Theorem mplval 17501

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 6116	. . . . . 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 6100	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( i mPwSer r ) = ( I mPwSer R ) )
6	4, 5	sylan9eqr 2497	. . . . . . 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 2493	. . . . . 6 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> s = S )
9	8	fveq2d 5695	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( Base ` s ) = ( Base ` S ) )
10		mplval.b	. . . . . . . . 9 |- B = ( Base ` S )
11	9, 10	syl6eqr 2493	. . . . . . . 8 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( Base ` s ) = B )
12		simplr 754	. . . . . . . . . . 11 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> r = R )
13	12	fveq2d 5695	. . . . . . . . . 10 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = ( 0g ` R ) )
14		mplval.z	. . . . . . . . . 10 |- .0. = ( 0g ` R )
15	13, 14	syl6eqr 2493	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = .0. )
16	15	breq2d 4304	. . . . . . . 8 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( f finSupp ( 0g ` r ) <-> f finSupp .0. ) )
17	11, 16	rabeqbidv 2967	. . . . . . 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 2493	. . . . . 6 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } = U )
20	8, 19	oveq12d 6109	. . . . 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 3314	. . . 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 17425	. . . 4 |- mPoly = ( i e. _V , r e. _V |-> [_ ( i mPwSer r ) / s ]_ ( s ↾s { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } ) )
23		ovex 6116	. . . 4 |- ( S ↾s U ) e. _V
24	21, 22, 23	ovmpt2a 6221	. . 3 |- ( ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = ( S ↾s U ) )
25		reldmmpl 17500	. . . . . 6 |- Rel dom mPoly
26	25	ovprc 6118	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = (/) )
27		ress0 14232	. . . . 5 |- ( (/) ↾s U ) = (/)
28	26, 27	syl6eqr 2493	. . . 4 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = ( (/) ↾s U ) )
29		reldmpsr 17428	. . . . . . 7 |- Rel dom mPwSer
30	29	ovprc 6118	. . . . . 6 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPwSer R ) = (/) )
31	7, 30	syl5eq 2487	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> S = (/) )
32	31	oveq1d 6106	. . . 4 |- ( -. ( I e. _V /\ R e. _V ) -> ( S ↾s U ) = ( (/) ↾s U ) )
33	28, 32	eqtr4d 2478	. . 3 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = ( S ↾s U ) )
34	24, 33	pm2.61i 164	. 2 |- ( I mPoly R ) = ( S ↾s U )
35	1, 34	eqtri 2463	1 |- P = ( S ↾s U )

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1369   e. wcel 1756   {crab 2719   _Vcvv 2972   [_csb 3288   (/)c0 3637   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   finSupp cfsupp 7620   Basecbs 14174   ↾_s cress 14175   0gc0g 14378   mPwSer cmps 17418   mPoly cmpl 17420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-slot 14178  df-base 14179  df-ress 14181  df-psr 17423  df-mpl 17425

This theorem is referenced by:  mplvalOLD  17502  mplbas  17503  mplval2  17507