Metamath Proof Explorer


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 ( ( 𝐾𝑉𝑊𝐻 ) → 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 ( ( 𝐾𝑉𝑊𝐻 ) → 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ } )
6 5 fveq2d ( ( 𝐾𝑉𝑊𝐻 ) → ( Base ‘ 𝐺 ) = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ } ) )
7 2 fvexi 𝑇 ∈ V
8 eqid { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ } = { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ }
9 8 grpbase ( 𝑇 ∈ V → 𝑇 = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ } ) )
10 7 9 ax-mp 𝑇 = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ } )
11 6 4 10 3eqtr4g ( ( 𝐾𝑉𝑊𝐻 ) → 𝐶 = 𝑇 )