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 4
|
tgrpopr |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ) |
6 |
5
|
3adant3 |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → + = ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ) |
7 |
6
|
oveqd |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → ( 𝑋 + 𝑌 ) = ( 𝑋 ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 𝑌 ) ) |
8 |
|
simp3l |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → 𝑋 ∈ 𝑇 ) |
9 |
|
simp3r |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → 𝑌 ∈ 𝑇 ) |
10 |
|
coexg |
⊢ ( ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) → ( 𝑋 ∘ 𝑌 ) ∈ V ) |
11 |
10
|
3ad2ant3 |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → ( 𝑋 ∘ 𝑌 ) ∈ V ) |
12 |
|
coeq1 |
⊢ ( 𝑓 = 𝑋 → ( 𝑓 ∘ 𝑔 ) = ( 𝑋 ∘ 𝑔 ) ) |
13 |
|
coeq2 |
⊢ ( 𝑔 = 𝑌 → ( 𝑋 ∘ 𝑔 ) = ( 𝑋 ∘ 𝑌 ) ) |
14 |
|
eqid |
⊢ ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) = ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) |
15 |
12 13 14
|
ovmpog |
⊢ ( ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ∧ ( 𝑋 ∘ 𝑌 ) ∈ V ) → ( 𝑋 ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 𝑌 ) = ( 𝑋 ∘ 𝑌 ) ) |
16 |
8 9 11 15
|
syl3anc |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → ( 𝑋 ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 𝑌 ) = ( 𝑋 ∘ 𝑌 ) ) |
17 |
7 16
|
eqtrd |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇 ) ) → ( 𝑋 + 𝑌 ) = ( 𝑋 ∘ 𝑌 ) ) |