Theorem mplvalcoe 15083

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 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 15053	. . . . . . 7 |- mPwSer Fn ( _V X. _V )
8	7	a1i 9	. . . . . 6 |- ( ( I e. V /\ R e. W ) -> mPwSer Fn ( _V X. _V ) )
9		fnovex 6118	. . . . . 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		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 13413	. . . . . . 7 |- Base Fn _V
15		funfvex 5712	. . . . . . . 8 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
16	15	funfni 5483	. . . . . . 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 4281	. . . 4 |- ( ( I e. V /\ R e. W ) -> U e. _V )
20		ressex 13421	. . . 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 6118	. . . . . . 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 6094	. . . . . . . 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 5699	. . . . . . . . 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		simpll 531	. . . . . . . . . 10 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> i = I )
34	33	oveq2d 6101	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( NN0 ^m i ) = ( NN0 ^m I ) )
35	33	raleqdv 2755	. . . . . . . . . . 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 533	. . . . . . . . . . . . . 14 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> r = R )
37	36	fveq2d 5699	. . . . . . . . . . . . 13 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = ( 0g ` R ) )
38		mplval.z	. . . . . . . . . . . . 13 |- .0. = ( 0g ` R )
39	37, 38	eqtr4di 2289	. . . . . . . . . . . 12 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = .0. )
40	39	eqeq2d 2250	. . . . . . . . . . 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 2765	. . . . . . . . 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 2766	. . . . . . . 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 2816	. . . . . . 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 2289	. . . . . 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 6103	. . . . 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 3194	. . . 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 15050	. . . 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 6218	. . 3 |- ( ( I e. _V /\ R e. _V /\ ( S ↾s U ) e. _V ) -> ( I mPoly R ) = ( S ↾s U ) )
50	3, 5, 21, 49	syl3anc 1278	. 2 |- ( ( I e. V /\ R e. W ) -> ( I mPoly R ) = ( S ↾s U ) )
51	1, 50	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   A.wral 2528   E.wrex 2529   {crab 2532   _Vcvv 2821   [_csb 3147   class class class wbr 4130   X. cxp 4772   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   ^m cmap 6922   < clt 8360   NN0cn0 9565   Basecbs 13354   ↾_s cress 13355   0gc0g 13612   mPwSer cmps 15047   mPoly cmpl 15048

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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-i2m1 8284

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 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-map 6924  df-ixp 6981  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-mulr 13447  df-sca 13449  df-vsca 13450  df-tset 13452  df-rest 13597  df-topn 13598  df-topgen 13616  df-pt 13617  df-psr 15049  df-mplcoe 15050

This theorem is used by:  mplbascoe  15084  mplval2g  15088