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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 ( ( 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 )