Theorem ltrncoval

Description: Two ways to express value of translation composition. (Contributed by NM, 31-May-2013)

	Ref	Expression
Hypotheses	ltrnel.l	⊢ ≤ = ( le ‘ 𝐾 )
	ltrnel.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	ltrnel.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ltrnel.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ltrncoval	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltrnel.l	⊢ ≤ = ( le ‘ 𝐾 )
2		ltrnel.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		ltrnel.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		ltrnel.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6		simp2r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → 𝐺 ∈ 𝑇 )
7		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8	7 3 4	ltrn1o	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
9	5 6 8	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
10		f1of	⊢ ( 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) → 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) )
11	9 10	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) )
12	7 2	atbase	⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
13	12	3ad2ant3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
14		fvco3	⊢ ( ( 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) )
15	11 13 14	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) )