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 ) )