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Metamath Proof Explorer

Theorem opsrbaslem

Description: Get a component of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015) (Revised by Mario Carneiro, 2-Oct-2015) (Revised by AV, 9-Sep-2021) (Revised by AV, 1-Nov-2024)

Ref Expression
Hypotheses opsrbas.s 
|- S = ( I mPwSer R )
opsrbas.o 
|- O = ( ( I ordPwSer R ) ` T )
opsrbas.t 
|- ( ph -> T C_ ( I X. I ) )
opsrbaslem.1 
|- E = Slot ( E ` ndx )
opsrbaslem.2 
|- ( E ` ndx ) =/= ( le ` ndx )
Assertion opsrbaslem 
|- ( ph -> ( E ` S ) = ( E ` O ) )

Proof

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 ( E ` ndx )
5 opsrbaslem.2 
 |-  ( E ` ndx ) =/= ( le ` ndx )
6 4 5 setsnid 
 |-  ( E ` S ) = ( E ` ( S sSet <. ( le ` ndx ) , ( le ` O ) >. ) )
7 eqid 
 |-  ( le ` O ) = ( le ` O )
8 simprl 
 |-  ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> I e. _V )
9 simprr 
 |-  ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> R e. _V )
10 3 adantr 
 |-  ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> T C_ ( I X. I ) )
11 1 2 7 8 9 10 opsrval2 
 |-  ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> O = ( S sSet <. ( le ` ndx ) , ( le ` O ) >. ) )
12 11 fveq2d 
 |-  ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> ( E ` O ) = ( E ` ( S sSet <. ( le ` ndx ) , ( le ` O ) >. ) ) )
13 6 12 eqtr4id 
 |-  ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> ( E ` S ) = ( E ` O ) )
14 0fv 
 |-  ( (/) ` T ) = (/)
15 14 eqcomi 
 |-  (/) = ( (/) ` T )
16 reldmpsr 
 |-  Rel dom mPwSer
17 16 ovprc 
 |-  ( -. ( I e. _V /\ R e. _V ) -> ( I mPwSer R ) = (/) )
18 reldmopsr 
 |-  Rel dom ordPwSer
19 18 ovprc 
 |-  ( -. ( I e. _V /\ R e. _V ) -> ( I ordPwSer R ) = (/) )
20 19 fveq1d 
 |-  ( -. ( I e. _V /\ R e. _V ) -> ( ( I ordPwSer R ) ` T ) = ( (/) ` T ) )
21 15 17 20 3eqtr4a 
 |-  ( -. ( I e. _V /\ R e. _V ) -> ( I mPwSer R ) = ( ( I ordPwSer R ) ` T ) )
22 21 1 2 3eqtr4g 
 |-  ( -. ( I e. _V /\ R e. _V ) -> S = O )
23 22 fveq2d 
 |-  ( -. ( I e. _V /\ R e. _V ) -> ( E ` S ) = ( E ` O ) )
24 23 adantl 
 |-  ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( E ` S ) = ( E ` O ) )
25 13 24 pm2.61dan 
 |-  ( ph -> ( E ` S ) = ( E ` O ) )