Theorem fnpsr 14746

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 14742	. 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 6867	. . . . 5 |- ^m Fn ( _V X. _V )
3		nn0ex 9450	. . . . 5 |- NN0 e. _V
4		vex 2806	. . . . 5 |- i e. _V
5		fnovex 6061	. . . . 5 |- ( ( ^m Fn ( _V X. _V ) /\ NN0 e. _V /\ i e. _V ) -> ( NN0 ^m i ) e. _V )
6	2, 3, 4, 5	mp3an 1374	. . . 4 |- ( NN0 ^m i ) e. _V
7	6	rabex 4239	. . 3 |- { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } e. _V
8		basfn 13204	. . . . . 6 |- Base Fn _V
9		vex 2806	. . . . . 6 |- r e. _V
10		funfvex 5665	. . . . . . 7 |- ( ( Fun Base /\ r e. dom Base ) -> ( Base ` r ) e. _V )
11	10	funfni 5439	. . . . . 6 |- ( ( Base Fn _V /\ r e. _V ) -> ( Base ` r ) e. _V )
12	8, 9, 11	mp2an 426	. . . . 5 |- ( Base ` r ) e. _V
13		vex 2806	. . . . 5 |- d e. _V
14		fnovex 6061	. . . . 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 1374	. . . 4 |- ( ( Base ` r ) ^m d ) e. _V
16		basendxnn 13201	. . . . . . 7 |- ( Base ` ndx ) e. NN
17		vex 2806	. . . . . . 7 |- b e. _V
18		opexg 4326	. . . . . . 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 13257	. . . . . . 7 |- ( +g ` ndx ) e. NN
21	17	a1i 9	. . . . . . . . 9 |- ( T. -> b e. _V )
22	21, 21	ofmresex 6308	. . . . . . . 8 |- ( T. -> ( oF ( +g ` r ) |` ( b X. b ) ) e. _V )
23	22	mptru 1407	. . . . . . 7 |- ( oF ( +g ` r ) |` ( b X. b ) ) e. _V
24		opexg 4326	. . . . . . 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 13278	. . . . . . . . 9 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
27	26	simpri 113	. . . . . . . 8 |- ( .r ` ndx ) e. NN
28	27	elexi 2816	. . . . . . 7 |- ( .r ` ndx ) e. _V
29	17, 17	mpoex 6388	. . . . . . 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 4327	. . . . . 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 4547	. . . . . 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 1374	. . . . 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 13299	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
34	33	simpri 113	. . . . . . . 8 |- (Scalar ` ndx ) e. NN
35	34	elexi 2816	. . . . . . 7 |- (Scalar ` ndx ) e. _V
36	35, 9	opex 4327	. . . . . 6 |- <. (Scalar ` ndx ) , r >. e. _V
37		vscaslid 13309	. . . . . . . . 9 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
38	37	simpri 113	. . . . . . . 8 |- ( .s ` ndx ) e. NN
39	38	elexi 2816	. . . . . . 7 |- ( .s ` ndx ) e. _V
40	12, 17	mpoex 6388	. . . . . . 7 |- ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) e. _V
41	39, 40	opex 4327	. . . . . 6 |- <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. e. _V
42		tsetndxnn 13335	. . . . . . . 8 |- (TopSet ` ndx ) e. NN
43	42	elexi 2816	. . . . . . 7 |- (TopSet ` ndx ) e. _V
44		topnfn 13390	. . . . . . . . . . 11 |- TopOpen Fn _V
45		funfvex 5665	. . . . . . . . . . . 12 |- ( ( Fun TopOpen /\ r e. dom TopOpen ) -> ( TopOpen ` r ) e. _V )
46	45	funfni 5439	. . . . . . . . . . 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 4281	. . . . . . . . 9 |- { ( TopOpen ` r ) } e. _V
49	13, 48	xpex 4848	. . . . . . . 8 |- ( d X. { ( TopOpen ` r ) } ) e. _V
50		ptex 13410	. . . . . . . 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 4327	. . . . . 6 |- <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. e. _V
53		tpexg 4547	. . . . . 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 1374	. . . . 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 4544	. . . 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 4223	. . 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 4223	. 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 6377	1 |- mPwSer Fn ( _V X. _V )

Syntax hints:   = wceq 1398   T. wtru 1399   e. wcel 2202   {crab 2515   _Vcvv 2803   [_csb 3128   u. cun 3199   {csn 3673   {ctp 3675   <.cop 3676   class class class wbr 4093   |-> cmpt 4155   X. cxp 4729   `'ccnv 4730   |` cres 4733   "cima 4734   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   oFcof 6242   oRcofr 6243   ^m cmap 6860   Fincfn 6952   <_ cle 8257   - cmin 8392   NNcn 9185   NN0cn0 9444   ndxcnx 13142  Slot cslot 13144   Basecbs 13145   +g cplusg 13223   .rcmulr 13224  Scalarcsca 13226   .scvsca 13227  TopSetcts 13229   TopOpenctopn 13386   Xt_cpt 13401   gsum_g cgsu 13403   mPwSer cmps 14740

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-i2m1 8180

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244  df-1st 6312  df-2nd 6313  df-map 6862  df-ixp 6911  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-mulr 13237  df-sca 13239  df-vsca 13240  df-tset 13242  df-rest 13387  df-topn 13388  df-topgen 13406  df-pt 13407  df-psr 14742

This theorem is referenced by:  psrelbas  14759  psrplusgg  14762  psradd  14763  psraddcl  14764  mplvalcoe  14774  mplbascoe  14775  fnmpl  14777  mplplusgg  14787