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 ) )

Metamath Proof Explorer

Theorem opsrbaslem

Proof