Theorem mplval 18700

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
mplval	|- P = ( S ↾s U )

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

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

Proof of Theorem mplval

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		ovex 6342	. . . . . 6 |- ( i mPwSer r ) e. _V
3	2	a1i 11	. . . . 5 |- ( ( i = I /\ r = R ) -> ( i mPwSer r ) e. _V )
4		id 22	. . . . . . . 8 |- ( s = ( i mPwSer r ) -> s = ( i mPwSer r ) )
5		oveq12 6323	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( i mPwSer r ) = ( I mPwSer R ) )
6	4, 5	sylan9eqr 2517	. . . . . . 7 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> s = ( I mPwSer R ) )
7		mplval.s	. . . . . . 7 |- S = ( I mPwSer R )
8	6, 7	syl6eqr 2513	. . . . . 6 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> s = S )
9	8	fveq2d 5891	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( Base ` s ) = ( Base ` S ) )
10		mplval.b	. . . . . . . . 9 |- B = ( Base ` S )
11	9, 10	syl6eqr 2513	. . . . . . . 8 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( Base ` s ) = B )
12		simplr 767	. . . . . . . . . . 11 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> r = R )
13	12	fveq2d 5891	. . . . . . . . . 10 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = ( 0g ` R ) )
14		mplval.z	. . . . . . . . . 10 |- .0. = ( 0g ` R )
15	13, 14	syl6eqr 2513	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = .0. )
16	15	breq2d 4427	. . . . . . . 8 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( f finSupp ( 0g ` r ) <-> f finSupp .0. ) )
17	11, 16	rabeqbidv 3051	. . . . . . 7 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } = { f e. B | f finSupp .0. } )
18		mplval.u	. . . . . . 7 |- U = { f e. B | f finSupp .0. }
19	17, 18	syl6eqr 2513	. . . . . 6 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } = U )
20	8, 19	oveq12d 6332	. . . . 5 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( s ↾s { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } ) = ( S ↾s U ) )
21	3, 20	csbied 3401	. . . 4 |- ( ( i = I /\ r = R ) -> [_ ( i mPwSer r ) / s ]_ ( s ↾s { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } ) = ( S ↾s U ) )
22		df-mpl 18630	. . . 4 |- mPoly = ( i e. _V , r e. _V |-> [_ ( i mPwSer r ) / s ]_ ( s ↾s { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } ) )
23		ovex 6342	. . . 4 |- ( S ↾s U ) e. _V
24	21, 22, 23	ovmpt2a 6453	. . 3 |- ( ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = ( S ↾s U ) )
25		reldmmpl 18699	. . . . . 6 |- Rel dom mPoly
26	25	ovprc 6344	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = (/) )
27		ress0 15231	. . . . 5 |- ( (/) ↾s U ) = (/)
28	26, 27	syl6eqr 2513	. . . 4 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = ( (/) ↾s U ) )
29		reldmpsr 18633	. . . . . . 7 |- Rel dom mPwSer
30	29	ovprc 6344	. . . . . 6 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPwSer R ) = (/) )
31	7, 30	syl5eq 2507	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> S = (/) )
32	31	oveq1d 6329	. . . 4 |- ( -. ( I e. _V /\ R e. _V ) -> ( S ↾s U ) = ( (/) ↾s U ) )
33	28, 32	eqtr4d 2498	. . 3 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = ( S ↾s U ) )
34	24, 33	pm2.61i 169	. 2 |- ( I mPoly R ) = ( S ↾s U )
35	1, 34	eqtri 2483	1 |- P = ( S ↾s U )

Syntax hints:   -. wn 3   /\ wa 375   = wceq 1454   e. wcel 1897   {crab 2752   _Vcvv 3056   [_csb 3374   (/)c0 3742   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   finSupp cfsupp 7908   Basecbs 15169   ↾_s cress 15170   0gc0g 15386   mPwSer cmps 18623   mPoly cmpl 18625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5564  df-fun 5602  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-slot 15173  df-base 15174  df-ress 15176  df-psr 18628  df-mpl 18630

This theorem is referenced by:  mplbas  18701  mplval2  18703