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 |
1
|
tgrpfset |
⊢ ( 𝐾 ∈ 𝑉 → ( TGrp ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 〈 ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 } ) ) |
5 |
4
|
fveq1d |
⊢ ( 𝐾 ∈ 𝑉 → ( ( TGrp ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ { 〈 ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 } ) ‘ 𝑊 ) ) |
6 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
7 |
6
|
opeq2d |
⊢ ( 𝑤 = 𝑊 → 〈 ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) 〉 = 〈 ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) 〉 ) |
8 |
|
eqidd |
⊢ ( 𝑤 = 𝑊 → ( 𝑓 ∘ 𝑔 ) = ( 𝑓 ∘ 𝑔 ) ) |
9 |
6 6 8
|
mpoeq123dv |
⊢ ( 𝑤 = 𝑊 → ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) |
10 |
9
|
opeq2d |
⊢ ( 𝑤 = 𝑊 → 〈 ( +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 |
⊢ ( 𝑤 = 𝑊 → { 〈 ( 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 |
|
eqid |
⊢ ( 𝑤 ∈ 𝐻 ↦ { 〈 ( 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 |
|
prex |
⊢ { 〈 ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 } ∈ V |
14 |
11 12 13
|
fvmpt |
⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ { 〈 ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 } ) ‘ 𝑊 ) = { 〈 ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 } ) |
15 |
2
|
opeq2i |
⊢ 〈 ( Base ‘ ndx ) , 𝑇 〉 = 〈 ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) 〉 |
16 |
|
eqid |
⊢ ( 𝑓 ∘ 𝑔 ) = ( 𝑓 ∘ 𝑔 ) |
17 |
2 2 16
|
mpoeq123i |
⊢ ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∘ 𝑔 ) ) |
18 |
17
|
opeq2i |
⊢ 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 = 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 |
19 |
15 18
|
preq12i |
⊢ { 〈 ( Base ‘ ndx ) , 𝑇 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 } = { 〈 ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 } |
20 |
14 19
|
eqtr4di |
⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ { 〈 ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 } ) ‘ 𝑊 ) = { 〈 ( Base ‘ ndx ) , 𝑇 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 } ) |
21 |
5 20
|
sylan9eq |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( ( TGrp ‘ 𝐾 ) ‘ 𝑊 ) = { 〈 ( Base ‘ ndx ) , 𝑇 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 } ) |
22 |
3 21
|
syl5eq |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐺 = { 〈 ( Base ‘ ndx ) , 𝑇 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 〉 } ) |