Theorem psrsca 19389

Description: The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)

Hypotheses

Ref	Expression
psrsca.s	|- S = ( I mPwSer R )
psrsca.i	|- ( ph -> I e. V )
psrsca.r	|- ( ph -> R e. W )

Assertion

Ref	Expression
psrsca	|- ( ph -> R = (Scalar ` S ) )

Proof of Theorem psrsca

Dummy variables f h w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrsca.r	. . 3 |- ( ph -> R e. W )
2		psrvalstr 19363	. . . 4 |- ( { <. ( Base ` ndx ) , ( Base ` S ) >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. ( Base ` S ) |-> ( ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { x } ) oF ( .r ` R ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( TopOpen ` R ) } ) ) >. } ) Struct <. 1 , 9 >.
3		scaid 16014	. . . 4 |- Scalar = Slot (Scalar ` ndx )
4		snsstp1 4347	. . . . 5 |- { <. (Scalar ` ndx ) , R >. } C_ { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. ( Base ` S ) |-> ( ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { x } ) oF ( .r ` R ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( TopOpen ` R ) } ) ) >. }
5		ssun2 3777	. . . . 5 |- { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. ( Base ` S ) |-> ( ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { x } ) oF ( .r ` R ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( TopOpen ` R ) } ) ) >. } C_ ( { <. ( Base ` ndx ) , ( Base ` S ) >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. ( Base ` S ) |-> ( ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { x } ) oF ( .r ` R ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( TopOpen ` R ) } ) ) >. } )
6	4, 5	sstri 3612	. . . 4 |- { <. (Scalar ` ndx ) , R >. } C_ ( { <. ( Base ` ndx ) , ( Base ` S ) >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. ( Base ` S ) |-> ( ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { x } ) oF ( .r ` R ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( TopOpen ` R ) } ) ) >. } )
7	2, 3, 6	strfv 15907	. . 3 |- ( R e. W -> R = (Scalar ` ( { <. ( Base ` ndx ) , ( Base ` S ) >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. ( Base ` S ) |-> ( ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { x } ) oF ( .r ` R ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( TopOpen ` R ) } ) ) >. } ) ) )
8	1, 7	syl 17	. 2 |- ( ph -> R = (Scalar ` ( { <. ( Base ` ndx ) , ( Base ` S ) >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. ( Base ` S ) |-> ( ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { x } ) oF ( .r ` R ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( TopOpen ` R ) } ) ) >. } ) ) )
9		psrsca.s	. . . 4 |- S = ( I mPwSer R )
10		eqid 2622	. . . 4 |- ( Base ` R ) = ( Base ` R )
11		eqid 2622	. . . 4 |- ( +g ` R ) = ( +g ` R )
12		eqid 2622	. . . 4 |- ( .r ` R ) = ( .r ` R )
13		eqid 2622	. . . 4 |- ( TopOpen ` R ) = ( TopOpen ` R )
14		eqid 2622	. . . 4 |- { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
15		eqid 2622	. . . . 5 |- ( Base ` S ) = ( Base ` S )
16		psrsca.i	. . . . 5 |- ( ph -> I e. V )
17	9, 10, 14, 15, 16	psrbas 19378	. . . 4 |- ( ph -> ( Base ` S ) = ( ( Base ` R ) ^m { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } ) )
18		eqid 2622	. . . . 5 |- ( +g ` S ) = ( +g ` S )
19	9, 15, 11, 18	psrplusg 19381	. . . 4 |- ( +g ` S ) = ( oF ( +g ` R ) |` ( ( Base ` S ) X. ( Base ` S ) ) )
20		eqid 2622	. . . . 5 |- ( .r ` S ) = ( .r ` S )
21	9, 15, 12, 20, 14	psrmulr 19384	. . . 4 |- ( .r ` S ) = ( f e. ( Base ` S ) , z e. ( Base ` S ) |-> ( w e. { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } |-> ( R gsumg ( x e. { y e. { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } | y oR <_ w } |-> ( ( f ` x ) ( .r ` R ) ( z ` ( w oF - x ) ) ) ) ) ) )
22		eqid 2622	. . . 4 |- ( x e. ( Base ` R ) , f e. ( Base ` S ) |-> ( ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { x } ) oF ( .r ` R ) f ) ) = ( x e. ( Base ` R ) , f e. ( Base ` S ) |-> ( ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { x } ) oF ( .r ` R ) f ) )
23		eqidd 2623	. . . 4 |- ( ph -> ( Xt_ ` ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( TopOpen ` R ) } ) ) = ( Xt_ ` ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( TopOpen ` R ) } ) ) )
24	9, 10, 11, 12, 13, 14, 17, 19, 21, 22, 23, 16, 1	psrval 19362	. . 3 |- ( ph -> S = ( { <. ( Base ` ndx ) , ( Base ` S ) >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. ( Base ` S ) |-> ( ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { x } ) oF ( .r ` R ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( TopOpen ` R ) } ) ) >. } ) )
25	24	fveq2d 6195	. 2 |- ( ph -> (Scalar ` S ) = (Scalar ` ( { <. ( Base ` ndx ) , ( Base ` S ) >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. ( Base ` S ) |-> ( ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { x } ) oF ( .r ` R ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( TopOpen ` R ) } ) ) >. } ) ) )
26	8, 25	eqtr4d 2659	1 |- ( ph -> R = (Scalar ` S ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   u. cun 3572   {csn 4177   {ctp 4181   <.cop 4183   X. cxp 5112   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   ^m cmap 7857   Fincfn 7955   1c1 9937   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   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:  psrlmod  19401  psrassa  19414  mpllsslem  19435  mplsca  19445  opsrsca  19483  opsrassa  19489  ply1lss  19566  opsrlmod  19616