Theorem psrmulrg 15126

Description: The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)

Hypotheses

Ref	Expression
psrmulr.s	|- S = ( I mPwSer R )
psrmulr.b	|- B = ( Base ` S )
psrmulr.m	|- .x. = ( .r ` R )
psrmulr.t	|- .xb = ( .r ` S )
psrmulr.d	|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }

Assertion

Ref	Expression
psrmulrg	|- ( ( I e. V /\ R e. W ) -> .xb = ( 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 ) ) ) ) ) ) ) )

Distinct variable groups:   .x. , f, g, k, x   B, f, g, k, x   D, f, g, k, x, y   f, I, g, h, k, x, y   R, f, g, k, x   f, V, g, k, x   f, W, g, k, x

Allowed substitution hints:   B( y, h)   D( h)   R( y, h)   S( x, y, f, g, h, k)   .xb ( x, y, f, g, h, k)   .x. ( y, h)   V( y, h)   W( y, h)

Proof of Theorem psrmulrg

Step	Hyp	Ref	Expression
1		psrmulr.s	. . . 4 |- S = ( I mPwSer R )
2		eqid 2238	. . . 4 |- ( Base ` R ) = ( Base ` R )
3		eqid 2238	. . . 4 |- ( +g ` R ) = ( +g ` R )
4		psrmulr.m	. . . 4 |- .x. = ( .r ` R )
5		eqid 2238	. . . 4 |- ( TopOpen ` R ) = ( TopOpen ` R )
6		psrmulr.d	. . . 4 |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
7		psrmulr.b	. . . . 5 |- B = ( Base ` S )
8		simpl 109	. . . . 5 |- ( ( I e. V /\ R e. W ) -> I e. V )
9		simpr 110	. . . . 5 |- ( ( I e. V /\ R e. W ) -> R e. W )
10	1, 2, 6, 7, 8, 9	psrbasg 15118	. . . 4 |- ( ( I e. V /\ R e. W ) -> B = ( ( Base ` R ) ^m D ) )
11		eqid 2238	. . . 4 |- ( oF ( +g ` R ) |` ( B X. B ) ) = ( oF ( +g ` R ) |` ( B X. B ) )
12		eqid 2238	. . . 4 |- ( 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 ) ) ) ) ) ) ) = ( 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 ) ) ) ) ) ) )
13		eqid 2238	. . . 4 |- ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) = ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) )
14		eqidd 2239	. . . 4 |- ( ( I e. V /\ R e. W ) -> ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) = ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) )
15	1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 8, 9	psrval 15102	. . 3 |- ( ( I e. V /\ R e. W ) -> S = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( B X. B ) ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) )
16		eqid 2238	. . . . . . 7 |- ( +g ` S ) = ( +g ` S )
17	1, 7, 3, 16	psrplusgg 15122	. . . . . 6 |- ( ( I e. V /\ R e. W ) -> ( +g ` S ) = ( oF ( +g ` R ) |` ( B X. B ) ) )
18	17	opeq2d 3911	. . . . 5 |- ( ( I e. V /\ R e. W ) -> <. ( +g ` ndx ) , ( +g ` S ) >. = <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( B X. B ) ) >. )
19	18	tpeq2d 3801	. . . 4 |- ( ( I e. V /\ R e. W ) -> { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( B X. B ) ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } )
20	19	uneq1d 3382	. . 3 |- ( ( I e. V /\ R e. W ) -> ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( B X. B ) ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) )
21	15, 20	eqtr4d 2274	. 2 |- ( ( I e. V /\ R e. W ) -> S = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) )
22		basfn 13463	. . . . 5 |- Base Fn _V
23		fnpsr 15103	. . . . . . 7 |- mPwSer Fn ( _V X. _V )
24	8	elexd 2835	. . . . . . 7 |- ( ( I e. V /\ R e. W ) -> I e. _V )
25	9	elexd 2835	. . . . . . 7 |- ( ( I e. V /\ R e. W ) -> R e. _V )
26		fnovex 6118	. . . . . . 7 |- ( ( mPwSer Fn ( _V X. _V ) /\ I e. _V /\ R e. _V ) -> ( I mPwSer R ) e. _V )
27	23, 24, 25, 26	mp3an2i 1383	. . . . . 6 |- ( ( I e. V /\ R e. W ) -> ( I mPwSer R ) e. _V )
28	1, 27	eqeltrid 2325	. . . . 5 |- ( ( I e. V /\ R e. W ) -> S e. _V )
29		funfvex 5712	. . . . . 6 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
30	29	funfni 5483	. . . . 5 |- ( ( Base Fn _V /\ S e. _V ) -> ( Base ` S ) e. _V )
31	22, 28, 30	sylancr 418	. . . 4 |- ( ( I e. V /\ R e. W ) -> ( Base ` S ) e. _V )
32	7, 31	eqeltrid 2325	. . 3 |- ( ( I e. V /\ R e. W ) -> B e. _V )
33		plusgslid 13518	. . . . 5 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
34	33	slotex 13431	. . . 4 |- ( S e. _V -> ( +g ` S ) e. _V )
35	28, 34	syl 14	. . 3 |- ( ( I e. V /\ R e. W ) -> ( +g ` S ) e. _V )
36		mpoexga 6448	. . . 4 |- ( ( B e. _V /\ B e. _V ) -> ( 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 ) ) ) ) ) ) ) e. _V )
37	32, 32, 36	syl2anc 415	. . 3 |- ( ( I e. V /\ R e. W ) -> ( 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 ) ) ) ) ) ) ) e. _V )
38		funfvex 5712	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
39	38	funfni 5483	. . . . 5 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
40	22, 25, 39	sylancr 418	. . . 4 |- ( ( I e. V /\ R e. W ) -> ( Base ` R ) e. _V )
41		mpoexga 6448	. . . 4 |- ( ( ( Base ` R ) e. _V /\ B e. _V ) -> ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) e. _V )
42	40, 32, 41	syl2anc 415	. . 3 |- ( ( I e. V /\ R e. W ) -> ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) e. _V )
43		fnmap 6929	. . . . . . 7 |- ^m Fn ( _V X. _V )
44		nn0ex 9574	. . . . . . . 8 |- NN0 e. _V
45	44	a1i 9	. . . . . . 7 |- ( ( I e. V /\ R e. W ) -> NN0 e. _V )
46		fnovex 6118	. . . . . . 7 |- ( ( ^m Fn ( _V X. _V ) /\ NN0 e. _V /\ I e. _V ) -> ( NN0 ^m I ) e. _V )
47	43, 45, 24, 46	mp3an2i 1383	. . . . . 6 |- ( ( I e. V /\ R e. W ) -> ( NN0 ^m I ) e. _V )
48	6, 47	rabexd 4281	. . . . 5 |- ( ( I e. V /\ R e. W ) -> D e. _V )
49		topnfn 13650	. . . . . . 7 |- TopOpen Fn _V
50		funfvex 5712	. . . . . . . 8 |- ( ( Fun TopOpen /\ R e. dom TopOpen ) -> ( TopOpen ` R ) e. _V )
51	50	funfni 5483	. . . . . . 7 |- ( ( TopOpen Fn _V /\ R e. _V ) -> ( TopOpen ` R ) e. _V )
52	49, 25, 51	sylancr 418	. . . . . 6 |- ( ( I e. V /\ R e. W ) -> ( TopOpen ` R ) e. _V )
53		snexg 4321	. . . . . 6 |- ( ( TopOpen ` R ) e. _V -> { ( TopOpen ` R ) } e. _V )
54	52, 53	syl 14	. . . . 5 |- ( ( I e. V /\ R e. W ) -> { ( TopOpen ` R ) } e. _V )
55	48, 54	xpexd 4890	. . . 4 |- ( ( I e. V /\ R e. W ) -> ( D X. { ( TopOpen ` R ) } ) e. _V )
56		ptex 13670	. . . 4 |- ( ( D X. { ( TopOpen ` R ) } ) e. _V -> ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) e. _V )
57	55, 56	syl 14	. . 3 |- ( ( I e. V /\ R e. W ) -> ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) e. _V )
58	32, 35, 37, 9, 42, 57	psrvalstrd 15104	. 2 |- ( ( I e. V /\ R e. W ) -> ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) Struct <. 1 , 9 >. )
59		mulrslid 13538	. 2 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
60		snsstp3 3867	. . . 4 |- { <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. }
61		ssun1 3392	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } )
62	60, 61	sstri 3257	. . 3 |- { <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } )
63	62	a1i 9	. 2 |- ( ( I e. V /\ R e. W ) -> { <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` S ) >. , <. ( .r ` ndx ) , ( 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 ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. ( Base ` R ) , f e. B |-> ( ( D X. { x } ) oF .x. f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) )
64		psrmulr.t	. 2 |- .xb = ( .r ` S )
65	21, 58, 59, 63, 37, 64	strslfv3 13450	1 |- ( ( I e. V /\ R e. W ) -> .xb = ( 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 ) ) ) ) ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   u. cun 3218   C_ wss 3220   {csn 3709   {ctp 3711   <.cop 3712   class class class wbr 4130   |-> cmpt 4192   X. cxp 4772   `'ccnv 4773   |` cres 4776   "cima 4777   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   oFcof 6300   oRcofr 6301   ^m cmap 6922   Fincfn 7022   1c1 8181   <_ cle 8362   - cmin 8499   NNcn 9307   9c9 9365   NN0cn0 9568   ndxcnx 13401   Basecbs 13404   +g cplusg 13483   .rcmulr 13484  Scalarcsca 13486   .scvsca 13487  TopSetcts 13489   TopOpenctopn 13646   Xt_cpt 13661   gsum_g cgsu 14202   mPwSer cmps 15097

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-map 6924  df-ixp 6981  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-7 9371  df-8 9372  df-9 9373  df-n0 9569  df-z 9650  df-uz 9932  df-fz 10423  df-struct 13406  df-ndx 13407  df-slot 13408  df-base 13410  df-plusg 13496  df-mulr 13497  df-sca 13499  df-vsca 13500  df-tset 13502  df-rest 13647  df-topn 13648  df-topgen 13666  df-pt 13667  df-psr 15099

This theorem is used by:  psrmulfval  15127