Theorem psrbasg 14511

Description: The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 8-Jul-2019.)

Hypotheses

Ref	Expression
psrbas.s	|- S = ( I mPwSer R )
psrbas.k	|- K = ( Base ` R )
psrbas.d	|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
psrbas.b	|- B = ( Base ` S )
psrbas.i	|- ( ph -> I e. V )
psrbasg.r	|- ( ph -> R e. W )

Assertion

Ref	Expression
psrbasg	|- ( ph -> B = ( K ^m D ) )

Distinct variable group:   f, I

Allowed substitution hints:   ph( f)   B( f)   D( f)   R( f)   S( f)   K( f)   V( f)   W( f)

Proof of Theorem psrbasg

Dummy variables g h k p x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrbas.s	. . . 4 |- S = ( I mPwSer R )
2		psrbas.k	. . . 4 |- K = ( Base ` R )
3		eqid 2206	. . . 4 |- ( +g ` R ) = ( +g ` R )
4		eqid 2206	. . . 4 |- ( .r ` R ) = ( .r ` R )
5		eqid 2206	. . . 4 |- ( TopOpen ` R ) = ( TopOpen ` R )
6		psrbas.d	. . . 4 |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
7		eqidd 2207	. . . 4 |- ( ph -> ( K ^m D ) = ( K ^m D ) )
8		eqid 2206	. . . 4 |- ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) = ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) )
9		eqid 2206	. . . 4 |- ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) = ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) )
10		eqid 2206	. . . 4 |- ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) = ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) )
11		eqidd 2207	. . . 4 |- ( ph -> ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) = ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) )
12		psrbas.i	. . . 4 |- ( ph -> I e. V )
13		psrbasg.r	. . . 4 |- ( ph -> R e. W )
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	psrval 14503	. . 3 |- ( ph -> S = ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) )
15	14	fveq2d 5593	. 2 |- ( ph -> ( Base ` S ) = ( Base ` ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) ) )
16		psrbas.b	. . 3 |- B = ( Base ` S )
17	16	a1i 9	. 2 |- ( ph -> B = ( Base ` S ) )
18		basfn 12965	. . . . . . . 8 |- Base Fn _V
19	13	elexd 2787	. . . . . . . 8 |- ( ph -> R e. _V )
20		funfvex 5606	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
21	20	funfni 5385	. . . . . . . 8 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
22	18, 19, 21	sylancr 414	. . . . . . 7 |- ( ph -> ( Base ` R ) e. _V )
23	2, 22	eqeltrid 2293	. . . . . 6 |- ( ph -> K e. _V )
24		nn0ex 9321	. . . . . . . . 9 |- NN0 e. _V
25		mapvalg 6758	. . . . . . . . 9 |- ( ( NN0 e. _V /\ I e. V ) -> ( NN0 ^m I ) = { p | p : I --> NN0 } )
26	24, 12, 25	sylancr 414	. . . . . . . 8 |- ( ph -> ( NN0 ^m I ) = { p | p : I --> NN0 } )
27	24	a1i 9	. . . . . . . . 9 |- ( ph -> NN0 e. _V )
28		mapex 6754	. . . . . . . . 9 |- ( ( I e. V /\ NN0 e. _V ) -> { p | p : I --> NN0 } e. _V )
29	12, 27, 28	syl2anc 411	. . . . . . . 8 |- ( ph -> { p | p : I --> NN0 } e. _V )
30	26, 29	eqeltrd 2283	. . . . . . 7 |- ( ph -> ( NN0 ^m I ) e. _V )
31	6, 30	rabexd 4197	. . . . . 6 |- ( ph -> D e. _V )
32		mapvalg 6758	. . . . . 6 |- ( ( K e. _V /\ D e. _V ) -> ( K ^m D ) = { p | p : D --> K } )
33	23, 31, 32	syl2anc 411	. . . . 5 |- ( ph -> ( K ^m D ) = { p | p : D --> K } )
34		mapex 6754	. . . . . 6 |- ( ( D e. _V /\ K e. _V ) -> { p | p : D --> K } e. _V )
35	31, 23, 34	syl2anc 411	. . . . 5 |- ( ph -> { p | p : D --> K } e. _V )
36	33, 35	eqeltrd 2283	. . . 4 |- ( ph -> ( K ^m D ) e. _V )
37	36, 36	ofmresex 6235	. . . 4 |- ( ph -> ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) e. _V )
38		mpoexga 6311	. . . . 5 |- ( ( ( K ^m D ) e. _V /\ ( K ^m D ) e. _V ) -> ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) e. _V )
39	36, 36, 38	syl2anc 411	. . . 4 |- ( ph -> ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) e. _V )
40		mpoexga 6311	. . . . 5 |- ( ( K e. _V /\ ( K ^m D ) e. _V ) -> ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) e. _V )
41	23, 36, 40	syl2anc 411	. . . 4 |- ( ph -> ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) e. _V )
42		topnfn 13151	. . . . . . . 8 |- TopOpen Fn _V
43		funfvex 5606	. . . . . . . . 9 |- ( ( Fun TopOpen /\ R e. dom TopOpen ) -> ( TopOpen ` R ) e. _V )
44	43	funfni 5385	. . . . . . . 8 |- ( ( TopOpen Fn _V /\ R e. _V ) -> ( TopOpen ` R ) e. _V )
45	42, 19, 44	sylancr 414	. . . . . . 7 |- ( ph -> ( TopOpen ` R ) e. _V )
46		snexg 4236	. . . . . . 7 |- ( ( TopOpen ` R ) e. _V -> { ( TopOpen ` R ) } e. _V )
47	45, 46	syl 14	. . . . . 6 |- ( ph -> { ( TopOpen ` R ) } e. _V )
48		xpexg 4797	. . . . . 6 |- ( ( D e. _V /\ { ( TopOpen ` R ) } e. _V ) -> ( D X. { ( TopOpen ` R ) } ) e. _V )
49	31, 47, 48	syl2anc 411	. . . . 5 |- ( ph -> ( D X. { ( TopOpen ` R ) } ) e. _V )
50		ptex 13171	. . . . 5 |- ( ( D X. { ( TopOpen ` R ) } ) e. _V -> ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) e. _V )
51	49, 50	syl 14	. . . 4 |- ( ph -> ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) e. _V )
52	36, 37, 39, 13, 41, 51	psrvalstrd 14505	. . 3 |- ( ph -> ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) Struct <. 1 , 9 >. )
53		basendxnn 12963	. . . . 5 |- ( Base ` ndx ) e. NN
54		opexg 4280	. . . . 5 |- ( ( ( Base ` ndx ) e. NN /\ ( K ^m D ) e. _V ) -> <. ( Base ` ndx ) , ( K ^m D ) >. e. _V )
55	53, 36, 54	sylancr 414	. . . 4 |- ( ph -> <. ( Base ` ndx ) , ( K ^m D ) >. e. _V )
56		tpid1g 3750	. . . 4 |- ( <. ( Base ` ndx ) , ( K ^m D ) >. e. _V -> <. ( Base ` ndx ) , ( K ^m D ) >. e. { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } )
57		elun1 3344	. . . 4 |- ( <. ( Base ` ndx ) , ( K ^m D ) >. e. { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } -> <. ( Base ` ndx ) , ( K ^m D ) >. e. ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) )
58	55, 56, 57	3syl 17	. . 3 |- ( ph -> <. ( Base ` ndx ) , ( K ^m D ) >. e. ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) )
59	52, 36, 58	opelstrbas 13022	. 2 |- ( ph -> ( K ^m D ) = ( Base ` ( { <. ( Base ` ndx ) , ( K ^m D ) >. , <. ( +g ` ndx ) , ( oF ( +g ` R ) |` ( ( K ^m D ) X. ( K ^m D ) ) ) >. , <. ( .r ` ndx ) , ( g e. ( K ^m D ) , h e. ( K ^m D ) |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( g ` x ) ( .r ` R ) ( h ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , ( x e. K , g e. ( K ^m D ) |-> ( ( D X. { x } ) oF ( .r ` R ) g ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( D X. { ( TopOpen ` R ) } ) ) >. } ) ) )
60	15, 17, 59	3eqtr4d 2249	1 |- ( ph -> B = ( K ^m D ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   {cab 2192   {crab 2489   _Vcvv 2773   u. cun 3168   {csn 3638   {ctp 3640   <.cop 3641   class class class wbr 4051   |-> cmpt 4113   X. cxp 4681   `'ccnv 4682   |` cres 4685   "cima 4686   Fn wfn 5275   -->wf 5276   ` cfv 5280  (class class class)co 5957   e. cmpo 5959   oFcof 6169   oRcofr 6170   ^m cmap 6748   Fincfn 6840   1c1 7946   <_ cle 8128   - cmin 8263   NNcn 9056   9c9 9114   NN0cn0 9315   ndxcnx 12904   Basecbs 12907   +g cplusg 12984   .rcmulr 12985  Scalarcsca 12987   .scvsca 12988  TopSetcts 12990   TopOpenctopn 13147   Xt_cpt 13162   gsum_g cgsu 13164   mPwSer cmps 14498

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-tp 3646  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-of 6171  df-1st 6239  df-2nd 6240  df-map 6750  df-ixp 6799  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-5 9118  df-6 9119  df-7 9120  df-8 9121  df-9 9122  df-n0 9316  df-z 9393  df-uz 9669  df-fz 10151  df-struct 12909  df-ndx 12910  df-slot 12911  df-base 12913  df-plusg 12997  df-mulr 12998  df-sca 13000  df-vsca 13001  df-tset 13003  df-rest 13148  df-topn 13149  df-topgen 13167  df-pt 13168  df-psr 14500

This theorem is referenced by:  psrelbas  14512  psrplusgg  14515  psraddcl  14517  psr0cl  14518  psrnegcl  14520  psrgrp  14522  psr1clfi  14525