Theorem mplvalg 15222

Description: Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.)

Hypotheses

Ref	Expression
mplval.p	|- P = ( I mPoly R )
mplval.s	|- S = ( I mPwSer R )
mplval.b	|- B = ( Base ` S )
mplval.z	|- .0. = ( 0g ` R )
mplval.u	|- U = { f e. B | f finSupp .0. }

Assertion

Ref	Expression
mplvalg	|- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )

Distinct variable groups:   B, f   f, I   R, f   .0. , f

Allowed substitution hints:   P( f)   S( f)   U( f)   V( f)   W( f)

Proof of Theorem mplvalg

Dummy variables i r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplval.p	. 2 |- P = ( I mPoly R )
2		elex 2833	. . . 4 |- ( I e. V -> I e. _V )
3	2	adantr 276	. . 3 |- ( ( I e. V /\ R e. W ) -> I e. _V )
4		elex 2833	. . . 4 |- ( R e. W -> R e. _V )
5	4	adantl 277	. . 3 |- ( ( I e. V /\ R e. W ) -> R e. _V )
6		mplval.s	. . . . 5 |- S = ( I mPwSer R )
7		fnpsr 15161	. . . . . . 7 |- mPwSer Fn ( _V X. _V )
8	7	a1i 9	. . . . . 6 |- ( ( I e. V /\ R e. W ) -> mPwSer Fn ( _V X. _V ) )
9		fnovex 6121	. . . . . 6 |- ( ( mPwSer Fn ( _V X. _V ) /\ I e. _V /\ R e. _V ) -> ( I mPwSer R ) e. _V )
10	8, 3, 5, 9	syl3anc 1278	. . . . 5 |- ( ( I e. V /\ R e. W ) -> ( I mPwSer R ) e. _V )
11	6, 10	eqeltrid 2325	. . . 4 |- ( ( I e. V /\ R e. W ) -> S e. _V )
12		mplval.u	. . . . 5 |- U = { f e. B | f finSupp .0. }
13		mplval.b	. . . . . 6 |- B = ( Base ` S )
14		basfn 13468	. . . . . . 7 |- Base Fn _V
15		funfvex 5715	. . . . . . . 8 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
16	15	funfni 5486	. . . . . . 7 |- ( ( Base Fn _V /\ S e. _V ) -> ( Base ` S ) e. _V )
17	14, 11, 16	sylancr 418	. . . . . 6 |- ( ( I e. V /\ R e. W ) -> ( Base ` S ) e. _V )
18	13, 17	eqeltrid 2325	. . . . 5 |- ( ( I e. V /\ R e. W ) -> B e. _V )
19	12, 18	rabexd 4283	. . . 4 |- ( ( I e. V /\ R e. W ) -> U e. _V )
20		ressex 13476	. . . 4 |- ( ( S e. _V /\ U e. _V ) -> ( S ↾s U ) e. _V )
21	11, 19, 20	syl2anc 415	. . 3 |- ( ( I e. V /\ R e. W ) -> ( S ↾s U ) e. _V )
22		vex 2824	. . . . . . 7 |- i e. _V
23		vex 2824	. . . . . . 7 |- r e. _V
24		fnovex 6121	. . . . . . 7 |- ( ( mPwSer Fn ( _V X. _V ) /\ i e. _V /\ r e. _V ) -> ( i mPwSer r ) e. _V )
25	7, 22, 23, 24	mp3an 1378	. . . . . 6 |- ( i mPwSer r ) e. _V
26	25	a1i 9	. . . . 5 |- ( ( i = I /\ r = R ) -> ( i mPwSer r ) e. _V )
27		id 19	. . . . . . . 8 |- ( s = ( i mPwSer r ) -> s = ( i mPwSer r ) )
28		oveq12 6097	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( i mPwSer r ) = ( I mPwSer R ) )
29	27, 28	sylan9eqr 2293	. . . . . . 7 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> s = ( I mPwSer R ) )
30	29, 6	eqtr4di 2289	. . . . . 6 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> s = S )
31	30	fveq2d 5702	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( Base ` s ) = ( Base ` S ) )
32	31, 13	eqtr4di 2289	. . . . . . . 8 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( Base ` s ) = B )
33		simplr 533	. . . . . . . . . . 11 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> r = R )
34	33	fveq2d 5702	. . . . . . . . . 10 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = ( 0g ` R ) )
35		mplval.z	. . . . . . . . . 10 |- .0. = ( 0g ` R )
36	34, 35	eqtr4di 2289	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = .0. )
37	36	breq2d 4143	. . . . . . . 8 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( f finSupp ( 0g ` r ) <-> f finSupp .0. ) )
38	32, 37	rabeqbidv 2816	. . . . . . 7 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } = { f e. B | f finSupp .0. } )
39	38, 12	eqtr4di 2289	. . . . . 6 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } = U )
40	30, 39	oveq12d 6106	. . . . 5 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( s ↾s { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } ) = ( S ↾s U ) )
41	26, 40	csbied 3194	. . . 4 |- ( ( i = I /\ r = R ) -> [_ ( i mPwSer r ) / s ]_ ( s ↾s { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } ) = ( S ↾s U ) )
42		df-mpl 15158	. . . 4 |- mPoly = ( i e. _V , r e. _V |-> [_ ( i mPwSer r ) / s ]_ ( s ↾s { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } ) )
43	41, 42	ovmpoga 6221	. . 3 |- ( ( I e. _V /\ R e. _V /\ ( S ↾s U ) e. _V ) -> ( I mPoly R ) = ( S ↾s U ) )
44	3, 5, 21, 43	syl3anc 1278	. 2 |- ( ( I e. V /\ R e. W ) -> ( I mPoly R ) = ( S ↾s U ) )
45	1, 44	eqtrid 2283	1 |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   [_csb 3147   class class class wbr 4131   X. cxp 4774   Fn wfn 5375   ` cfv 5380  (class class class)co 6088   finSupp cfsupp 7292   Basecbs 13409   ↾_s cress 13410   0gc0g 13668   mPwSer cmps 15153   mPoly cmpl 15155

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-i2m1 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-map 6929  df-ixp 6987  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-ndx 13412  df-slot 13413  df-base 13415  df-sets 13416  df-iress 13417  df-plusg 13502  df-mulr 13503  df-sca 13505  df-vsca 13506  df-tset 13508  df-rest 13653  df-topn 13654  df-topgen 13672  df-pt 13673  df-psr 15156  df-mpl 15158

This theorem is used by:  mplbas  15223  mplval2g  15228