Theorem psrvscafval 19390

Description: The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
psrvsca.s	|- S = ( I mPwSer R )
psrvsca.n	|- .xb = ( .s ` S )
psrvsca.k	|- K = ( Base ` R )
psrvsca.b	|- B = ( Base ` S )
psrvsca.m	|- .x. = ( .r ` R )
psrvsca.d	|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }

Assertion

Ref	Expression
psrvscafval	|- .xb = ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) )

Distinct variable groups:   x, f, B   f, h, I, x   f, K, x   D, f, x   R, f, x   .x. , f, x   .xb , f, x

Allowed substitution hints:   B( h)   D( h)   R( h)   S( x, f, h)   .xb ( h)   .x. ( h)   K( h)

Proof of Theorem psrvscafval

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

Step	Hyp	Ref	Expression
1		psrvsca.s	. . . . 5 |- S = ( I mPwSer R )
2		psrvsca.k	. . . . 5 |- K = ( Base ` R )
3		eqid 2622	. . . . 5 |- ( +g ` R ) = ( +g ` R )
4		psrvsca.m	. . . . 5 |- .x. = ( .r ` R )
5		eqid 2622	. . . . 5 |- ( TopOpen ` R ) = ( TopOpen ` R )
6		psrvsca.d	. . . . 5 |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
7		psrvsca.b	. . . . . 6 |- B = ( Base ` S )
8		simpl 473	. . . . . 6 |- ( ( I e. _V /\ R e. _V ) -> I e. _V )
9	1, 2, 6, 7, 8	psrbas 19378	. . . . 5 |- ( ( I e. _V /\ R e. _V ) -> B = ( K ^m D ) )
10		eqid 2622	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
11	1, 7, 3, 10	psrplusg 19381	. . . . 5 |- ( +g ` S ) = ( oF ( +g ` R ) |` ( B X. B ) )
12		eqid 2622	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
13	1, 7, 4, 12, 6	psrmulr 19384	. . . . 5 |- ( .r ` S ) = ( f e. B , g e. B |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( f ` x ) .x. ( g ` ( k oF - x ) ) ) ) ) ) )
14		eqid 2622	. . . . 5 |- ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) = ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) )
15		eqidd 2623	. . . . 5 |- ( ( I e. _V /\ R e. _V ) -> ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) = ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) )
16		simpr 477	. . . . 5 |- ( ( I e. _V /\ R e. _V ) -> R e. _V )
17	1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 15, 8, 16	psrval 19362	. . . 4 |- ( ( I e. _V /\ R e. _V ) -> S = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) )
18	17	fveq2d 6195	. . 3 |- ( ( I e. _V /\ R e. _V ) -> ( .s ` S ) = ( .s ` ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) ) )
19		psrvsca.n	. . 3 |- .xb = ( .s ` S )
20		fvex 6201	. . . . . 6 |- ( Base ` R ) e. _V
21	2, 20	eqeltri 2697	. . . . 5 |- K e. _V
22		fvex 6201	. . . . . 6 |- ( Base ` S ) e. _V
23	7, 22	eqeltri 2697	. . . . 5 |- B e. _V
24	21, 23	mpt2ex 7247	. . . 4 |- ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) e. _V
25		psrvalstr 19363	. . . . 5 |- ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) Struct <. 1 , 9 >.
26		vscaid 16016	. . . . 5 |- .s = Slot ( .s ` ndx )
27		snsstp2 4348	. . . . . 6 |- { <. ( .s ` ndx ) , ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. } C_ { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. }
28		ssun2 3777	. . . . . 6 |- { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } )
29	27, 28	sstri 3612	. . . . 5 |- { <. ( .s ` ndx ) , ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } )
30	25, 26, 29	strfv 15907	. . . 4 |- ( ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) e. _V -> ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) = ( .s ` ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) ) )
31	24, 30	ax-mp 5	. . 3 |- ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) = ( .s ` ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( .r ` S ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) )
32	18, 19, 31	3eqtr4g 2681	. 2 |- ( ( I e. _V /\ R e. _V ) -> .xb = ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) )
33		eqid 2622	. . . . . 6 |- (/) = (/)
34		fn0 6011	. . . . . 6 |- ( (/) Fn (/) <-> (/) = (/) )
35	33, 34	mpbir 221	. . . . 5 |- (/) Fn (/)
36		reldmpsr 19361	. . . . . . . . . 10 |- Rel dom mPwSer
37	36	ovprc 6683	. . . . . . . . 9 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPwSer R ) = (/) )
38	1, 37	syl5eq 2668	. . . . . . . 8 |- ( -. ( I e. _V /\ R e. _V ) -> S = (/) )
39	38	fveq2d 6195	. . . . . . 7 |- ( -. ( I e. _V /\ R e. _V ) -> ( .s ` S ) = ( .s ` (/) ) )
40		df-vsca 15958	. . . . . . . 8 |- .s = Slot 6
41	40	str0 15911	. . . . . . 7 |- (/) = ( .s ` (/) )
42	39, 19, 41	3eqtr4g 2681	. . . . . 6 |- ( -. ( I e. _V /\ R e. _V ) -> .xb = (/) )
43	36, 1, 7	elbasov 15921	. . . . . . . . . 10 |- ( f e. B -> ( I e. _V /\ R e. _V ) )
44	43	con3i 150	. . . . . . . . 9 |- ( -. ( I e. _V /\ R e. _V ) -> -. f e. B )
45	44	eq0rdv 3979	. . . . . . . 8 |- ( -. ( I e. _V /\ R e. _V ) -> B = (/) )
46	45	xpeq2d 5139	. . . . . . 7 |- ( -. ( I e. _V /\ R e. _V ) -> ( K X. B ) = ( K X. (/) ) )
47		xp0 5552	. . . . . . 7 |- ( K X. (/) ) = (/)
48	46, 47	syl6eq 2672	. . . . . 6 |- ( -. ( I e. _V /\ R e. _V ) -> ( K X. B ) = (/) )
49	42, 48	fneq12d 5983	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> ( .xb Fn ( K X. B ) <-> (/) Fn (/) ) )
50	35, 49	mpbiri 248	. . . 4 |- ( -. ( I e. _V /\ R e. _V ) -> .xb Fn ( K X. B ) )
51		fnov 6768	. . . 4 |- ( .xb Fn ( K X. B ) <-> .xb = ( x e. K , f e. B |-> ( x .xb f ) ) )
52	50, 51	sylib 208	. . 3 |- ( -. ( I e. _V /\ R e. _V ) -> .xb = ( x e. K , f e. B |-> ( x .xb f ) ) )
53	44	pm2.21d 118	. . . . . 6 |- ( -. ( I e. _V /\ R e. _V ) -> ( f e. B -> ( ( D X. { x } ) oF .x. f ) = ( x .xb f ) ) )
54	53	a1d 25	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> ( x e. K -> ( f e. B -> ( ( D X. { x } ) oF .x. f ) = ( x .xb f ) ) ) )
55	54	3imp 1256	. . . 4 |- ( ( -. ( I e. _V /\ R e. _V ) /\ x e. K /\ f e. B ) -> ( ( D X. { x } ) oF .x. f ) = ( x .xb f ) )
56	55	mpt2eq3dva 6719	. . 3 |- ( -. ( I e. _V /\ R e. _V ) -> ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) = ( x e. K , f e. B |-> ( x .xb f ) ) )
57	52, 56	eqtr4d 2659	. 2 |- ( -. ( I e. _V /\ R e. _V ) -> .xb = ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) )
58	32, 57	pm2.61i 176	1 |- .xb = ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   u. cun 3572   (/)c0 3915   {csn 4177   {ctp 4181   <.cop 4183   X. cxp 5112   `'ccnv 5113   "cima 5117   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   ^m cmap 7857   Fincfn 7955   1c1 9937   NNcn 11020   6c6 11074   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:  psrvsca  19391