Theorem mplvalcoe 14496

Description: Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)

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 )
mplvalcoe.u	|- U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) }

Assertion

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

Distinct variable groups:   B, f   f, a, b, k, I   R, f, a, b, k   .0. , f

Allowed substitution hints:   B( k, a, b)   P( f, k, a, b)   S( f, k, a, b)   U( f, k, a, b)   V( f, k, a, b)   W( f, k, a, b)   .0. ( k, a, b)

Proof of Theorem mplvalcoe

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 2784	. . . 4 |- ( I e. V -> I e. _V )
3	2	adantr 276	. . 3 |- ( ( I e. V /\ R e. W ) -> I e. _V )
4		elex 2784	. . . 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 14473	. . . . . . 7 |- mPwSer Fn ( _V X. _V )
8	7	a1i 9	. . . . . 6 |- ( ( I e. V /\ R e. W ) -> mPwSer Fn ( _V X. _V ) )
9		fnovex 5984	. . . . . 6 |- ( ( mPwSer Fn ( _V X. _V ) /\ I e. _V /\ R e. _V ) -> ( I mPwSer R ) e. _V )
10	8, 3, 5, 9	syl3anc 1250	. . . . 5 |- ( ( I e. V /\ R e. W ) -> ( I mPwSer R ) e. _V )
11	6, 10	eqeltrid 2293	. . . 4 |- ( ( I e. V /\ R e. W ) -> S e. _V )
12		mplvalcoe.u	. . . . 5 |- U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) }
13		mplval.b	. . . . . 6 |- B = ( Base ` S )
14		basfn 12934	. . . . . . 7 |- Base Fn _V
15		funfvex 5600	. . . . . . . 8 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
16	15	funfni 5381	. . . . . . 7 |- ( ( Base Fn _V /\ S e. _V ) -> ( Base ` S ) e. _V )
17	14, 11, 16	sylancr 414	. . . . . 6 |- ( ( I e. V /\ R e. W ) -> ( Base ` S ) e. _V )
18	13, 17	eqeltrid 2293	. . . . 5 |- ( ( I e. V /\ R e. W ) -> B e. _V )
19	12, 18	rabexd 4193	. . . 4 |- ( ( I e. V /\ R e. W ) -> U e. _V )
20		ressex 12941	. . . 4 |- ( ( S e. _V /\ U e. _V ) -> ( S ↾s U ) e. _V )
21	11, 19, 20	syl2anc 411	. . 3 |- ( ( I e. V /\ R e. W ) -> ( S ↾s U ) e. _V )
22		vex 2776	. . . . . . 7 |- i e. _V
23		vex 2776	. . . . . . 7 |- r e. _V
24		fnovex 5984	. . . . . . 7 |- ( ( mPwSer Fn ( _V X. _V ) /\ i e. _V /\ r e. _V ) -> ( i mPwSer r ) e. _V )
25	7, 22, 23, 24	mp3an 1350	. . . . . 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 5960	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( i mPwSer r ) = ( I mPwSer R ) )
29	27, 28	sylan9eqr 2261	. . . . . . 7 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> s = ( I mPwSer R ) )
30	29, 6	eqtr4di 2257	. . . . . 6 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> s = S )
31	30	fveq2d 5587	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( Base ` s ) = ( Base ` S ) )
32	31, 13	eqtr4di 2257	. . . . . . . 8 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( Base ` s ) = B )
33		simpll 527	. . . . . . . . . 10 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> i = I )
34	33	oveq2d 5967	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( NN0 ^m i ) = ( NN0 ^m I ) )
35	33	raleqdv 2709	. . . . . . . . . . 11 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( A. k e. i ( a ` k ) < ( b ` k ) <-> A. k e. I ( a ` k ) < ( b ` k ) ) )
36		simplr 528	. . . . . . . . . . . . . 14 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> r = R )
37	36	fveq2d 5587	. . . . . . . . . . . . 13 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = ( 0g ` R ) )
38		mplval.z	. . . . . . . . . . . . 13 |- .0. = ( 0g ` R )
39	37, 38	eqtr4di 2257	. . . . . . . . . . . 12 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = .0. )
40	39	eqeq2d 2218	. . . . . . . . . . 11 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( ( f ` b ) = ( 0g ` r ) <-> ( f ` b ) = .0. ) )
41	35, 40	imbi12d 234	. . . . . . . . . 10 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) <-> ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) ) )
42	34, 41	raleqbidv 2719	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( A. b e. ( NN0 ^m i ) ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) <-> A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) ) )
43	34, 42	rexeqbidv 2720	. . . . . . . 8 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( E. a e. ( NN0 ^m i ) A. b e. ( NN0 ^m i ) ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) <-> E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) ) )
44	32, 43	rabeqbidv 2768	. . . . . . 7 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> { f e. ( Base ` s ) | E. a e. ( NN0 ^m i ) A. b e. ( NN0 ^m i ) ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) } = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) } )
45	44, 12	eqtr4di 2257	. . . . . 6 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> { f e. ( Base ` s ) | E. a e. ( NN0 ^m i ) A. b e. ( NN0 ^m i ) ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) } = U )
46	30, 45	oveq12d 5969	. . . . 5 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( s ↾s { f e. ( Base ` s ) | E. a e. ( NN0 ^m i ) A. b e. ( NN0 ^m i ) ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) } ) = ( S ↾s U ) )
47	26, 46	csbied 3141	. . . 4 |- ( ( i = I /\ r = R ) -> [_ ( i mPwSer r ) / s ]_ ( s ↾s { f e. ( Base ` s ) | E. a e. ( NN0 ^m i ) A. b e. ( NN0 ^m i ) ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) } ) = ( S ↾s U ) )
48		df-mplcoe 14470	. . . 4 |- mPoly = ( i e. _V , r e. _V |-> [_ ( i mPwSer r ) / s ]_ ( s ↾s { f e. ( Base ` s ) | E. a e. ( NN0 ^m i ) A. b e. ( NN0 ^m i ) ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) } ) )
49	47, 48	ovmpoga 6082	. . 3 |- ( ( I e. _V /\ R e. _V /\ ( S ↾s U ) e. _V ) -> ( I mPoly R ) = ( S ↾s U ) )
50	3, 5, 21, 49	syl3anc 1250	. 2 |- ( ( I e. V /\ R e. W ) -> ( I mPoly R ) = ( S ↾s U ) )
51	1, 50	eqtrid 2251	1 |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   A.wral 2485   E.wrex 2486   {crab 2489   _Vcvv 2773   [_csb 3094   class class class wbr 4047   X. cxp 4677   Fn wfn 5271   ` cfv 5276  (class class class)co 5951   ^m cmap 6742   < clt 8114   NN0cn0 9302   Basecbs 12876   ↾_s cress 12877   0gc0g 13132   mPwSer cmps 14467   mPoly cmpl 14468

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 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-i2m1 8037

This theorem depends on definitions:  df-bi 117  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-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-tp 3642  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-of 6165  df-1st 6233  df-2nd 6234  df-map 6744  df-ixp 6793  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-9 9109  df-n0 9303  df-ndx 12879  df-slot 12880  df-base 12882  df-sets 12883  df-iress 12884  df-plusg 12966  df-mulr 12967  df-sca 12969  df-vsca 12970  df-tset 12972  df-rest 13117  df-topn 13118  df-topgen 13136  df-pt 13137  df-psr 14469  df-mplcoe 14470

This theorem is referenced by:  mplbascoe  14497  mplval2g  14501