Theorem ig1pval 21649

Description: Substitutions for the polynomial ideal generator function. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Hypotheses

Ref	Expression
ig1pval.p	|- P = (Poly1 ` R )
ig1pval.g	|- G = (idlGen1p ` R )
ig1pval.z	|- .0. = ( 0g ` P )
ig1pval.u	|- U = (LIdeal ` P )
ig1pval.d	|- D = ( deg1 ` R )
ig1pval.m	|- M = (Monic1p ` R )

Assertion

Ref	Expression
ig1pval	|- ( ( R e. V /\ I e. U ) -> ( G ` I ) = if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) ) )

Distinct variable groups:   g, I   g, M   R, g

Allowed substitution hints:   D( g)   P( g)   U( g)   G( g)   V( g)   .0. ( g)

Proof of Theorem ig1pval

Dummy variables i r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ig1pval.g	. . . 4 |- G = (idlGen1p ` R )
2		elex 2986	. . . . 5 |- ( R e. V -> R e. _V )
3		fveq2 5696	. . . . . . . . . 10 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
4		ig1pval.p	. . . . . . . . . 10 |- P = (Poly1 ` R )
5	3, 4	syl6eqr 2493	. . . . . . . . 9 |- ( r = R -> (Poly1 ` r ) = P )
6	5	fveq2d 5700	. . . . . . . 8 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = (LIdeal ` P ) )
7		ig1pval.u	. . . . . . . 8 |- U = (LIdeal ` P )
8	6, 7	syl6eqr 2493	. . . . . . 7 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = U )
9	5	fveq2d 5700	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = ( 0g ` P ) )
10		ig1pval.z	. . . . . . . . . . 11 |- .0. = ( 0g ` P )
11	9, 10	syl6eqr 2493	. . . . . . . . . 10 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = .0. )
12	11	sneqd 3894	. . . . . . . . 9 |- ( r = R -> { ( 0g ` (Poly1 ` r ) ) } = { .0. } )
13	12	eqeq2d 2454	. . . . . . . 8 |- ( r = R -> ( i = { ( 0g ` (Poly1 ` r ) ) } <-> i = { .0. } ) )
14		fveq2 5696	. . . . . . . . . . 11 |- ( r = R -> (Monic1p ` r ) = (Monic1p ` R ) )
15		ig1pval.m	. . . . . . . . . . 11 |- M = (Monic1p ` R )
16	14, 15	syl6eqr 2493	. . . . . . . . . 10 |- ( r = R -> (Monic1p ` r ) = M )
17	16	ineq2d 3557	. . . . . . . . 9 |- ( r = R -> ( i i^i (Monic1p ` r ) ) = ( i i^i M ) )
18		fveq2 5696	. . . . . . . . . . . 12 |- ( r = R -> ( deg1 ` r ) = ( deg1 ` R ) )
19		ig1pval.d	. . . . . . . . . . . 12 |- D = ( deg1 ` R )
20	18, 19	syl6eqr 2493	. . . . . . . . . . 11 |- ( r = R -> ( deg1 ` r ) = D )
21	20	fveq1d 5698	. . . . . . . . . 10 |- ( r = R -> ( ( deg1 ` r ) ` g ) = ( D ` g ) )
22	12	difeq2d 3479	. . . . . . . . . . . 12 |- ( r = R -> ( i \ { ( 0g ` (Poly1 ` r ) ) } ) = ( i \ { .0. } ) )
23	20, 22	imaeq12d 5175	. . . . . . . . . . 11 |- ( r = R -> ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) = ( D " ( i \ { .0. } ) ) )
24	23	supeq1d 7701	. . . . . . . . . 10 |- ( r = R -> sup ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , `' < ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) )
25	21, 24	eqeq12d 2457	. . . . . . . . 9 |- ( r = R -> ( ( ( deg1 ` r ) ` g ) = sup ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , `' < ) <-> ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) )
26	17, 25	riotaeqbidv 6060	. . . . . . . 8 |- ( r = R -> ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = sup ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , `' < ) ) = ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) )
27	13, 11, 26	ifbieq12d 3821	. . . . . . 7 |- ( r = R -> if ( i = { ( 0g ` (Poly1 ` r ) ) } , ( 0g ` (Poly1 ` r ) ) , ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = sup ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , `' < ) ) ) = if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) ) )
28	8, 27	mpteq12dv 4375	. . . . . 6 |- ( r = R -> ( i e. (LIdeal ` (Poly1 ` r ) ) |-> if ( i = { ( 0g ` (Poly1 ` r ) ) } , ( 0g ` (Poly1 ` r ) ) , ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = sup ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , `' < ) ) ) ) = ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) ) ) )
29		df-ig1p 21611	. . . . . 6 |- idlGen1p = ( r e. _V |-> ( i e. (LIdeal ` (Poly1 ` r ) ) |-> if ( i = { ( 0g ` (Poly1 ` r ) ) } , ( 0g ` (Poly1 ` r ) ) , ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = sup ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , `' < ) ) ) ) )
30		fvex 5706	. . . . . . . 8 |- (LIdeal ` P ) e. _V
31	7, 30	eqeltri 2513	. . . . . . 7 |- U e. _V
32	31	mptex 5953	. . . . . 6 |- ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) ) ) e. _V
33	28, 29, 32	fvmpt 5779	. . . . 5 |- ( R e. _V -> (idlGen1p ` R ) = ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) ) ) )
34	2, 33	syl 16	. . . 4 |- ( R e. V -> (idlGen1p ` R ) = ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) ) ) )
35	1, 34	syl5eq 2487	. . 3 |- ( R e. V -> G = ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) ) ) )
36	35	fveq1d 5698	. 2 |- ( R e. V -> ( G ` I ) = ( ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) ) ) ` I ) )
37		eqeq1 2449	. . . 4 |- ( i = I -> ( i = { .0. } <-> I = { .0. } ) )
38		ineq1 3550	. . . . 5 |- ( i = I -> ( i i^i M ) = ( I i^i M ) )
39		difeq1 3472	. . . . . . . 8 |- ( i = I -> ( i \ { .0. } ) = ( I \ { .0. } ) )
40	39	imaeq2d 5174	. . . . . . 7 |- ( i = I -> ( D " ( i \ { .0. } ) ) = ( D " ( I \ { .0. } ) ) )
41	40	supeq1d 7701	. . . . . 6 |- ( i = I -> sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) )
42	41	eqeq2d 2454	. . . . 5 |- ( i = I -> ( ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) <-> ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
43	38, 42	riotaeqbidv 6060	. . . 4 |- ( i = I -> ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) = ( iota_ g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
44	37, 43	ifbieq2d 3819	. . 3 |- ( i = I -> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) ) = if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) ) )
45		eqid 2443	. . 3 |- ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) ) ) = ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) ) )
46		fvex 5706	. . . . 5 |- ( 0g ` P ) e. _V
47	10, 46	eqeltri 2513	. . . 4 |- .0. e. _V
48		riotaex 6061	. . . 4 |- ( iota_ g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) e. _V
49	47, 48	ifex 3863	. . 3 |- if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) ) e. _V
50	44, 45, 49	fvmpt 5779	. 2 |- ( I e. U -> ( ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) ) ) ) ` I ) = if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) ) )
51	36, 50	sylan9eq 2495	1 |- ( ( R e. V /\ I e. U ) -> ( G ` I ) = if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2977   \ cdif 3330   i^i cin 3332   ifcif 3796   {csn 3882   e. cmpt 4355   `'ccnv 4844   "cima 4848   ` cfv 5423   iota_crio 6056   supcsup 7695   RRcr 9286   < clt 9423   0gc0g 14383  LIdealclidl 17256  Poly₁cpl1 17638   deg₁ cdg1 21528  Monic_1pcmn1 21602  idlGen_1pcig1p 21606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-sup 7696  df-ig1p 21611

This theorem is referenced by:  ig1pval2  21650  ig1pval3  21651