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)

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 N
opsrbaslem.2
|- N e. NN
opsrbaslem.3
|- N < ; 1 0
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 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 ) )