Theorem fnpsr 14646

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 14642	. 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 6810	. . . . 5 |- ^m Fn ( _V X. _V )
3		nn0ex 9386	. . . . 5 |- NN0 e. _V
4		vex 2802	. . . . 5 |- i e. _V
5		fnovex 6040	. . . . 5 |- ( ( ^m Fn ( _V X. _V ) /\ NN0 e. _V /\ i e. _V ) -> ( NN0 ^m i ) e. _V )
6	2, 3, 4, 5	mp3an 1371	. . . 4 |- ( NN0 ^m i ) e. _V
7	6	rabex 4228	. . 3 |- { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } e. _V
8		basfn 13106	. . . . . 6 |- Base Fn _V
9		vex 2802	. . . . . 6 |- r e. _V
10		funfvex 5646	. . . . . . 7 |- ( ( Fun Base /\ r e. dom Base ) -> ( Base ` r ) e. _V )
11	10	funfni 5423	. . . . . 6 |- ( ( Base Fn _V /\ r e. _V ) -> ( Base ` r ) e. _V )
12	8, 9, 11	mp2an 426	. . . . 5 |- ( Base ` r ) e. _V
13		vex 2802	. . . . 5 |- d e. _V
14		fnovex 6040	. . . . 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 1371	. . . 4 |- ( ( Base ` r ) ^m d ) e. _V
16		basendxnn 13103	. . . . . . 7 |- ( Base ` ndx ) e. NN
17		vex 2802	. . . . . . 7 |- b e. _V
18		opexg 4314	. . . . . . 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 13159	. . . . . . 7 |- ( +g ` ndx ) e. NN
21	17	a1i 9	. . . . . . . . 9 |- ( T. -> b e. _V )
22	21, 21	ofmresex 6288	. . . . . . . 8 |- ( T. -> ( oF ( +g ` r ) |` ( b X. b ) ) e. _V )
23	22	mptru 1404	. . . . . . 7 |- ( oF ( +g ` r ) |` ( b X. b ) ) e. _V
24		opexg 4314	. . . . . . 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 13180	. . . . . . . . 9 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
27	26	simpri 113	. . . . . . . 8 |- ( .r ` ndx ) e. NN
28	27	elexi 2812	. . . . . . 7 |- ( .r ` ndx ) e. _V
29	17, 17	mpoex 6366	. . . . . . 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 4315	. . . . . 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 4535	. . . . . 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 1371	. . . . 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 13201	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
34	33	simpri 113	. . . . . . . 8 |- (Scalar ` ndx ) e. NN
35	34	elexi 2812	. . . . . . 7 |- (Scalar ` ndx ) e. _V
36	35, 9	opex 4315	. . . . . 6 |- <. (Scalar ` ndx ) , r >. e. _V
37		vscaslid 13211	. . . . . . . . 9 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
38	37	simpri 113	. . . . . . . 8 |- ( .s ` ndx ) e. NN
39	38	elexi 2812	. . . . . . 7 |- ( .s ` ndx ) e. _V
40	12, 17	mpoex 6366	. . . . . . 7 |- ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) e. _V
41	39, 40	opex 4315	. . . . . 6 |- <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. e. _V
42		tsetndxnn 13237	. . . . . . . 8 |- (TopSet ` ndx ) e. NN
43	42	elexi 2812	. . . . . . 7 |- (TopSet ` ndx ) e. _V
44		topnfn 13292	. . . . . . . . . . 11 |- TopOpen Fn _V
45		funfvex 5646	. . . . . . . . . . . 12 |- ( ( Fun TopOpen /\ r e. dom TopOpen ) -> ( TopOpen ` r ) e. _V )
46	45	funfni 5423	. . . . . . . . . . 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 4269	. . . . . . . . 9 |- { ( TopOpen ` r ) } e. _V
49	13, 48	xpex 4834	. . . . . . . 8 |- ( d X. { ( TopOpen ` r ) } ) e. _V
50		ptex 13312	. . . . . . . 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 4315	. . . . . 6 |- <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. e. _V
53		tpexg 4535	. . . . . 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 1371	. . . . 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 4532	. . . 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 4213	. . 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 4213	. 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 6355	1 |- mPwSer Fn ( _V X. _V )

Syntax hints:   = wceq 1395   T. wtru 1396   e. wcel 2200   {crab 2512   _Vcvv 2799   [_csb 3124   u. cun 3195   {csn 3666   {ctp 3668   <.cop 3669   class class class wbr 4083   |-> cmpt 4145   X. cxp 4717   `'ccnv 4718   |` cres 4721   "cima 4722   Fn wfn 5313   ` cfv 5318  (class class class)co 6007   e. cmpo 6009   oFcof 6222   oRcofr 6223   ^m cmap 6803   Fincfn 6895   <_ cle 8193   - cmin 8328   NNcn 9121   NN0cn0 9380   ndxcnx 13044  Slot cslot 13046   Basecbs 13047   +g cplusg 13125   .rcmulr 13126  Scalarcsca 13128   .scvsca 13129  TopSetcts 13131   TopOpenctopn 13288   Xt_cpt 13303   gsum_g cgsu 13305   mPwSer cmps 14640

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-i2m1 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-of 6224  df-1st 6292  df-2nd 6293  df-map 6805  df-ixp 6854  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-ndx 13050  df-slot 13051  df-base 13053  df-plusg 13138  df-mulr 13139  df-sca 13141  df-vsca 13142  df-tset 13144  df-rest 13289  df-topn 13290  df-topgen 13308  df-pt 13309  df-psr 14642

This theorem is referenced by:  psrelbas  14654  psrplusgg  14657  psradd  14658  psraddcl  14659  mplvalcoe  14669  mplbascoe  14670  fnmpl  14672  mplplusgg  14682