Theorem mplval 19428

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		ovexd 6680	. . . . 5 |- ( ( i = I /\ r = R ) -> ( i mPwSer r ) e. _V )
3		id 22	. . . . . . . 8 |- ( s = ( i mPwSer r ) -> s = ( i mPwSer r ) )
4		oveq12 6659	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( i mPwSer r ) = ( I mPwSer R ) )
5	3, 4	sylan9eqr 2678	. . . . . . 7 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> s = ( I mPwSer R ) )
6		mplval.s	. . . . . . 7 |- S = ( I mPwSer R )
7	5, 6	syl6eqr 2674	. . . . . 6 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> s = S )
8	7	fveq2d 6195	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( Base ` s ) = ( Base ` S ) )
9		mplval.b	. . . . . . . . 9 |- B = ( Base ` S )
10	8, 9	syl6eqr 2674	. . . . . . . 8 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( Base ` s ) = B )
11		simplr 792	. . . . . . . . . . 11 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> r = R )
12	11	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = ( 0g ` R ) )
13		mplval.z	. . . . . . . . . 10 |- .0. = ( 0g ` R )
14	12, 13	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = .0. )
15	14	breq2d 4665	. . . . . . . 8 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( f finSupp ( 0g ` r ) <-> f finSupp .0. ) )
16	10, 15	rabeqbidv 3195	. . . . . . 7 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } = { f e. B | f finSupp .0. } )
17		mplval.u	. . . . . . 7 |- U = { f e. B | f finSupp .0. }
18	16, 17	syl6eqr 2674	. . . . . 6 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } = U )
19	7, 18	oveq12d 6668	. . . . 5 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( s ↾s { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } ) = ( S ↾s U ) )
20	2, 19	csbied 3560	. . . 4 |- ( ( i = I /\ r = R ) -> [_ ( i mPwSer r ) / s ]_ ( s ↾s { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } ) = ( S ↾s U ) )
21		df-mpl 19358	. . . 4 |- mPoly = ( i e. _V , r e. _V |-> [_ ( i mPwSer r ) / s ]_ ( s ↾s { f e. ( Base ` s ) | f finSupp ( 0g ` r ) } ) )
22		ovex 6678	. . . 4 |- ( S ↾s U ) e. _V
23	20, 21, 22	ovmpt2a 6791	. . 3 |- ( ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = ( S ↾s U ) )
24		reldmmpl 19427	. . . . . 6 |- Rel dom mPoly
25	24	ovprc 6683	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = (/) )
26		ress0 15934	. . . . 5 |- ( (/) ↾s U ) = (/)
27	25, 26	syl6eqr 2674	. . . 4 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = ( (/) ↾s U ) )
28		reldmpsr 19361	. . . . . . 7 |- Rel dom mPwSer
29	28	ovprc 6683	. . . . . 6 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPwSer R ) = (/) )
30	6, 29	syl5eq 2668	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> S = (/) )
31	30	oveq1d 6665	. . . 4 |- ( -. ( I e. _V /\ R e. _V ) -> ( S ↾s U ) = ( (/) ↾s U ) )
32	27, 31	eqtr4d 2659	. . 3 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = ( S ↾s U ) )
33	23, 32	pm2.61i 176	. 2 |- ( I mPoly R ) = ( S ↾s U )
34	1, 33	eqtri 2644	1 |- P = ( S ↾s U )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   [_csb 3533   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   finSupp cfsupp 8275   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   mPwSer cmps 19351   mPoly cmpl 19353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-slot 15861  df-base 15863  df-ress 15865  df-psr 19356  df-mpl 19358

This theorem is referenced by:  mplbas  19429  mplval2  19431