Theorem fnpsr 14941

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 14937	. 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 6902	. . . . 5 |- ^m Fn ( _V X. _V )
3		nn0ex 9519	. . . . 5 |- NN0 e. _V
4		vex 2818	. . . . 5 |- i e. _V
5		fnovex 6091	. . . . 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 4261	. . 3 |- { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } e. _V
8		basfn 13355	. . . . . 6 |- Base Fn _V
9		vex 2818	. . . . . 6 |- r e. _V
10		funfvex 5692	. . . . . . 7 |- ( ( Fun Base /\ r e. dom Base ) -> ( Base ` r ) e. _V )
11	10	funfni 5463	. . . . . 6 |- ( ( Base Fn _V /\ r e. _V ) -> ( Base ` r ) e. _V )
12	8, 9, 11	mp2an 426	. . . . 5 |- ( Base ` r ) e. _V
13		vex 2818	. . . . 5 |- d e. _V
14		fnovex 6091	. . . . 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 13352	. . . . . . 7 |- ( Base ` ndx ) e. NN
17		vex 2818	. . . . . . 7 |- b e. _V
18		opexg 4349	. . . . . . 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 13408	. . . . . . 7 |- ( +g ` ndx ) e. NN
21	17	a1i 9	. . . . . . . . 9 |- ( T. -> b e. _V )
22	21, 21	ofmresex 6343	. . . . . . . 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 4349	. . . . . . 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 13429	. . . . . . . . 9 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
27	26	simpri 113	. . . . . . . 8 |- ( .r ` ndx ) e. NN
28	27	elexi 2828	. . . . . . 7 |- ( .r ` ndx ) e. _V
29	17, 17	mpoex 6423	. . . . . . 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 4350	. . . . . 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 4570	. . . . . 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 13450	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
34	33	simpri 113	. . . . . . . 8 |- (Scalar ` ndx ) e. NN
35	34	elexi 2828	. . . . . . 7 |- (Scalar ` ndx ) e. _V
36	35, 9	opex 4350	. . . . . 6 |- <. (Scalar ` ndx ) , r >. e. _V
37		vscaslid 13460	. . . . . . . . 9 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
38	37	simpri 113	. . . . . . . 8 |- ( .s ` ndx ) e. NN
39	38	elexi 2828	. . . . . . 7 |- ( .s ` ndx ) e. _V
40	12, 17	mpoex 6423	. . . . . . 7 |- ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) e. _V
41	39, 40	opex 4350	. . . . . 6 |- <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. e. _V
42		tsetndxnn 13486	. . . . . . . 8 |- (TopSet ` ndx ) e. NN
43	42	elexi 2828	. . . . . . 7 |- (TopSet ` ndx ) e. _V
44		topnfn 13541	. . . . . . . . . . 11 |- TopOpen Fn _V
45		funfvex 5692	. . . . . . . . . . . 12 |- ( ( Fun TopOpen /\ r e. dom TopOpen ) -> ( TopOpen ` r ) e. _V )
46	45	funfni 5463	. . . . . . . . . . 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 4303	. . . . . . . . 9 |- { ( TopOpen ` r ) } e. _V
49	13, 48	xpex 4871	. . . . . . . 8 |- ( d X. { ( TopOpen ` r ) } ) e. _V
50		ptex 13561	. . . . . . . 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 4350	. . . . . 6 |- <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. e. _V
53		tpexg 4570	. . . . . 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 4567	. . . 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 4244	. . 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 4244	. 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 6412	1 |- mPwSer Fn ( _V X. _V )

Syntax hints:   = wceq 1398   T. wtru 1399   e. wcel 2205   {crab 2526   _Vcvv 2815   [_csb 3141   u. cun 3212   {csn 3694   {ctp 3696   <.cop 3697   class class class wbr 4114   |-> cmpt 4176   X. cxp 4752   `'ccnv 4753   |` cres 4756   "cima 4757   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   e. cmpo 6060   oFcof 6273   oRcofr 6274   ^m cmap 6895   Fincfn 6988   <_ cle 8325   - cmin 8460   NNcn 9254   NN0cn0 9513   ndxcnx 13293  Slot cslot 13295   Basecbs 13296   +g cplusg 13374   .rcmulr 13375  Scalarcsca 13377   .scvsca 13378  TopSetcts 13380   TopOpenctopn 13537   Xt_cpt 13552   gsum_g cgsu 13554   mPwSer cmps 14935

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-i2m1 8248

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-map 6897  df-ixp 6947  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-mulr 13388  df-sca 13390  df-vsca 13391  df-tset 13393  df-rest 13538  df-topn 13539  df-topgen 13557  df-pt 13558  df-psr 14937

This theorem is referenced by:  psrelbas  14956  psrplusgg  14959  psradd  14960  psraddcl  14961  mplvalcoe  14971  mplbascoe  14972  fnmpl  14974  mplplusgg  14984