Theorem fnpsr 15160

Description: The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)

Assertion

Ref	Expression
fnpsr	|- mPwSer Fn ( _V X. _V )

Proof of Theorem fnpsr

Dummy variables b d f g h i k r x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-psr 15156	. 2 |- mPwSer = ( i e. _V , r e. _V |-> [_ { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]_ [_ ( ( Base ` r ) ^m d ) / b ]_ ( { <. ( 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 ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , r >. , <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. } ) )
2		fnmap 6933	. . . . 5 |- ^m Fn ( _V X. _V )
3		nn0ex 9580	. . . . 5 |- NN0 e. _V
4		vex 2824	. . . . 5 |- i e. _V
5		fnovex 6121	. . . . 5 |- ( ( ^m Fn ( _V X. _V ) /\ NN0 e. _V /\ i e. _V ) -> ( NN0 ^m i ) e. _V )
6	2, 3, 4, 5	mp3an 1378	. . . 4 |- ( NN0 ^m i ) e. _V
7	6	rabex 4282	. . 3 |- { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } e. _V
8		basfn 13469	. . . . . 6 |- Base Fn _V
9		vex 2824	. . . . . 6 |- r e. _V
10		funfvex 5715	. . . . . . 7 |- ( ( Fun Base /\ r e. dom Base ) -> ( Base ` r ) e. _V )
11	10	funfni 5486	. . . . . 6 |- ( ( Base Fn _V /\ r e. _V ) -> ( Base ` r ) e. _V )
12	8, 9, 11	mp2an 430	. . . . 5 |- ( Base ` r ) e. _V
13		vex 2824	. . . . 5 |- d e. _V
14		fnovex 6121	. . . . 5 |- ( ( ^m Fn ( _V X. _V ) /\ ( Base ` r ) e. _V /\ d e. _V ) -> ( ( Base ` r ) ^m d ) e. _V )
15	2, 12, 13, 14	mp3an 1378	. . . 4 |- ( ( Base ` r ) ^m d ) e. _V
16		basendxnn 13466	. . . . . . 7 |- ( Base ` ndx ) e. NN
17		vex 2824	. . . . . . 7 |- b e. _V
18		opexg 4370	. . . . . . 7 |- ( ( ( Base ` ndx ) e. NN /\ b e. _V ) -> <. ( Base ` ndx ) , b >. e. _V )
19	16, 17, 18	mp2an 430	. . . . . 6 |- <. ( Base ` ndx ) , b >. e. _V
20		plusgndxnn 13524	. . . . . . 7 |- ( +g ` ndx ) e. NN
21	17	a1i 9	. . . . . . . . 9 |- ( T. -> b e. _V )
22	21, 21	ofmresex 6374	. . . . . . . 8 |- ( T. -> ( oF ( +g ` r ) |` ( b X. b ) ) e. _V )
23	22	mptru 1411	. . . . . . 7 |- ( oF ( +g ` r ) |` ( b X. b ) ) e. _V
24		opexg 4370	. . . . . . 7 |- ( ( ( +g ` ndx ) e. NN /\ ( oF ( +g ` r ) |` ( b X. b ) ) e. _V ) -> <. ( +g ` ndx ) , ( oF ( +g ` r ) |` ( b X. b ) ) >. e. _V )
25	20, 23, 24	mp2an 430	. . . . . 6 |- <. ( +g ` ndx ) , ( oF ( +g ` r ) |` ( b X. b ) ) >. e. _V
26		mulrslid 13545	. . . . . . . . 9 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
27	26	simpri 113	. . . . . . . 8 |- ( .r ` ndx ) e. NN
28	27	elexi 2834	. . . . . . 7 |- ( .r ` ndx ) e. _V
29	17, 17	mpoex 6454	. . . . . . 7 |- ( f e. b , g e. b |-> ( k e. d |-> ( r gsumg ( x e. { y e. d | y oR <_ k } |-> ( ( f ` x ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) e. _V
30	28, 29	opex 4371	. . . . . 6 |- <. ( .r ` ndx ) , ( f e. b , g e. b |-> ( k e. d |-> ( r gsumg ( x e. { y e. d | y oR <_ k } |-> ( ( f ` x ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) >. e. _V
31		tpexg 4592	. . . . . 6 |- ( ( <. ( Base ` ndx ) , b >. e. _V /\ <. ( +g ` ndx ) , ( oF ( +g ` r ) |` ( b X. b ) ) >. e. _V /\ <. ( .r ` ndx ) , ( f e. b , g e. b |-> ( k e. d |-> ( r gsumg ( x e. { y e. d | y oR <_ k } |-> ( ( f ` x ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) >. e. _V ) -> { <. ( 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 ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) >. } e. _V )
32	19, 25, 30, 31	mp3an 1378	. . . . 5 |- { <. ( 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 ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) >. } e. _V
33		scaslid 13566	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
34	33	simpri 113	. . . . . . . 8 |- (Scalar ` ndx ) e. NN
35	34	elexi 2834	. . . . . . 7 |- (Scalar ` ndx ) e. _V
36	35, 9	opex 4371	. . . . . 6 |- <. (Scalar ` ndx ) , r >. e. _V
37		vscaslid 13576	. . . . . . . . 9 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
38	37	simpri 113	. . . . . . . 8 |- ( .s ` ndx ) e. NN
39	38	elexi 2834	. . . . . . 7 |- ( .s ` ndx ) e. _V
40	12, 17	mpoex 6454	. . . . . . 7 |- ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) e. _V
41	39, 40	opex 4371	. . . . . 6 |- <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. e. _V
42		tsetndxnn 13602	. . . . . . . 8 |- (TopSet ` ndx ) e. NN
43	42	elexi 2834	. . . . . . 7 |- (TopSet ` ndx ) e. _V
44		topnfn 13657	. . . . . . . . . . 11 |- TopOpen Fn _V
45		funfvex 5715	. . . . . . . . . . . 12 |- ( ( Fun TopOpen /\ r e. dom TopOpen ) -> ( TopOpen ` r ) e. _V )
46	45	funfni 5486	. . . . . . . . . . 11 |- ( ( TopOpen Fn _V /\ r e. _V ) -> ( TopOpen ` r ) e. _V )
47	44, 9, 46	mp2an 430	. . . . . . . . . 10 |- ( TopOpen ` r ) e. _V
48	47	snex 4324	. . . . . . . . 9 |- { ( TopOpen ` r ) } e. _V
49	13, 48	xpex 4893	. . . . . . . 8 |- ( d X. { ( TopOpen ` r ) } ) e. _V
50		ptex 13677	. . . . . . . 8 |- ( ( d X. { ( TopOpen ` r ) } ) e. _V -> ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) e. _V )
51	49, 50	ax-mp 5	. . . . . . 7 |- ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) e. _V
52	43, 51	opex 4371	. . . . . 6 |- <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. e. _V
53		tpexg 4592	. . . . . 6 |- ( ( <. (Scalar ` ndx ) , r >. e. _V /\ <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. e. _V /\ <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. e. _V ) -> { <. (Scalar ` ndx ) , r >. , <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. } e. _V )
54	36, 41, 52, 53	mp3an 1378	. . . . 5 |- { <. (Scalar ` ndx ) , r >. , <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. } e. _V
55	32, 54	unex 4589	. . . 4 |- ( { <. ( 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 ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , r >. , <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. } ) e. _V
56	15, 55	csbexa 4264	. . 3 |- [_ ( ( Base ` r ) ^m d ) / b ]_ ( { <. ( 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 ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , r >. , <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. } ) e. _V
57	7, 56	csbexa 4264	. 2 |- [_ { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]_ [_ ( ( Base ` r ) ^m d ) / b ]_ ( { <. ( 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 ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , r >. , <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. } ) e. _V
58	1, 57	fnmpoi 6443	1 |- mPwSer Fn ( _V X. _V )

This proof depends on syntax axioms:   = wceq 1402   T. wtru 1403   e. wcel 2209   {crab 2532   _Vcvv 2821   [_csb 3147   u. cun 3218   {csn 3709   {ctp 3711   <.cop 3712   class class class wbr 4131   |-> cmpt 4193   X. cxp 4774   `'ccnv 4775   |` cres 4778   "cima 4779   Fn wfn 5375   ` cfv 5380  (class class class)co 6088   e. cmpo 6090   oFcof 6303   oRcofr 6304   ^m cmap 6926   Fincfn 7026   <_ cle 8367   - cmin 8504   NNcn 9313   NN0cn0 9574   ndxcnx 13407  Slot cslot 13409   Basecbs 13410   +g cplusg 13490   .rcmulr 13491  Scalarcsca 13493   .scvsca 13494  TopSetcts 13496   TopOpenctopn 13653   Xt_cpt 13668   gsum_g cgsu 14240   mPwSer cmps 15154

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 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-i2m1 8290

This proof depends on definitions:  df-bi 117  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-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-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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-map 6928  df-ixp 6985  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-mulr 13504  df-sca 13506  df-vsca 13507  df-tset 13509  df-rest 13654  df-topn 13655  df-topgen 13673  df-pt 13674  df-psr 15156

This theorem is used by:  psrelbas  15179  psrplusgg  15182  psradd  15183  psraddcl  15184  psrmulrg  15186  psrmulfval  15187  mplvalcoe  15206  mplbascoe  15207  fnmpl  15209  mplplusgg  15219