Theorem ig1pval 22336

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 3122	. . . . 5 |- ( R e. V -> R e. _V )
3		fveq2 5866	. . . . . . . . . 10 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
4		ig1pval.p	. . . . . . . . . 10 |- P = (Poly1 ` R )
5	3, 4	syl6eqr 2526	. . . . . . . . 9 |- ( r = R -> (Poly1 ` r ) = P )
6	5	fveq2d 5870	. . . . . . . 8 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = (LIdeal ` P ) )
7		ig1pval.u	. . . . . . . 8 |- U = (LIdeal ` P )
8	6, 7	syl6eqr 2526	. . . . . . 7 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = U )
9	5	fveq2d 5870	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = ( 0g ` P ) )
10		ig1pval.z	. . . . . . . . . . 11 |- .0. = ( 0g ` P )
11	9, 10	syl6eqr 2526	. . . . . . . . . 10 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = .0. )
12	11	sneqd 4039	. . . . . . . . 9 |- ( r = R -> { ( 0g ` (Poly1 ` r ) ) } = { .0. } )
13	12	eqeq2d 2481	. . . . . . . 8 |- ( r = R -> ( i = { ( 0g ` (Poly1 ` r ) ) } <-> i = { .0. } ) )
14		fveq2 5866	. . . . . . . . . . 11 |- ( r = R -> (Monic1p ` r ) = (Monic1p ` R ) )
15		ig1pval.m	. . . . . . . . . . 11 |- M = (Monic1p ` R )
16	14, 15	syl6eqr 2526	. . . . . . . . . 10 |- ( r = R -> (Monic1p ` r ) = M )
17	16	ineq2d 3700	. . . . . . . . 9 |- ( r = R -> ( i i^i (Monic1p ` r ) ) = ( i i^i M ) )
18		fveq2 5866	. . . . . . . . . . . 12 |- ( r = R -> ( deg1 ` r ) = ( deg1 ` R ) )
19		ig1pval.d	. . . . . . . . . . . 12 |- D = ( deg1 ` R )
20	18, 19	syl6eqr 2526	. . . . . . . . . . 11 |- ( r = R -> ( deg1 ` r ) = D )
21	20	fveq1d 5868	. . . . . . . . . 10 |- ( r = R -> ( ( deg1 ` r ) ` g ) = ( D ` g ) )
22	12	difeq2d 3622	. . . . . . . . . . . 12 |- ( r = R -> ( i \ { ( 0g ` (Poly1 ` r ) ) } ) = ( i \ { .0. } ) )
23	20, 22	imaeq12d 5338	. . . . . . . . . . 11 |- ( r = R -> ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) = ( D " ( i \ { .0. } ) ) )
24	23	supeq1d 7906	. . . . . . . . . 10 |- ( r = R -> sup ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , `' < ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) )
25	21, 24	eqeq12d 2489	. . . . . . . . 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 6248	. . . . . . . 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 3966	. . . . . . 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 4525	. . . . . 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 22298	. . . . . 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 5876	. . . . . . . 8 |- (LIdeal ` P ) e. _V
31	7, 30	eqeltri 2551	. . . . . . 7 |- U e. _V
32	31	mptex 6131	. . . . . 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 5950	. . . . 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 2520	. . 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 5868	. 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 2471	. . . 4 |- ( i = I -> ( i = { .0. } <-> I = { .0. } ) )
38		ineq1 3693	. . . . 5 |- ( i = I -> ( i i^i M ) = ( I i^i M ) )
39		difeq1 3615	. . . . . . . 8 |- ( i = I -> ( i \ { .0. } ) = ( I \ { .0. } ) )
40	39	imaeq2d 5337	. . . . . . 7 |- ( i = I -> ( D " ( i \ { .0. } ) ) = ( D " ( I \ { .0. } ) ) )
41	40	supeq1d 7906	. . . . . 6 |- ( i = I -> sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) )
42	41	eqeq2d 2481	. . . . 5 |- ( i = I -> ( ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) <-> ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
43	38, 42	riotaeqbidv 6248	. . . 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 3964	. . 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 2467	. . 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 5876	. . . . 5 |- ( 0g ` P ) e. _V
47	10, 46	eqeltri 2551	. . . 4 |- .0. e. _V
48		riotaex 6249	. . . 4 |- ( iota_ g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) e. _V
49	47, 48	ifex 4008	. . 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 5950	. 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 2528	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 1379   e. wcel 1767   _Vcvv 3113   \ cdif 3473   i^i cin 3475   ifcif 3939   {csn 4027   |-> cmpt 4505   `'ccnv 4998   "cima 5002   ` cfv 5588   iota_crio 6244   supcsup 7900   RRcr 9491   < clt 9628   0gc0g 14695  LIdealclidl 17616  Poly₁cpl1 18015   deg₁ cdg1 22215  Monic_1pcmn1 22289  idlGen_1pcig1p 22293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-sup 7901  df-ig1p 22298

This theorem is referenced by:  ig1pval2  22337  ig1pval3  22338