Theorem ig1peu 23931

Description: There is a unique monic polynomial of minimal degree in any nonzero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)

Hypotheses

Ref	Expression
ig1peu.p	|- P = (Poly1 ` R )
ig1peu.u	|- U = (LIdeal ` P )
ig1peu.z	|- .0. = ( 0g ` P )
ig1peu.m	|- M = (Monic1p ` R )
ig1peu.d	|- D = ( deg1 ` R )

Assertion

Ref	Expression
ig1peu	|- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> E! g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) )

Distinct variable groups:   D, g   g, I   g, M   P, g   R, g   U, g   .0. , g

Proof of Theorem ig1peu

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . . . . 11 |- ( Base ` P ) = ( Base ` P )
2		ig1peu.u	. . . . . . . . . . 11 |- U = (LIdeal ` P )
3	1, 2	lidlss 19210	. . . . . . . . . 10 |- ( I e. U -> I C_ ( Base ` P ) )
4	3	3ad2ant2 1083	. . . . . . . . 9 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> I C_ ( Base ` P ) )
5	4	ssdifd 3746	. . . . . . . 8 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( I \ { .0. } ) C_ ( ( Base ` P ) \ { .0. } ) )
6		imass2 5501	. . . . . . . 8 |- ( ( I \ { .0. } ) C_ ( ( Base ` P ) \ { .0. } ) -> ( D " ( I \ { .0. } ) ) C_ ( D " ( ( Base ` P ) \ { .0. } ) ) )
7	5, 6	syl 17	. . . . . . 7 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( D " ( I \ { .0. } ) ) C_ ( D " ( ( Base ` P ) \ { .0. } ) ) )
8		drngring 18754	. . . . . . . . 9 |- ( R e. DivRing -> R e. Ring )
9	8	3ad2ant1 1082	. . . . . . . 8 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> R e. Ring )
10		ig1peu.d	. . . . . . . . 9 |- D = ( deg1 ` R )
11		ig1peu.p	. . . . . . . . 9 |- P = (Poly1 ` R )
12		ig1peu.z	. . . . . . . . 9 |- .0. = ( 0g ` P )
13	10, 11, 12, 1	deg1n0ima 23849	. . . . . . . 8 |- ( R e. Ring -> ( D " ( ( Base ` P ) \ { .0. } ) ) C_ NN0 )
14	9, 13	syl 17	. . . . . . 7 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( D " ( ( Base ` P ) \ { .0. } ) ) C_ NN0 )
15	7, 14	sstrd 3613	. . . . . 6 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( D " ( I \ { .0. } ) ) C_ NN0 )
16		nn0uz 11722	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
17	15, 16	syl6sseq 3651	. . . . 5 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( D " ( I \ { .0. } ) ) C_ ( ZZ>= ` 0 ) )
18	11	ply1ring 19618	. . . . . . . . . 10 |- ( R e. Ring -> P e. Ring )
19	9, 18	syl 17	. . . . . . . . 9 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> P e. Ring )
20		simp2 1062	. . . . . . . . 9 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> I e. U )
21	2, 12	lidl0cl 19212	. . . . . . . . 9 |- ( ( P e. Ring /\ I e. U ) -> .0. e. I )
22	19, 20, 21	syl2anc 693	. . . . . . . 8 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> .0. e. I )
23	22	snssd 4340	. . . . . . 7 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> { .0. } C_ I )
24		simp3 1063	. . . . . . . 8 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> I =/= { .0. } )
25	24	necomd 2849	. . . . . . 7 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> { .0. } =/= I )
26		pssdifn0 3944	. . . . . . 7 |- ( ( { .0. } C_ I /\ { .0. } =/= I ) -> ( I \ { .0. } ) =/= (/) )
27	23, 25, 26	syl2anc 693	. . . . . 6 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( I \ { .0. } ) =/= (/) )
28	10, 11, 1	deg1xrf 23841	. . . . . . . . . 10 |- D : ( Base ` P ) --> RR*
29		ffn 6045	. . . . . . . . . 10 |- ( D : ( Base ` P ) --> RR* -> D Fn ( Base ` P ) )
30	28, 29	ax-mp 5	. . . . . . . . 9 |- D Fn ( Base ` P )
31	30	a1i 11	. . . . . . . 8 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> D Fn ( Base ` P ) )
32	4	ssdifssd 3748	. . . . . . . 8 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( I \ { .0. } ) C_ ( Base ` P ) )
33		fnimaeq0 6013	. . . . . . . 8 |- ( ( D Fn ( Base ` P ) /\ ( I \ { .0. } ) C_ ( Base ` P ) ) -> ( ( D " ( I \ { .0. } ) ) = (/) <-> ( I \ { .0. } ) = (/) ) )
34	31, 32, 33	syl2anc 693	. . . . . . 7 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( ( D " ( I \ { .0. } ) ) = (/) <-> ( I \ { .0. } ) = (/) ) )
35	34	necon3bid 2838	. . . . . 6 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( ( D " ( I \ { .0. } ) ) =/= (/) <-> ( I \ { .0. } ) =/= (/) ) )
36	27, 35	mpbird 247	. . . . 5 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( D " ( I \ { .0. } ) ) =/= (/) )
37		infssuzcl 11772	. . . . 5 |- ( ( ( D " ( I \ { .0. } ) ) C_ ( ZZ>= ` 0 ) /\ ( D " ( I \ { .0. } ) ) =/= (/) ) -> inf ( ( D " ( I \ { .0. } ) ) , RR , < ) e. ( D " ( I \ { .0. } ) ) )
38	17, 36, 37	syl2anc 693	. . . 4 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> inf ( ( D " ( I \ { .0. } ) ) , RR , < ) e. ( D " ( I \ { .0. } ) ) )
39		fvelimab 6253	. . . . 5 |- ( ( D Fn ( Base ` P ) /\ ( I \ { .0. } ) C_ ( Base ` P ) ) -> (inf ( ( D " ( I \ { .0. } ) ) , RR , < ) e. ( D " ( I \ { .0. } ) ) <-> E. h e. ( I \ { .0. } ) ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
40	31, 32, 39	syl2anc 693	. . . 4 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> (inf ( ( D " ( I \ { .0. } ) ) , RR , < ) e. ( D " ( I \ { .0. } ) ) <-> E. h e. ( I \ { .0. } ) ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
41	38, 40	mpbid 222	. . 3 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> E. h e. ( I \ { .0. } ) ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) )
42	19	adantr 481	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> P e. Ring )
43		simpl2 1065	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> I e. U )
44	9	adantr 481	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> R e. Ring )
45		eqid 2622	. . . . . . . . . . 11 |- (algSc ` P ) = (algSc ` P )
46		eqid 2622	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
47	11, 45, 46, 1	ply1sclf 19655	. . . . . . . . . 10 |- ( R e. Ring -> (algSc ` P ) : ( Base ` R ) --> ( Base ` P ) )
48	44, 47	syl 17	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> (algSc ` P ) : ( Base ` R ) --> ( Base ` P ) )
49		simpl1 1064	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> R e. DivRing )
50	32	sselda 3603	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> h e. ( Base ` P ) )
51		eldifsni 4320	. . . . . . . . . . . . . 14 |- ( h e. ( I \ { .0. } ) -> h =/= .0. )
52	51	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> h =/= .0. )
53		eqid 2622	. . . . . . . . . . . . . 14 |- (Unic1p ` R ) = (Unic1p ` R )
54	11, 1, 12, 53	drnguc1p 23930	. . . . . . . . . . . . 13 |- ( ( R e. DivRing /\ h e. ( Base ` P ) /\ h =/= .0. ) -> h e. (Unic1p ` R ) )
55	49, 50, 52, 54	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> h e. (Unic1p ` R ) )
56		eqid 2622	. . . . . . . . . . . . 13 |- (Unit ` R ) = (Unit ` R )
57	10, 56, 53	uc1pldg 23908	. . . . . . . . . . . 12 |- ( h e. (Unic1p ` R ) -> ( (coe1 ` h ) ` ( D ` h ) ) e. (Unit ` R ) )
58	55, 57	syl 17	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> ( (coe1 ` h ) ` ( D ` h ) ) e. (Unit ` R ) )
59		eqid 2622	. . . . . . . . . . . 12 |- ( invr ` R ) = ( invr ` R )
60	56, 59	unitinvcl 18674	. . . . . . . . . . 11 |- ( ( R e. Ring /\ ( (coe1 ` h ) ` ( D ` h ) ) e. (Unit ` R ) ) -> ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) e. (Unit ` R ) )
61	44, 58, 60	syl2anc 693	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) e. (Unit ` R ) )
62	46, 56	unitcl 18659	. . . . . . . . . 10 |- ( ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) e. (Unit ` R ) -> ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) e. ( Base ` R ) )
63	61, 62	syl 17	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) e. ( Base ` R ) )
64	48, 63	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) e. ( Base ` P ) )
65		eldifi 3732	. . . . . . . . 9 |- ( h e. ( I \ { .0. } ) -> h e. I )
66	65	adantl 482	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> h e. I )
67		eqid 2622	. . . . . . . . 9 |- ( .r ` P ) = ( .r ` P )
68	2, 1, 67	lidlmcl 19217	. . . . . . . 8 |- ( ( ( P e. Ring /\ I e. U ) /\ ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) e. ( Base ` P ) /\ h e. I ) ) -> ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) e. I )
69	42, 43, 64, 66, 68	syl22anc 1327	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) e. I )
70		ig1peu.m	. . . . . . . . 9 |- M = (Monic1p ` R )
71	53, 70, 11, 67, 45, 10, 59	uc1pmon1p 23911	. . . . . . . 8 |- ( ( R e. Ring /\ h e. (Unic1p ` R ) ) -> ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) e. M )
72	44, 55, 71	syl2anc 693	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) e. M )
73	69, 72	elind 3798	. . . . . 6 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) e. ( I i^i M ) )
74		eqid 2622	. . . . . . . . . 10 |- (RLReg ` R ) = (RLReg ` R )
75	74, 56	unitrrg 19293	. . . . . . . . 9 |- ( R e. Ring -> (Unit ` R ) C_ (RLReg ` R ) )
76	44, 75	syl 17	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> (Unit ` R ) C_ (RLReg ` R ) )
77	76, 61	sseldd 3604	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) e. (RLReg ` R ) )
78	10, 11, 74, 1, 67, 45	deg1mul3 23875	. . . . . . 7 |- ( ( R e. Ring /\ ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) e. (RLReg ` R ) /\ h e. ( Base ` P ) ) -> ( D ` ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) ) = ( D ` h ) )
79	44, 77, 50, 78	syl3anc 1326	. . . . . 6 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> ( D ` ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) ) = ( D ` h ) )
80		fveq2 6191	. . . . . . . 8 |- ( g = ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) -> ( D ` g ) = ( D ` ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) ) )
81	80	eqeq1d 2624	. . . . . . 7 |- ( g = ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) -> ( ( D ` g ) = ( D ` h ) <-> ( D ` ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) ) = ( D ` h ) ) )
82	81	rspcev 3309	. . . . . 6 |- ( ( ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) e. ( I i^i M ) /\ ( D ` ( ( (algSc ` P ) ` ( ( invr ` R ) ` ( (coe1 ` h ) ` ( D ` h ) ) ) ) ( .r ` P ) h ) ) = ( D ` h ) ) -> E. g e. ( I i^i M ) ( D ` g ) = ( D ` h ) )
83	73, 79, 82	syl2anc 693	. . . . 5 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> E. g e. ( I i^i M ) ( D ` g ) = ( D ` h ) )
84		eqeq2 2633	. . . . . 6 |- ( ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) -> ( ( D ` g ) = ( D ` h ) <-> ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
85	84	rexbidv 3052	. . . . 5 |- ( ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) -> ( E. g e. ( I i^i M ) ( D ` g ) = ( D ` h ) <-> E. g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
86	83, 85	syl5ibcom 235	. . . 4 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> ( ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) -> E. g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
87	86	rexlimdva 3031	. . 3 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( E. h e. ( I \ { .0. } ) ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) -> E. g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
88	41, 87	mpd 15	. 2 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> E. g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) )
89		eqid 2622	. . . . . . 7 |- ( -g ` P ) = ( -g ` P )
90	9	ad2antrr 762	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) -> R e. Ring )
91		inss2 3834	. . . . . . . . 9 |- ( I i^i M ) C_ M
92		simprl 794	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> g e. ( I i^i M ) )
93	91, 92	sseldi 3601	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> g e. M )
94	93	adantr 481	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) -> g e. M )
95		simprl 794	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) -> ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) )
96		simprr 796	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> h e. ( I i^i M ) )
97	91, 96	sseldi 3601	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> h e. M )
98	97	adantr 481	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) -> h e. M )
99		simprr 796	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) -> ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) )
100	10, 70, 11, 89, 90, 94, 95, 98, 99	deg1submon1p 23912	. . . . . 6 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) -> ( D ` ( g ( -g ` P ) h ) ) < inf ( ( D " ( I \ { .0. } ) ) , RR , < ) )
101	100	ex 450	. . . . 5 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) -> ( D ` ( g ( -g ` P ) h ) ) < inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
102	17	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( g ( -g ` P ) h ) =/= .0. ) -> ( D " ( I \ { .0. } ) ) C_ ( ZZ>= ` 0 ) )
103	30	a1i 11	. . . . . . . . . 10 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( g ( -g ` P ) h ) =/= .0. ) -> D Fn ( Base ` P ) )
104	32	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( g ( -g ` P ) h ) =/= .0. ) -> ( I \ { .0. } ) C_ ( Base ` P ) )
105	19	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> P e. Ring )
106		simpl2 1065	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> I e. U )
107		inss1 3833	. . . . . . . . . . . . . 14 |- ( I i^i M ) C_ I
108	107, 92	sseldi 3601	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> g e. I )
109	107, 96	sseldi 3601	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> h e. I )
110	2, 89	lidlsubcl 19216	. . . . . . . . . . . . 13 |- ( ( ( P e. Ring /\ I e. U ) /\ ( g e. I /\ h e. I ) ) -> ( g ( -g ` P ) h ) e. I )
111	105, 106, 108, 109, 110	syl22anc 1327	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( g ( -g ` P ) h ) e. I )
112	111	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( g ( -g ` P ) h ) =/= .0. ) -> ( g ( -g ` P ) h ) e. I )
113		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( g ( -g ` P ) h ) =/= .0. ) -> ( g ( -g ` P ) h ) =/= .0. )
114		eldifsn 4317	. . . . . . . . . . 11 |- ( ( g ( -g ` P ) h ) e. ( I \ { .0. } ) <-> ( ( g ( -g ` P ) h ) e. I /\ ( g ( -g ` P ) h ) =/= .0. ) )
115	112, 113, 114	sylanbrc 698	. . . . . . . . . 10 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( g ( -g ` P ) h ) =/= .0. ) -> ( g ( -g ` P ) h ) e. ( I \ { .0. } ) )
116		fnfvima 6496	. . . . . . . . . 10 |- ( ( D Fn ( Base ` P ) /\ ( I \ { .0. } ) C_ ( Base ` P ) /\ ( g ( -g ` P ) h ) e. ( I \ { .0. } ) ) -> ( D ` ( g ( -g ` P ) h ) ) e. ( D " ( I \ { .0. } ) ) )
117	103, 104, 115, 116	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( g ( -g ` P ) h ) =/= .0. ) -> ( D ` ( g ( -g ` P ) h ) ) e. ( D " ( I \ { .0. } ) ) )
118		infssuzle 11771	. . . . . . . . 9 |- ( ( ( D " ( I \ { .0. } ) ) C_ ( ZZ>= ` 0 ) /\ ( D ` ( g ( -g ` P ) h ) ) e. ( D " ( I \ { .0. } ) ) ) -> inf ( ( D " ( I \ { .0. } ) ) , RR , < ) <_ ( D ` ( g ( -g ` P ) h ) ) )
119	102, 117, 118	syl2anc 693	. . . . . . . 8 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( g ( -g ` P ) h ) =/= .0. ) -> inf ( ( D " ( I \ { .0. } ) ) , RR , < ) <_ ( D ` ( g ( -g ` P ) h ) ) )
120	119	ex 450	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( ( g ( -g ` P ) h ) =/= .0. -> inf ( ( D " ( I \ { .0. } ) ) , RR , < ) <_ ( D ` ( g ( -g ` P ) h ) ) ) )
121		imassrn 5477	. . . . . . . . . . 11 |- ( D " ( I \ { .0. } ) ) C_ ran D
122		frn 6053	. . . . . . . . . . . 12 |- ( D : ( Base ` P ) --> RR* -> ran D C_ RR* )
123	28, 122	ax-mp 5	. . . . . . . . . . 11 |- ran D C_ RR*
124	121, 123	sstri 3612	. . . . . . . . . 10 |- ( D " ( I \ { .0. } ) ) C_ RR*
125	124, 38	sseldi 3601	. . . . . . . . 9 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> inf ( ( D " ( I \ { .0. } ) ) , RR , < ) e. RR* )
126	125	adantr 481	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> inf ( ( D " ( I \ { .0. } ) ) , RR , < ) e. RR* )
127		ringgrp 18552	. . . . . . . . . . . 12 |- ( P e. Ring -> P e. Grp )
128	19, 127	syl 17	. . . . . . . . . . 11 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> P e. Grp )
129	128	adantr 481	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> P e. Grp )
130	107, 4	syl5ss 3614	. . . . . . . . . . . 12 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( I i^i M ) C_ ( Base ` P ) )
131	130	adantr 481	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( I i^i M ) C_ ( Base ` P ) )
132	131, 92	sseldd 3604	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> g e. ( Base ` P ) )
133	131, 96	sseldd 3604	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> h e. ( Base ` P ) )
134	1, 89	grpsubcl 17495	. . . . . . . . . 10 |- ( ( P e. Grp /\ g e. ( Base ` P ) /\ h e. ( Base ` P ) ) -> ( g ( -g ` P ) h ) e. ( Base ` P ) )
135	129, 132, 133, 134	syl3anc 1326	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( g ( -g ` P ) h ) e. ( Base ` P ) )
136	10, 11, 1	deg1xrcl 23842	. . . . . . . . 9 |- ( ( g ( -g ` P ) h ) e. ( Base ` P ) -> ( D ` ( g ( -g ` P ) h ) ) e. RR* )
137	135, 136	syl 17	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( D ` ( g ( -g ` P ) h ) ) e. RR* )
138		xrlenlt 10103	. . . . . . . 8 |- ( (inf ( ( D " ( I \ { .0. } ) ) , RR , < ) e. RR* /\ ( D ` ( g ( -g ` P ) h ) ) e. RR* ) -> (inf ( ( D " ( I \ { .0. } ) ) , RR , < ) <_ ( D ` ( g ( -g ` P ) h ) ) <-> -. ( D ` ( g ( -g ` P ) h ) ) < inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
139	126, 137, 138	syl2anc 693	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> (inf ( ( D " ( I \ { .0. } ) ) , RR , < ) <_ ( D ` ( g ( -g ` P ) h ) ) <-> -. ( D ` ( g ( -g ` P ) h ) ) < inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
140	120, 139	sylibd 229	. . . . . 6 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( ( g ( -g ` P ) h ) =/= .0. -> -. ( D ` ( g ( -g ` P ) h ) ) < inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
141	140	necon4ad 2813	. . . . 5 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( ( D ` ( g ( -g ` P ) h ) ) < inf ( ( D " ( I \ { .0. } ) ) , RR , < ) -> ( g ( -g ` P ) h ) = .0. ) )
142	101, 141	syld 47	. . . 4 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) -> ( g ( -g ` P ) h ) = .0. ) )
143	1, 12, 89	grpsubeq0 17501	. . . . 5 |- ( ( P e. Grp /\ g e. ( Base ` P ) /\ h e. ( Base ` P ) ) -> ( ( g ( -g ` P ) h ) = .0. <-> g = h ) )
144	129, 132, 133, 143	syl3anc 1326	. . . 4 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( ( g ( -g ` P ) h ) = .0. <-> g = h ) )
145	142, 144	sylibd 229	. . 3 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) -> g = h ) )
146	145	ralrimivva 2971	. 2 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> A. g e. ( I i^i M ) A. h e. ( I i^i M ) ( ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) -> g = h ) )
147		fveq2 6191	. . . 4 |- ( g = h -> ( D ` g ) = ( D ` h ) )
148	147	eqeq1d 2624	. . 3 |- ( g = h -> ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) <-> ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
149	148	reu4 3400	. 2 |- ( E! g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) <-> ( E. g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ A. g e. ( I i^i M ) A. h e. ( I i^i M ) ( ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) /\ ( D ` h ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) -> g = h ) ) )
150	88, 146, 149	sylanbrc 698	1 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> E! g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   RR*cxr 10073   < clt 10074   <_ cle 10075   NN0cn0 11292   ZZ>=cuz 11687   Basecbs 15857   .rcmulr 15942   0gc0g 16100   Grpcgrp 17422   -gcsg 17424   Ringcrg 18547  Unitcui 18639   invrcinvr 18671   DivRingcdr 18747  LIdealclidl 19170  RLRegcrlreg 19279  algSccascl 19311  Poly₁cpl1 19547  coe₁cco1 19548   deg₁ cdg1 23814  Monic_1pcmn1 23885  Unic_1pcuc1p 23886

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rlreg 19283  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-cnfld 19747  df-mdeg 23815  df-deg1 23816  df-mon1 23890  df-uc1p 23891

This theorem is referenced by:  ig1pval3  23934