This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem tgrpfset

Description: The translation group maps for a lattice K . (Contributed by NM, 5-Jun-2013)

Ref Expression
Hypothesis tgrpset.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion tgrpfset ( 𝐾 ∈ 𝑉 → ( TGrp ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } ) )

Proof

Step Hyp Ref Expression
1 tgrpset.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 elex ⊢ ( 𝐾 ∈ 𝑉 → 𝐾 ∈ V )
3 fveq2 ⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = ( LHyp ‘ 𝐾 ) )
4 3 1 eqtr4di ⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = 𝐻 )
5 fveq2 ⊢ ( 𝑘 = 𝐾 → ( LTrn ‘ 𝑘 ) = ( LTrn ‘ 𝐾 ) )
6 5 fveq1d ⊢ ( 𝑘 = 𝐾 → ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) )
7 6 opeq2d ⊢ ( 𝑘 = 𝐾 → ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟩ = ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟩ )
8 eqidd ⊢ ( 𝑘 = 𝐾 → ( 𝑓 ∘ 𝑔 ) = ( 𝑓 ∘ 𝑔 ) )
9 6 6 8 mpoeq123dv ⊢ ( 𝑘 = 𝐾 → ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) )
10 9 opeq2d ⊢ ( 𝑘 = 𝐾 → ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ = ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ )
11 7 10 preq12d ⊢ ( 𝑘 = 𝐾 → { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } = { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } )
12 4 11 mpteq12dv ⊢ ( 𝑘 = 𝐾 → ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } ) = ( 𝑤 ∈ 𝐻 ↦ { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } ) )
13 df-tgrp ⊢ TGrp = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } ) )
14 12 13 1 mptfvmpt ⊢ ( 𝐾 ∈ V → ( TGrp ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } ) )
15 2 14 syl ⊢ ( 𝐾 ∈ 𝑉 → ( TGrp ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ } ) )