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 ) )

Metamath Proof Explorer

Theorem opsrbaslem

Proof