Theorem fnpsr 14297

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 14294	. 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 6723	. . . . 5 |- ^m Fn ( _V X. _V )
3		nn0ex 9272	. . . . 5 |- NN0 e. _V
4		vex 2766	. . . . 5 |- i e. _V
5		fnovex 5958	. . . . 5 |- ( ( ^m Fn ( _V X. _V ) /\ NN0 e. _V /\ i e. _V ) -> ( NN0 ^m i ) e. _V )
6	2, 3, 4, 5	mp3an 1348	. . . 4 |- ( NN0 ^m i ) e. _V
7	6	rabex 4178	. . 3 |- { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } e. _V
8		basfn 12761	. . . . . 6 |- Base Fn _V
9		vex 2766	. . . . . 6 |- r e. _V
10		funfvex 5578	. . . . . . 7 |- ( ( Fun Base /\ r e. dom Base ) -> ( Base ` r ) e. _V )
11	10	funfni 5361	. . . . . 6 |- ( ( Base Fn _V /\ r e. _V ) -> ( Base ` r ) e. _V )
12	8, 9, 11	mp2an 426	. . . . 5 |- ( Base ` r ) e. _V
13		vex 2766	. . . . 5 |- d e. _V
14		fnovex 5958	. . . . 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 1348	. . . 4 |- ( ( Base ` r ) ^m d ) e. _V
16		basendxnn 12759	. . . . . . 7 |- ( Base ` ndx ) e. NN
17		vex 2766	. . . . . . 7 |- b e. _V
18		opexg 4262	. . . . . . 7 |- ( ( ( Base ` ndx ) e. NN /\ b e. _V ) -> <. ( Base ` ndx ) , b >. e. _V )
19	16, 17, 18	mp2an 426	. . . . . 6 |- <. ( Base ` ndx ) , b >. e. _V
20		plusgndxnn 12814	. . . . . . 7 |- ( +g ` ndx ) e. NN
21	17	a1i 9	. . . . . . . . 9 |- ( T. -> b e. _V )
22	21, 21	ofmresex 6203	. . . . . . . 8 |- ( T. -> ( oF ( +g ` r ) |` ( b X. b ) ) e. _V )
23	22	mptru 1373	. . . . . . 7 |- ( oF ( +g ` r ) |` ( b X. b ) ) e. _V
24		opexg 4262	. . . . . . 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 426	. . . . . 6 |- <. ( +g ` ndx ) , ( oF ( +g ` r ) |` ( b X. b ) ) >. e. _V
26		mulrslid 12834	. . . . . . . . 9 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
27	26	simpri 113	. . . . . . . 8 |- ( .r ` ndx ) e. NN
28	27	elexi 2775	. . . . . . 7 |- ( .r ` ndx ) e. _V
29	17, 17	mpoex 6281	. . . . . . 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 4263	. . . . . 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 4480	. . . . . 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 1348	. . . . 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 12855	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
34	33	simpri 113	. . . . . . . 8 |- (Scalar ` ndx ) e. NN
35	34	elexi 2775	. . . . . . 7 |- (Scalar ` ndx ) e. _V
36	35, 9	opex 4263	. . . . . 6 |- <. (Scalar ` ndx ) , r >. e. _V
37		vscaslid 12865	. . . . . . . . 9 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
38	37	simpri 113	. . . . . . . 8 |- ( .s ` ndx ) e. NN
39	38	elexi 2775	. . . . . . 7 |- ( .s ` ndx ) e. _V
40	12, 17	mpoex 6281	. . . . . . 7 |- ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) e. _V
41	39, 40	opex 4263	. . . . . 6 |- <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. e. _V
42		tsetndxnn 12891	. . . . . . . 8 |- (TopSet ` ndx ) e. NN
43	42	elexi 2775	. . . . . . 7 |- (TopSet ` ndx ) e. _V
44		topnfn 12946	. . . . . . . . . . 11 |- TopOpen Fn _V
45		funfvex 5578	. . . . . . . . . . . 12 |- ( ( Fun TopOpen /\ r e. dom TopOpen ) -> ( TopOpen ` r ) e. _V )
46	45	funfni 5361	. . . . . . . . . . 11 |- ( ( TopOpen Fn _V /\ r e. _V ) -> ( TopOpen ` r ) e. _V )
47	44, 9, 46	mp2an 426	. . . . . . . . . 10 |- ( TopOpen ` r ) e. _V
48	47	snex 4219	. . . . . . . . 9 |- { ( TopOpen ` r ) } e. _V
49	13, 48	xpex 4779	. . . . . . . 8 |- ( d X. { ( TopOpen ` r ) } ) e. _V
50		ptex 12966	. . . . . . . 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 4263	. . . . . 6 |- <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. e. _V
53		tpexg 4480	. . . . . 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 1348	. . . . 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 4477	. . . 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 4163	. . 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 4163	. 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 6270	1 |- mPwSer Fn ( _V X. _V )

Syntax hints:   = wceq 1364   T. wtru 1365   e. wcel 2167   {crab 2479   _Vcvv 2763   [_csb 3084   u. cun 3155   {csn 3623   {ctp 3625   <.cop 3626   class class class wbr 4034   |-> cmpt 4095   X. cxp 4662   `'ccnv 4663   |` cres 4666   "cima 4667   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   oFcof 6137   oRcofr 6138   ^m cmap 6716   Fincfn 6808   <_ cle 8079   - cmin 8214   NNcn 9007   NN0cn0 9266   ndxcnx 12700  Slot cslot 12702   Basecbs 12703   +g cplusg 12780   .rcmulr 12781  Scalarcsca 12783   .scvsca 12784  TopSetcts 12786   TopOpenctopn 12942   Xt_cpt 12957   gsum_g cgsu 12959   mPwSer cmps 14293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-i2m1 8001

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-of 6139  df-1st 6207  df-2nd 6208  df-map 6718  df-ixp 6767  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-9 9073  df-n0 9267  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-mulr 12794  df-sca 12796  df-vsca 12797  df-tset 12799  df-rest 12943  df-topn 12944  df-topgen 12962  df-pt 12963  df-psr 14294

This theorem is referenced by:  psrelbas  14304  psrplusgg  14306  psradd  14307  psraddcl  14308