Theorem tgrpov

Description: The group operation value of the translation group is the composition of translations. (Contributed by NM, 5-Jun-2013)

	Ref	Expression
Hypotheses	tgrpset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	tgrpset.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	tgrpset.g	⊢ 𝐺 = ( ( TGrp ‘ 𝐾 ) ‘ 𝑊 )
	tgrp.o	⊢ + = ( +g ‘ 𝐺 )
Assertion	tgrpov	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → ( 𝑋 + 𝑌 ) = ( 𝑋 ∘ 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		tgrpset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		tgrpset.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		tgrpset.g	⊢ 𝐺 = ( ( TGrp ‘ 𝐾 ) ‘ 𝑊 )
4		tgrp.o	⊢ + = ( +g ‘ 𝐺 )
5	1 2 3 4	tgrpopr	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) )
6	5	3adant3	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → + = ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) )
7	6	oveqd	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → ( 𝑋 + 𝑌 ) = ( 𝑋 ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 𝑌 ) )
8		simp3l	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → 𝑋 ∈ 𝑇 )
9		simp3r	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → 𝑌 ∈ 𝑇 )
10		coexg	⊢ ( ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) → ( 𝑋 ∘ 𝑌 ) ∈ V )
11	10	3ad2ant3	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → ( 𝑋 ∘ 𝑌 ) ∈ V )
12		coeq1	⊢ ( 𝑓 = 𝑋 → ( 𝑓 ∘ 𝑔 ) = ( 𝑋 ∘ 𝑔 ) )
13		coeq2	⊢ ( 𝑔 = 𝑌 → ( 𝑋 ∘ 𝑔 ) = ( 𝑋 ∘ 𝑌 ) )
14		eqid	⊢ ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) = ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) )
15	12 13 14	ovmpog	⊢ ( ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ∧ ( 𝑋 ∘ 𝑌 ) ∈ V ) → ( 𝑋 ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 𝑌 ) = ( 𝑋 ∘ 𝑌 ) )
16	8 9 11 15	syl3anc	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → ( 𝑋 ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 𝑌 ) = ( 𝑋 ∘ 𝑌 ) )
17	7 16	eqtrd	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → ( 𝑋 + 𝑌 ) = ( 𝑋 ∘ 𝑌 ) )