Theorem mplvalcoe 15064

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 15034	. . . . . . 7 |- mPwSer Fn ( _V X. _V )
8	7	a1i 9	. . . . . 6 |- ( ( I e. V /\ R e. W ) -> mPwSer Fn ( _V X. _V ) )
9		fnovex 6112	. . . . . 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 13394	. . . . . . 7 |- Base Fn _V
15		funfvex 5710	. . . . . . . 8 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
16	15	funfni 5481	. . . . . . 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 4279	. . . 4 |- ( ( I e. V /\ R e. W ) -> U e. _V )
20		ressex 13402	. . . 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 6112	. . . . . . 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 6088	. . . . . . . 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 5697	. . . . . . . . 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 6095	. . . . . . . . 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 5697	. . . . . . . . . . . . 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 6097	. . . . 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 15031	. . . 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 6212	. . 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 ) )

Syntax hints:   -> 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 4128   X. cxp 4770   Fn wfn 5370   ` cfv 5375  (class class class)co 6079   ^m cmap 6916   < clt 8354   NN0cn0 9546   Basecbs 13335   ↾_s cress 13336   0gc0g 13593   mPwSer cmps 15028   mPoly cmpl 15029

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 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 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-i2m1 8278

This theorem 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 3714  df-pr 3715  df-tp 3716  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-of 6296  df-1st 6368  df-2nd 6369  df-map 6918  df-ixp 6975  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-tset 13433  df-rest 13578  df-topn 13579  df-topgen 13597  df-pt 13598  df-psr 15030  df-mplcoe 15031

This theorem is referenced by:  mplbascoe  15065  mplval2g  15069