Metamath Proof Explorer

Theorem opsrbaslemOLD

Description: Obsolete version of opsrbaslem as of 1-Nov-2024. 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) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypotheses opsrbas.s 
|- S = ( I mPwSer R )
opsrbas.o 
|- O = ( ( I ordPwSer R ) ` T )
opsrbas.t 
|- ( ph -> T C_ ( I X. I ) )
opsrbaslemOLD.1 
|- E = Slot N
opsrbaslemOLD.2 
|- N e. NN
opsrbaslemOLD.3 
|- N < ; 1 0
Assertion opsrbaslemOLD 
|- ( 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 opsrbaslemOLD.1 
 |-  E = Slot N
5 opsrbaslemOLD.2 
 |-  N e. NN
6 opsrbaslemOLD.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 ) )