Theorem tgrpbase

Description: The base set of the translation group is the set of all translations (for a fiducial co-atom W ). (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.c	⊢ 𝐶 = ( Base ‘ 𝐺 )
Assertion	tgrpbase	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐶 = 𝑇 )

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.c	⊢ 𝐶 = ( Base ‘ 𝐺 )
5	1 2 3	tgrpset	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } )
6	5	fveq2d	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ 𝐺 ) = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } ) )
7	2	fvexi	⊢ 𝑇 ∈ V
8		eqid	⊢ { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } = { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ }
9	8	grpbase	⊢ ( 𝑇 ∈ V → 𝑇 = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } ) )
10	7 9	ax-mp	⊢ 𝑇 = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } )
11	6 4 10	3eqtr4g	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐶 = 𝑇 )