝐹 ∘ 𝐺 ) = 𝐺 )
18 11 13 17 3eqtr4rd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ 𝐹 ) )