Step |
Hyp |
Ref |
Expression |
1 |
|
opsrbas.s |
|- S = ( I mPwSer R ) |
2 |
|
opsrbas.o |
|- O = ( ( I ordPwSer R ) ` T ) |
3 |
|
opsrbas.t |
|- ( ph -> T C_ ( I X. I ) ) |
4 |
|
opsrbaslem.1 |
|- E = Slot N |
5 |
|
opsrbaslem.2 |
|- N e. NN |
6 |
|
opsrbaslem.3 |
|- N < ; 1 0 |
7 |
4 5
|
ndxid |
|- E = Slot ( E ` ndx ) |
8 |
5
|
nnrei |
|- N e. RR |
9 |
8 6
|
ltneii |
|- N =/= ; 1 0 |
10 |
4 5
|
ndxarg |
|- ( E ` ndx ) = N |
11 |
|
plendx |
|- ( le ` ndx ) = ; 1 0 |
12 |
10 11
|
neeq12i |
|- ( ( E ` ndx ) =/= ( le ` ndx ) <-> N =/= ; 1 0 ) |
13 |
9 12
|
mpbir |
|- ( E ` ndx ) =/= ( le ` ndx ) |
14 |
7 13
|
setsnid |
|- ( E ` S ) = ( E ` ( S sSet <. ( le ` ndx ) , ( le ` O ) >. ) ) |
15 |
|
eqid |
|- ( le ` O ) = ( le ` O ) |
16 |
|
simprl |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> I e. _V ) |
17 |
|
simprr |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> R e. _V ) |
18 |
3
|
adantr |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> T C_ ( I X. I ) ) |
19 |
1 2 15 16 17 18
|
opsrval2 |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> O = ( S sSet <. ( le ` ndx ) , ( le ` O ) >. ) ) |
20 |
19
|
fveq2d |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> ( E ` O ) = ( E ` ( S sSet <. ( le ` ndx ) , ( le ` O ) >. ) ) ) |
21 |
14 20
|
eqtr4id |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> ( E ` S ) = ( E ` O ) ) |
22 |
|
0fv |
|- ( (/) ` T ) = (/) |
23 |
22
|
eqcomi |
|- (/) = ( (/) ` T ) |
24 |
|
reldmpsr |
|- Rel dom mPwSer |
25 |
24
|
ovprc |
|- ( -. ( I e. _V /\ R e. _V ) -> ( I mPwSer R ) = (/) ) |
26 |
|
reldmopsr |
|- Rel dom ordPwSer |
27 |
26
|
ovprc |
|- ( -. ( I e. _V /\ R e. _V ) -> ( I ordPwSer R ) = (/) ) |
28 |
27
|
fveq1d |
|- ( -. ( I e. _V /\ R e. _V ) -> ( ( I ordPwSer R ) ` T ) = ( (/) ` T ) ) |
29 |
23 25 28
|
3eqtr4a |
|- ( -. ( I e. _V /\ R e. _V ) -> ( I mPwSer R ) = ( ( I ordPwSer R ) ` T ) ) |
30 |
29
|
adantl |
|- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( I mPwSer R ) = ( ( I ordPwSer R ) ` T ) ) |
31 |
30 1 2
|
3eqtr4g |
|- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> S = O ) |
32 |
31
|
fveq2d |
|- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( E ` S ) = ( E ` O ) ) |
33 |
21 32
|
pm2.61dan |
|- ( ph -> ( E ` S ) = ( E ` O ) ) |