Theorem ig1pval 23124

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 3054	. . . . 5 |- ( R e. V -> R e. _V )
3		fveq2 5865	. . . . . . . . . 10 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
4		ig1pval.p	. . . . . . . . . 10 |- P = (Poly1 ` R )
5	3, 4	syl6eqr 2503	. . . . . . . . 9 |- ( r = R -> (Poly1 ` r ) = P )
6	5	fveq2d 5869	. . . . . . . 8 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = (LIdeal ` P ) )
7		ig1pval.u	. . . . . . . 8 |- U = (LIdeal ` P )
8	6, 7	syl6eqr 2503	. . . . . . 7 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = U )
9	5	fveq2d 5869	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = ( 0g ` P ) )
10		ig1pval.z	. . . . . . . . . . 11 |- .0. = ( 0g ` P )
11	9, 10	syl6eqr 2503	. . . . . . . . . 10 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = .0. )
12	11	sneqd 3980	. . . . . . . . 9 |- ( r = R -> { ( 0g ` (Poly1 ` r ) ) } = { .0. } )
13	12	eqeq2d 2461	. . . . . . . 8 |- ( r = R -> ( i = { ( 0g ` (Poly1 ` r ) ) } <-> i = { .0. } ) )
14		fveq2 5865	. . . . . . . . . . 11 |- ( r = R -> (Monic1p ` r ) = (Monic1p ` R ) )
15		ig1pval.m	. . . . . . . . . . 11 |- M = (Monic1p ` R )
16	14, 15	syl6eqr 2503	. . . . . . . . . 10 |- ( r = R -> (Monic1p ` r ) = M )
17	16	ineq2d 3634	. . . . . . . . 9 |- ( r = R -> ( i i^i (Monic1p ` r ) ) = ( i i^i M ) )
18		fveq2 5865	. . . . . . . . . . . 12 |- ( r = R -> ( deg1 ` r ) = ( deg1 ` R ) )
19		ig1pval.d	. . . . . . . . . . . 12 |- D = ( deg1 ` R )
20	18, 19	syl6eqr 2503	. . . . . . . . . . 11 |- ( r = R -> ( deg1 ` r ) = D )
21	20	fveq1d 5867	. . . . . . . . . 10 |- ( r = R -> ( ( deg1 ` r ) ` g ) = ( D ` g ) )
22	12	difeq2d 3551	. . . . . . . . . . . 12 |- ( r = R -> ( i \ { ( 0g ` (Poly1 ` r ) ) } ) = ( i \ { .0. } ) )
23	20, 22	imaeq12d 5169	. . . . . . . . . . 11 |- ( r = R -> ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) = ( D " ( i \ { .0. } ) ) )
24	23	infeq1d 7993	. . . . . . . . . 10 |- ( r = R -> inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) )
25	21, 24	eqeq12d 2466	. . . . . . . . 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 6255	. . . . . . . 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 3908	. . . . . . 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 4481	. . . . . 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 23084	. . . . . 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 5875	. . . . . . . 8 |- (LIdeal ` P ) e. _V
31	7, 30	eqeltri 2525	. . . . . . 7 |- U e. _V
32	31	mptex 6136	. . . . . 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 5948	. . . . 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 2497	. . 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 5867	. 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 2455	. . . 4 |- ( i = I -> ( i = { .0. } <-> I = { .0. } ) )
38		ineq1 3627	. . . . 5 |- ( i = I -> ( i i^i M ) = ( I i^i M ) )
39		difeq1 3544	. . . . . . . 8 |- ( i = I -> ( i \ { .0. } ) = ( I \ { .0. } ) )
40	39	imaeq2d 5168	. . . . . . 7 |- ( i = I -> ( D " ( i \ { .0. } ) ) = ( D " ( I \ { .0. } ) ) )
41	40	infeq1d 7993	. . . . . 6 |- ( i = I -> inf ( ( D " ( i \ { .0. } ) ) , RR , < ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) )
42	41	eqeq2d 2461	. . . . 5 |- ( i = I -> ( ( D ` g ) = inf ( ( D " ( i \ { .0. } ) ) , RR , < ) <-> ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
43	38, 42	riotaeqbidv 6255	. . . 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 3906	. . 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 2451	. . 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 5875	. . . . 5 |- ( 0g ` P ) e. _V
47	10, 46	eqeltri 2525	. . . 4 |- .0. e. _V
48		riotaex 6256	. . . 4 |- ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) e. _V
49	47, 48	ifex 3949	. . 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 5948	. 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 2505	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 371   = wceq 1444   e. wcel 1887   _Vcvv 3045   \ cdif 3401   i^i cin 3403   ifcif 3881   {csn 3968   |-> cmpt 4461   "cima 4837   ` cfv 5582   iota_crio 6251  infcinf 7955   RRcr 9538   < clt 9675   0gc0g 15338  LIdealclidl 18393  Poly₁cpl1 18770   deg₁ cdg1 23003  Monic_1pcmn1 23074  idlGen_1pcig1p 23078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-sup 7956  df-inf 7957  df-ig1p 23084

This theorem is referenced by:  ig1pval2  23125  ig1pval3  23126