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Metamath Proof Explorer

Theorem tgrpopr

Description: The group operation of the translation group is function composition. (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 tgrpopr ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) )

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 tgrpset ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } )
6 5 fveq2d ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ 𝐺 ) = ( +g ‘ { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } ) )
7 2 fvexi ⊢ 𝑇 ∈ V
8 7 7 mpoex ⊢ ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ∈ V
9 eqid ⊢ { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } = { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ }
10 9 grpplusg ⊢ ( ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ∈ V → ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) = ( +g ‘ { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } ) )
11 8 10 ax-mp ⊢ ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) = ( +g ‘ { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } )
12 6 4 11 3eqtr4g ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) )