Step |
Hyp |
Ref |
Expression |
1 |
|
pmtridf1o.a |
|- ( ph -> A e. V ) |
2 |
|
pmtridf1o.x |
|- ( ph -> X e. A ) |
3 |
|
pmtridf1o.y |
|- ( ph -> Y e. A ) |
4 |
|
pmtridf1o.t |
|- T = if ( X = Y , ( _I |` A ) , ( ( pmTrsp ` A ) ` { X , Y } ) ) |
5 |
|
simpr |
|- ( ( ph /\ X = Y ) -> X = Y ) |
6 |
5
|
iftrued |
|- ( ( ph /\ X = Y ) -> if ( X = Y , ( _I |` A ) , ( ( pmTrsp ` A ) ` { X , Y } ) ) = ( _I |` A ) ) |
7 |
4 6
|
syl5eq |
|- ( ( ph /\ X = Y ) -> T = ( _I |` A ) ) |
8 |
7
|
fveq1d |
|- ( ( ph /\ X = Y ) -> ( T ` X ) = ( ( _I |` A ) ` X ) ) |
9 |
|
fvresi |
|- ( X e. A -> ( ( _I |` A ) ` X ) = X ) |
10 |
2 9
|
syl |
|- ( ph -> ( ( _I |` A ) ` X ) = X ) |
11 |
10
|
adantr |
|- ( ( ph /\ X = Y ) -> ( ( _I |` A ) ` X ) = X ) |
12 |
8 11 5
|
3eqtrd |
|- ( ( ph /\ X = Y ) -> ( T ` X ) = Y ) |
13 |
|
simpr |
|- ( ( ph /\ X =/= Y ) -> X =/= Y ) |
14 |
13
|
neneqd |
|- ( ( ph /\ X =/= Y ) -> -. X = Y ) |
15 |
14
|
iffalsed |
|- ( ( ph /\ X =/= Y ) -> if ( X = Y , ( _I |` A ) , ( ( pmTrsp ` A ) ` { X , Y } ) ) = ( ( pmTrsp ` A ) ` { X , Y } ) ) |
16 |
4 15
|
syl5eq |
|- ( ( ph /\ X =/= Y ) -> T = ( ( pmTrsp ` A ) ` { X , Y } ) ) |
17 |
16
|
fveq1d |
|- ( ( ph /\ X =/= Y ) -> ( T ` X ) = ( ( ( pmTrsp ` A ) ` { X , Y } ) ` X ) ) |
18 |
1
|
adantr |
|- ( ( ph /\ X =/= Y ) -> A e. V ) |
19 |
2
|
adantr |
|- ( ( ph /\ X =/= Y ) -> X e. A ) |
20 |
3
|
adantr |
|- ( ( ph /\ X =/= Y ) -> Y e. A ) |
21 |
|
eqid |
|- ( pmTrsp ` A ) = ( pmTrsp ` A ) |
22 |
21
|
pmtrprfv |
|- ( ( A e. V /\ ( X e. A /\ Y e. A /\ X =/= Y ) ) -> ( ( ( pmTrsp ` A ) ` { X , Y } ) ` X ) = Y ) |
23 |
18 19 20 13 22
|
syl13anc |
|- ( ( ph /\ X =/= Y ) -> ( ( ( pmTrsp ` A ) ` { X , Y } ) ` X ) = Y ) |
24 |
17 23
|
eqtrd |
|- ( ( ph /\ X =/= Y ) -> ( T ` X ) = Y ) |
25 |
12 24
|
pm2.61dane |
|- ( ph -> ( T ` X ) = Y ) |