Theorem ig1pval 22739

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 3115	. . . . 5 |- ( R e. V -> R e. _V )
3		fveq2 5848	. . . . . . . . . 10 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
4		ig1pval.p	. . . . . . . . . 10 |- P = (Poly1 ` R )
5	3, 4	syl6eqr 2513	. . . . . . . . 9 |- ( r = R -> (Poly1 ` r ) = P )
6	5	fveq2d 5852	. . . . . . . 8 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = (LIdeal ` P ) )
7		ig1pval.u	. . . . . . . 8 |- U = (LIdeal ` P )
8	6, 7	syl6eqr 2513	. . . . . . 7 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = U )
9	5	fveq2d 5852	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = ( 0g ` P ) )
10		ig1pval.z	. . . . . . . . . . 11 |- .0. = ( 0g ` P )
11	9, 10	syl6eqr 2513	. . . . . . . . . 10 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = .0. )
12	11	sneqd 4028	. . . . . . . . 9 |- ( r = R -> { ( 0g ` (Poly1 ` r ) ) } = { .0. } )
13	12	eqeq2d 2468	. . . . . . . 8 |- ( r = R -> ( i = { ( 0g ` (Poly1 ` r ) ) } <-> i = { .0. } ) )
14		fveq2 5848	. . . . . . . . . . 11 |- ( r = R -> (Monic1p ` r ) = (Monic1p ` R ) )
15		ig1pval.m	. . . . . . . . . . 11 |- M = (Monic1p ` R )
16	14, 15	syl6eqr 2513	. . . . . . . . . 10 |- ( r = R -> (Monic1p ` r ) = M )
17	16	ineq2d 3686	. . . . . . . . 9 |- ( r = R -> ( i i^i (Monic1p ` r ) ) = ( i i^i M ) )
18		fveq2 5848	. . . . . . . . . . . 12 |- ( r = R -> ( deg1 ` r ) = ( deg1 ` R ) )
19		ig1pval.d	. . . . . . . . . . . 12 |- D = ( deg1 ` R )
20	18, 19	syl6eqr 2513	. . . . . . . . . . 11 |- ( r = R -> ( deg1 ` r ) = D )
21	20	fveq1d 5850	. . . . . . . . . 10 |- ( r = R -> ( ( deg1 ` r ) ` g ) = ( D ` g ) )
22	12	difeq2d 3608	. . . . . . . . . . . 12 |- ( r = R -> ( i \ { ( 0g ` (Poly1 ` r ) ) } ) = ( i \ { .0. } ) )
23	20, 22	imaeq12d 5326	. . . . . . . . . . 11 |- ( r = R -> ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) = ( D " ( i \ { .0. } ) ) )
24	23	supeq1d 7897	. . . . . . . . . 10 |- ( r = R -> sup ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , `' < ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) )
25	21, 24	eqeq12d 2476	. . . . . . . . 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 6235	. . . . . . . 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 3956	. . . . . . 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 4517	. . . . . 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 22701	. . . . . 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 5858	. . . . . . . 8 |- (LIdeal ` P ) e. _V
31	7, 30	eqeltri 2538	. . . . . . 7 |- U e. _V
32	31	mptex 6118	. . . . . 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 5931	. . . . 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 2507	. . 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 5850	. 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 2458	. . . 4 |- ( i = I -> ( i = { .0. } <-> I = { .0. } ) )
38		ineq1 3679	. . . . 5 |- ( i = I -> ( i i^i M ) = ( I i^i M ) )
39		difeq1 3601	. . . . . . . 8 |- ( i = I -> ( i \ { .0. } ) = ( I \ { .0. } ) )
40	39	imaeq2d 5325	. . . . . . 7 |- ( i = I -> ( D " ( i \ { .0. } ) ) = ( D " ( I \ { .0. } ) ) )
41	40	supeq1d 7897	. . . . . 6 |- ( i = I -> sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) )
42	41	eqeq2d 2468	. . . . 5 |- ( i = I -> ( ( D ` g ) = sup ( ( D " ( i \ { .0. } ) ) , RR , `' < ) <-> ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
43	38, 42	riotaeqbidv 6235	. . . 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 3954	. . 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 2454	. . 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 5858	. . . . 5 |- ( 0g ` P ) e. _V
47	10, 46	eqeltri 2538	. . . 4 |- .0. e. _V
48		riotaex 6236	. . . 4 |- ( iota_ g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) e. _V
49	47, 48	ifex 3997	. . 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 5931	. 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 2515	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 367   = wceq 1398   e. wcel 1823   _Vcvv 3106   \ cdif 3458   i^i cin 3460   ifcif 3929   {csn 4016   |-> cmpt 4497   `'ccnv 4987   "cima 4991   ` cfv 5570   iota_crio 6231   supcsup 7892   RRcr 9480   < clt 9617   0gc0g 14929  LIdealclidl 18011  Poly₁cpl1 18411   deg₁ cdg1 22618  Monic_1pcmn1 22692  idlGen_1pcig1p 22696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-sup 7893  df-ig1p 22701

This theorem is referenced by:  ig1pval2  22740  ig1pval3  22741