Step |
Hyp |
Ref |
Expression |
1 |
|
tcphval.n |
β’ πΊ = ( toβPreHil β π ) |
2 |
|
eqidd |
β’ ( β€ β ( Base β π ) = ( Base β π ) ) |
3 |
|
eqid |
β’ ( Base β π ) = ( Base β π ) |
4 |
1 3
|
tcphbas |
β’ ( Base β π ) = ( Base β πΊ ) |
5 |
4
|
a1i |
β’ ( β€ β ( Base β π ) = ( Base β πΊ ) ) |
6 |
|
eqid |
β’ ( +g β π ) = ( +g β π ) |
7 |
1 6
|
tchplusg |
β’ ( +g β π ) = ( +g β πΊ ) |
8 |
7
|
a1i |
β’ ( β€ β ( +g β π ) = ( +g β πΊ ) ) |
9 |
8
|
oveqdr |
β’ ( ( β€ β§ ( π₯ β ( Base β π ) β§ π¦ β ( Base β π ) ) ) β ( π₯ ( +g β π ) π¦ ) = ( π₯ ( +g β πΊ ) π¦ ) ) |
10 |
|
eqidd |
β’ ( β€ β ( Scalar β π ) = ( Scalar β π ) ) |
11 |
|
eqid |
β’ ( Scalar β π ) = ( Scalar β π ) |
12 |
1 11
|
tcphsca |
β’ ( Scalar β π ) = ( Scalar β πΊ ) |
13 |
12
|
a1i |
β’ ( β€ β ( Scalar β π ) = ( Scalar β πΊ ) ) |
14 |
|
eqid |
β’ ( Base β ( Scalar β π ) ) = ( Base β ( Scalar β π ) ) |
15 |
|
eqid |
β’ ( Β·π β π ) = ( Β·π β π ) |
16 |
1 15
|
tcphvsca |
β’ ( Β·π β π ) = ( Β·π β πΊ ) |
17 |
16
|
a1i |
β’ ( β€ β ( Β·π β π ) = ( Β·π β πΊ ) ) |
18 |
17
|
oveqdr |
β’ ( ( β€ β§ ( π₯ β ( Base β ( Scalar β π ) ) β§ π¦ β ( Base β π ) ) ) β ( π₯ ( Β·π β π ) π¦ ) = ( π₯ ( Β·π β πΊ ) π¦ ) ) |
19 |
|
eqid |
β’ ( Β·π β π ) = ( Β·π β π ) |
20 |
1 19
|
tcphip |
β’ ( Β·π β π ) = ( Β·π β πΊ ) |
21 |
20
|
a1i |
β’ ( β€ β ( Β·π β π ) = ( Β·π β πΊ ) ) |
22 |
21
|
oveqdr |
β’ ( ( β€ β§ ( π₯ β ( Base β π ) β§ π¦ β ( Base β π ) ) ) β ( π₯ ( Β·π β π ) π¦ ) = ( π₯ ( Β·π β πΊ ) π¦ ) ) |
23 |
2 5 9 10 13 14 18 22
|
phlpropd |
β’ ( β€ β ( π β PreHil β πΊ β PreHil ) ) |
24 |
23
|
mptru |
β’ ( π β PreHil β πΊ β PreHil ) |