Theorem psrbas 19378

Description: The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 8-Jul-2019.)

Hypotheses

Ref	Expression
psrbas.s	|- S = ( I mPwSer R )
psrbas.k	|- K = ( Base ` R )
psrbas.d	|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
psrbas.b	|- B = ( Base ` S )
psrbas.i	|- ( ph -> I e. V )

Assertion

Ref	Expression
psrbas	|- ( ph -> B = ( K ^m D ) )

Distinct variable group:   f, I

Allowed substitution hints:   ph( f)   B( f)   D( f)   R( f)   S( f)   K( f)   V( f)

Proof of Theorem psrbas

Dummy variables g h k x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrbas.s	. . . . 5 |- S = ( I mPwSer R )
2		psrbas.k	. . . . 5 |- K = ( Base ` R )
3		eqid 2622	. . . . 5 |- ( +g ` R ) = ( +g ` R )
4		eqid 2622	. . . . 5 |- ( .r ` R ) = ( .r ` R )
5		eqid 2622	. . . . 5 |- ( TopOpen ` R ) = ( TopOpen ` R )
6		psrbas.d	. . . . 5 |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
7		eqidd 2623	. . . . 5 |- ( ( ph /\ R e. _V ) -> ( K ^m D ) = ( K ^m D ) )
8		eqid 2622	. . . . 5 |- ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) = ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) )
9		eqid 2622	. . . . 5 |- ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) = ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) )
10		eqid 2622	. . . . 5 |- ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) = ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) )
11		eqidd 2623	. . . . 5 |- ( ( ph /\ R e. _V ) -> ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) = ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) )
12		psrbas.i	. . . . . 6 |- ( ph -> I e. V )
13	12	adantr 481	. . . . 5 |- ( ( ph /\ R e. _V ) -> I e. V )
14		simpr 477	. . . . 5 |- ( ( ph /\ R e. _V ) -> R e. _V )
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14	psrval 19362	. . . 4 |- ( ( ph /\ R e. _V ) -> S = ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) )
16	15	fveq2d 6195	. . 3 |- ( ( ph /\ R e. _V ) -> ( Base ` S ) = ( Base ` ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) ) )
17		psrbas.b	. . 3 |- B = ( Base ` S )
18		ovex 6678	. . . 4 |- ( K ^m D ) e. _V
19		psrvalstr 19363	. . . . 5 |- ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) Struct <. 1 , 9 >.
20		baseid 15919	. . . . 5 |- Base = Slot ( Base ` ndx )
21		snsstp1 4347	. . . . . 6 |- { <. ( Base ` ndx ) , ( K ^m D ) >. } C_ { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. }
22		ssun1 3776	. . . . . 6 |- { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } C_ ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } )
23	21, 22	sstri 3612	. . . . 5 |- { <. ( Base ` ndx ) , ( K ^m D ) >. } C_ ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } )
24	19, 20, 23	strfv 15907	. . . 4 |- ( ( K ^m D ) e. _V -> ( K ^m D ) = ( Base ` ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) ) )
25	18, 24	ax-mp 5	. . 3 |- ( K ^m D ) = ( Base ` ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) )
26	16, 17, 25	3eqtr4g 2681	. 2 |- ( ( ph /\ R e. _V ) -> B = ( K ^m D ) )
27		reldmpsr 19361	. . . . . . . 8 |- Rel dom mPwSer
28	27	ovprc2 6685	. . . . . . 7 |- ( -. R e. _V -> ( I mPwSer R ) = (/) )
29	28	adantl 482	. . . . . 6 |- ( ( ph /\ -. R e. _V ) -> ( I mPwSer R ) = (/) )
30	1, 29	syl5eq 2668	. . . . 5 |- ( ( ph /\ -. R e. _V ) -> S = (/) )
31	30	fveq2d 6195	. . . 4 |- ( ( ph /\ -. R e. _V ) -> ( Base ` S ) = ( Base ` (/) ) )
32		base0 15912	. . . 4 |- (/) = ( Base ` (/) )
33	31, 17, 32	3eqtr4g 2681	. . 3 |- ( ( ph /\ -. R e. _V ) -> B = (/) )
34		fvprc 6185	. . . . . 6 |- ( -. R e. _V -> ( Base ` R ) = (/) )
35	34	adantl 482	. . . . 5 |- ( ( ph /\ -. R e. _V ) -> ( Base ` R ) = (/) )
36	2, 35	syl5eq 2668	. . . 4 |- ( ( ph /\ -. R e. _V ) -> K = (/) )
37	6	fczpsrbag 19367	. . . . . . 7 |- ( I e. V -> ( x e. I |-> 0 ) e. D )
38	12, 37	syl 17	. . . . . 6 |- ( ph -> ( x e. I |-> 0 ) e. D )
39	38	adantr 481	. . . . 5 |- ( ( ph /\ -. R e. _V ) -> ( x e. I |-> 0 ) e. D )
40		ne0i 3921	. . . . 5 |- ( ( x e. I |-> 0 ) e. D -> D =/= (/) )
41	39, 40	syl 17	. . . 4 |- ( ( ph /\ -. R e. _V ) -> D =/= (/) )
42		fvex 6201	. . . . . 6 |- ( Base ` R ) e. _V
43	2, 42	eqeltri 2697	. . . . 5 |- K e. _V
44		ovex 6678	. . . . . 6 |- ( NN0 ^m I ) e. _V
45	6, 44	rabex2 4815	. . . . 5 |- D e. _V
46	43, 45	map0 7898	. . . 4 |- ( ( K ^m D ) = (/) <-> ( K = (/) /\ D =/= (/) ) )
47	36, 41, 46	sylanbrc 698	. . 3 |- ( ( ph /\ -. R e. _V ) -> ( K ^m D ) = (/) )
48	33, 47	eqtr4d 2659	. 2 |- ( ( ph /\ -. R e. _V ) -> B = ( K ^m D ) )
49	26, 48	pm2.61dan 832	1 |- ( ph -> B = ( K ^m D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   u. cun 3572   (/)c0 3915   {csn 4177   {ctp 4181   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   oRcofr 6896   ^m cmap 7857   Fincfn 7955   0cc0 9936   1c1 9937   <_ cle 10075   - cmin 10266   NNcn 11020   9c9 11077   NN0cn0 11292   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945  TopSetcts 15947   TopOpenctopn 16082   Xt_cpt 16099   gsum_g cgsu 16101   mPwSer cmps 19351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-psr 19356

This theorem is referenced by:  psrelbas  19379  psrplusg  19381  psraddcl  19383  psrmulr  19384  psrmulcllem  19387  psrsca  19389  psrvscafval  19390  psrvscacl  19393  psr0cl  19394  psrnegcl  19396  psr1cl  19402  resspsrbas  19415  resspsradd  19416  resspsrmul  19417  subrgpsr  19419  mvrf  19424  mplmon  19463  mplcoe1  19465  opsrtoslem2  19485  psr1bas  19561  psrbaspropd  19605  ply1plusgfvi  19612