Step |
Hyp |
Ref |
Expression |
1 |
|
tgplacthmeo.1 |
⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 + 𝑥 ) ) |
2 |
|
tgplacthmeo.2 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
3 |
|
tgplacthmeo.3 |
⊢ + = ( +g ‘ 𝐺 ) |
4 |
|
tgplacthmeo.4 |
⊢ 𝐽 = ( TopOpen ‘ 𝐺 ) |
5 |
|
tgptmd |
⊢ ( 𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd ) |
6 |
1 2 3 4
|
tmdlactcn |
⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn 𝐽 ) ) |
7 |
5 6
|
sylan |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn 𝐽 ) ) |
8 |
|
tgpgrp |
⊢ ( 𝐺 ∈ TopGrp → 𝐺 ∈ Grp ) |
9 |
|
eqid |
⊢ ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) = ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) |
10 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
11 |
9 2 3 10
|
grplactcnv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ 𝐴 ) : 𝑋 –1-1-onto→ 𝑋 ∧ ◡ ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ 𝐴 ) = ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) ) ) ) |
12 |
8 11
|
sylan |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ 𝐴 ) : 𝑋 –1-1-onto→ 𝑋 ∧ ◡ ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ 𝐴 ) = ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) ) ) ) |
13 |
12
|
simprd |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → ◡ ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ 𝐴 ) = ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) ) ) |
14 |
9 2
|
grplactfval |
⊢ ( 𝐴 ∈ 𝑋 → ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ 𝐴 ) = ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 + 𝑥 ) ) ) |
15 |
14
|
adantl |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ 𝐴 ) = ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 + 𝑥 ) ) ) |
16 |
15 1
|
eqtr4di |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ 𝐴 ) = 𝐹 ) |
17 |
16
|
cnveqd |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → ◡ ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ 𝐴 ) = ◡ 𝐹 ) |
18 |
2 10
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) ∈ 𝑋 ) |
19 |
8 18
|
sylan |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) ∈ 𝑋 ) |
20 |
9 2
|
grplactfval |
⊢ ( ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) ∈ 𝑋 → ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ ( ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) + 𝑥 ) ) ) |
21 |
19 20
|
syl |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑔 ∈ 𝑋 ↦ ( 𝑥 ∈ 𝑋 ↦ ( 𝑔 + 𝑥 ) ) ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ ( ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) + 𝑥 ) ) ) |
22 |
13 17 21
|
3eqtr3d |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → ◡ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) + 𝑥 ) ) ) |
23 |
|
eqid |
⊢ ( 𝑥 ∈ 𝑋 ↦ ( ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) + 𝑥 ) ) = ( 𝑥 ∈ 𝑋 ↦ ( ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) + 𝑥 ) ) |
24 |
23 2 3 4
|
tmdlactcn |
⊢ ( ( 𝐺 ∈ TopMnd ∧ ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) ∈ 𝑋 ) → ( 𝑥 ∈ 𝑋 ↦ ( ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) + 𝑥 ) ) ∈ ( 𝐽 Cn 𝐽 ) ) |
25 |
5 19 24
|
syl2an2r |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑥 ∈ 𝑋 ↦ ( ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) + 𝑥 ) ) ∈ ( 𝐽 Cn 𝐽 ) ) |
26 |
22 25
|
eqeltrd |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → ◡ 𝐹 ∈ ( 𝐽 Cn 𝐽 ) ) |
27 |
|
ishmeo |
⊢ ( 𝐹 ∈ ( 𝐽 Homeo 𝐽 ) ↔ ( 𝐹 ∈ ( 𝐽 Cn 𝐽 ) ∧ ◡ 𝐹 ∈ ( 𝐽 Cn 𝐽 ) ) ) |
28 |
7 26 27
|
sylanbrc |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋 ) → 𝐹 ∈ ( 𝐽 Homeo 𝐽 ) ) |