Theorem mplvalcoe 14697

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 2812	. . . 4 |- ( I e. V -> I e. _V )
3	2	adantr 276	. . 3 |- ( ( I e. V /\ R e. W ) -> I e. _V )
4		elex 2812	. . . 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 14674	. . . . . . 7 |- mPwSer Fn ( _V X. _V )
8	7	a1i 9	. . . . . 6 |- ( ( I e. V /\ R e. W ) -> mPwSer Fn ( _V X. _V ) )
9		fnovex 6046	. . . . . 6 |- ( ( mPwSer Fn ( _V X. _V ) /\ I e. _V /\ R e. _V ) -> ( I mPwSer R ) e. _V )
10	8, 3, 5, 9	syl3anc 1271	. . . . 5 |- ( ( I e. V /\ R e. W ) -> ( I mPwSer R ) e. _V )
11	6, 10	eqeltrid 2316	. . . 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 13134	. . . . . . 7 |- Base Fn _V
15		funfvex 5652	. . . . . . . 8 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
16	15	funfni 5429	. . . . . . 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 2316	. . . . 5 |- ( ( I e. V /\ R e. W ) -> B e. _V )
19	12, 18	rabexd 4233	. . . 4 |- ( ( I e. V /\ R e. W ) -> U e. _V )
20		ressex 13141	. . . 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 2803	. . . . . . 7 |- i e. _V
23		vex 2803	. . . . . . 7 |- r e. _V
24		fnovex 6046	. . . . . . 7 |- ( ( mPwSer Fn ( _V X. _V ) /\ i e. _V /\ r e. _V ) -> ( i mPwSer r ) e. _V )
25	7, 22, 23, 24	mp3an 1371	. . . . . 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 6022	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( i mPwSer r ) = ( I mPwSer R ) )
29	27, 28	sylan9eqr 2284	. . . . . . 7 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> s = ( I mPwSer R ) )
30	29, 6	eqtr4di 2280	. . . . . 6 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> s = S )
31	30	fveq2d 5639	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( Base ` s ) = ( Base ` S ) )
32	31, 13	eqtr4di 2280	. . . . . . . 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 6029	. . . . . . . . 9 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( NN0 ^m i ) = ( NN0 ^m I ) )
35	33	raleqdv 2734	. . . . . . . . . . 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 5639	. . . . . . . . . . . . 13 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = ( 0g ` R ) )
38		mplval.z	. . . . . . . . . . . . 13 |- .0. = ( 0g ` R )
39	37, 38	eqtr4di 2280	. . . . . . . . . . . 12 |- ( ( ( i = I /\ r = R ) /\ s = ( i mPwSer r ) ) -> ( 0g ` r ) = .0. )
40	39	eqeq2d 2241	. . . . . . . . . . 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 2744	. . . . . . . . 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 2745	. . . . . . . 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 2795	. . . . . . 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 2280	. . . . . 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 6031	. . . . 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 3172	. . . 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 14671	. . . 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 6146	. . 3 |- ( ( I e. _V /\ R e. _V /\ ( S ↾s U ) e. _V ) -> ( I mPoly R ) = ( S ↾s U ) )
50	3, 5, 21, 49	syl3anc 1271	. 2 |- ( ( I e. V /\ R e. W ) -> ( I mPoly R ) = ( S ↾s U ) )
51	1, 50	eqtrid 2274	1 |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   {crab 2512   _Vcvv 2800   [_csb 3125   class class class wbr 4086   X. cxp 4721   Fn wfn 5319   ` cfv 5324  (class class class)co 6013   ^m cmap 6812   < clt 8207   NN0cn0 9395   Basecbs 13075   ↾_s cress 13076   0gc0g 13332   mPwSer cmps 14668   mPoly cmpl 14669

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-i2m1 8130

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-tp 3675  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230  df-1st 6298  df-2nd 6299  df-map 6814  df-ixp 6863  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082  df-iress 13083  df-plusg 13166  df-mulr 13167  df-sca 13169  df-vsca 13170  df-tset 13172  df-rest 13317  df-topn 13318  df-topgen 13336  df-pt 13337  df-psr 14670  df-mplcoe 14671

This theorem is referenced by:  mplbascoe  14698  mplval2g  14702