Theorem ig1pval 23932

Description: Substitutions for the polynomial ideal generator function. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)

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 ) = inf ( ( 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 3212	. . . . 5 |- ( R e. V -> R e. _V )
3		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
4		ig1pval.p	. . . . . . . . . 10 |- P = (Poly1 ` R )
5	3, 4	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> (Poly1 ` r ) = P )
6	5	fveq2d 6195	. . . . . . . 8 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = (LIdeal ` P ) )
7		ig1pval.u	. . . . . . . 8 |- U = (LIdeal ` P )
8	6, 7	syl6eqr 2674	. . . . . . 7 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = U )
9	5	fveq2d 6195	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = ( 0g ` P ) )
10		ig1pval.z	. . . . . . . . . . 11 |- .0. = ( 0g ` P )
11	9, 10	syl6eqr 2674	. . . . . . . . . 10 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = .0. )
12	11	sneqd 4189	. . . . . . . . 9 |- ( r = R -> { ( 0g ` (Poly1 ` r ) ) } = { .0. } )
13	12	eqeq2d 2632	. . . . . . . 8 |- ( r = R -> ( i = { ( 0g ` (Poly1 ` r ) ) } <-> i = { .0. } ) )
14		fveq2 6191	. . . . . . . . . . 11 |- ( r = R -> (Monic1p ` r ) = (Monic1p ` R ) )
15		ig1pval.m	. . . . . . . . . . 11 |- M = (Monic1p ` R )
16	14, 15	syl6eqr 2674	. . . . . . . . . 10 |- ( r = R -> (Monic1p ` r ) = M )
17	16	ineq2d 3814	. . . . . . . . 9 |- ( r = R -> ( i i^i (Monic1p ` r ) ) = ( i i^i M ) )
18		fveq2 6191	. . . . . . . . . . . 12 |- ( r = R -> ( deg1 ` r ) = ( deg1 ` R ) )
19		ig1pval.d	. . . . . . . . . . . 12 |- D = ( deg1 ` R )
20	18, 19	syl6eqr 2674	. . . . . . . . . . 11 |- ( r = R -> ( deg1 ` r ) = D )
21	20	fveq1d 6193	. . . . . . . . . 10 |- ( r = R -> ( ( deg1 ` r ) ` g ) = ( D ` g ) )
22	12	difeq2d 3728	. . . . . . . . . . . 12 |- ( r = R -> ( i \ { ( 0g ` (Poly1 ` r ) ) } ) = ( i \ { .0. } ) )
23	20, 22	imaeq12d 5467	. . . . . . . . . . 11 |- ( r = R -> ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) = ( D " ( i \ { .0. } ) ) )
24	23	infeq1d 8383	. . . . . . . . . 10 |- ( r = R -> inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) )
25	21, 24	eqeq12d 2637	. . . . . . . . 9 |- ( r = R -> ( ( ( deg1 ` r ) ` g ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) <-> ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) )
26	17, 25	riotaeqbidv 6614	. . . . . . . 8 |- ( r = R -> ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) ) = ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) )
27	13, 11, 26	ifbieq12d 4113	. . . . . . 7 |- ( r = R -> if ( i = { ( 0g ` (Poly1 ` r ) ) } , ( 0g ` (Poly1 ` r ) ) , ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) ) ) = if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) ) )
28	8, 27	mpteq12dv 4733	. . . . . 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 ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) ) ) ) = ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) ) ) )
29		df-ig1p 23894	. . . . . 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 ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) ) ) ) )
30		fvex 6201	. . . . . . . 8 |- (LIdeal ` P ) e. _V
31	7, 30	eqeltri 2697	. . . . . . 7 |- U e. _V
32	31	mptex 6486	. . . . . 6 |- ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) ) ) e. _V
33	28, 29, 32	fvmpt 6282	. . . . 5 |- ( R e. _V -> (idlGen1p ` R ) = ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) ) ) )
34	2, 33	syl 17	. . . 4 |- ( R e. V -> (idlGen1p ` R ) = ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) ) ) )
35	1, 34	syl5eq 2668	. . 3 |- ( R e. V -> G = ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) ) ) )
36	35	fveq1d 6193	. 2 |- ( R e. V -> ( G ` I ) = ( ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) ) ) ` I ) )
37		eqeq1 2626	. . . 4 |- ( i = I -> ( i = { .0. } <-> I = { .0. } ) )
38		ineq1 3807	. . . . 5 |- ( i = I -> ( i i^i M ) = ( I i^i M ) )
39		difeq1 3721	. . . . . . . 8 |- ( i = I -> ( i \ { .0. } ) = ( I \ { .0. } ) )
40	39	imaeq2d 5466	. . . . . . 7 |- ( i = I -> ( D " ( i \ { .0. } ) ) = ( D " ( I \ { .0. } ) ) )
41	40	infeq1d 8383	. . . . . 6 |- ( i = I -> inf ( ( D " ( i \ { .0. } ) ) , RR , < ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) )
42	41	eqeq2d 2632	. . . . 5 |- ( i = I -> ( ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) <-> ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
43	38, 42	riotaeqbidv 6614	. . . 4 |- ( i = I -> ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) = ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
44	37, 43	ifbieq2d 4111	. . 3 |- ( i = I -> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) ) = if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) )
45		eqid 2622	. . 3 |- ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) ) ) = ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) ) )
46		fvex 6201	. . . . 5 |- ( 0g ` P ) e. _V
47	10, 46	eqeltri 2697	. . . 4 |- .0. e. _V
48		riotaex 6615	. . . 4 |- ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) e. _V
49	47, 48	ifex 4156	. . 3 |- if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) e. _V
50	44, 45, 49	fvmpt 6282	. 2 |- ( I e. U -> ( ( i e. U |-> if ( i = { .0. } , .0. , ( iota_ g e. ( i i^i M ) ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) ) ) ) ` I ) = if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) )
51	36, 50	sylan9eq 2676	1 |- ( ( R e. V /\ I e. U ) -> ( G ` I ) = if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   i^i cin 3573   ifcif 4086   {csn 4177   |-> cmpt 4729   "cima 5117   ` cfv 5888   iota_crio 6610  infcinf 8347   RRcr 9935   < clt 10074   0gc0g 16100  LIdealclidl 19170  Poly₁cpl1 19547   deg₁ cdg1 23814  Monic_1pcmn1 23885  idlGen_1pcig1p 23889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-sup 8348  df-inf 8349  df-ig1p 23894

This theorem is referenced by:  ig1pval2  23933  ig1pval3  23934