Step |
Hyp |
Ref |
Expression |
1 |
|
mirval.p |
|- P = ( Base ` G ) |
2 |
|
mirval.d |
|- .- = ( dist ` G ) |
3 |
|
mirval.i |
|- I = ( Itv ` G ) |
4 |
|
mirval.l |
|- L = ( LineG ` G ) |
5 |
|
mirval.s |
|- S = ( pInvG ` G ) |
6 |
|
mirval.g |
|- ( ph -> G e. TarskiG ) |
7 |
|
miduniq2.a |
|- ( ph -> A e. P ) |
8 |
|
miduniq2.b |
|- ( ph -> B e. P ) |
9 |
|
miduniq2.x |
|- ( ph -> X e. P ) |
10 |
|
miduniq2.e |
|- ( ph -> ( ( S ` A ) ` ( ( S ` B ) ` X ) ) = ( ( S ` B ) ` ( ( S ` A ) ` X ) ) ) |
11 |
|
eqid |
|- ( S ` B ) = ( S ` B ) |
12 |
1 2 3 4 5 6 8 11
|
mirf |
|- ( ph -> ( S ` B ) : P --> P ) |
13 |
12 7
|
ffvelrnd |
|- ( ph -> ( ( S ` B ) ` A ) e. P ) |
14 |
|
eqid |
|- ( ( S ` B ) ` A ) = ( ( S ` B ) ` A ) |
15 |
|
eqid |
|- ( ( S ` B ) ` ( ( S ` B ) ` X ) ) = ( ( S ` B ) ` ( ( S ` B ) ` X ) ) |
16 |
|
eqid |
|- ( ( S ` B ) ` ( ( S ` B ) ` ( ( S ` A ) ` X ) ) ) = ( ( S ` B ) ` ( ( S ` B ) ` ( ( S ` A ) ` X ) ) ) |
17 |
12 9
|
ffvelrnd |
|- ( ph -> ( ( S ` B ) ` X ) e. P ) |
18 |
|
eqid |
|- ( S ` A ) = ( S ` A ) |
19 |
1 2 3 4 5 6 7 18 9
|
mircl |
|- ( ph -> ( ( S ` A ) ` X ) e. P ) |
20 |
12 19
|
ffvelrnd |
|- ( ph -> ( ( S ` B ) ` ( ( S ` A ) ` X ) ) e. P ) |
21 |
1 2 3 4 5 6 11 14 15 16 8 7 17 20 10
|
mirauto |
|- ( ph -> ( ( S ` ( ( S ` B ) ` A ) ) ` ( ( S ` B ) ` ( ( S ` B ) ` X ) ) ) = ( ( S ` B ) ` ( ( S ` B ) ` ( ( S ` A ) ` X ) ) ) ) |
22 |
1 2 3 4 5 6 8 11 9
|
mirmir |
|- ( ph -> ( ( S ` B ) ` ( ( S ` B ) ` X ) ) = X ) |
23 |
22
|
fveq2d |
|- ( ph -> ( ( S ` ( ( S ` B ) ` A ) ) ` ( ( S ` B ) ` ( ( S ` B ) ` X ) ) ) = ( ( S ` ( ( S ` B ) ` A ) ) ` X ) ) |
24 |
1 2 3 4 5 6 8 11 19
|
mirmir |
|- ( ph -> ( ( S ` B ) ` ( ( S ` B ) ` ( ( S ` A ) ` X ) ) ) = ( ( S ` A ) ` X ) ) |
25 |
21 23 24
|
3eqtr3d |
|- ( ph -> ( ( S ` ( ( S ` B ) ` A ) ) ` X ) = ( ( S ` A ) ` X ) ) |
26 |
1 2 3 4 5 6 13 7 9 25
|
miduniq1 |
|- ( ph -> ( ( S ` B ) ` A ) = A ) |
27 |
1 2 3 4 5 6 8 11 7
|
mirinv |
|- ( ph -> ( ( ( S ` B ) ` A ) = A <-> B = A ) ) |
28 |
26 27
|
mpbid |
|- ( ph -> B = A ) |
29 |
28
|
eqcomd |
|- ( ph -> A = B ) |