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 ) ) ) |