Theorem ig1peuOLD 23123

Description: There is a unique monic polynomial of minimal degree in any nonzero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) Obsolete version of ig1peu 23122 as of 25-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

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
ig1peuOLD	|- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> E! g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) )

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

Proof of Theorem ig1peuOLD

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2451	. . . . . . . . . . 11 |- ( Base ` P ) = ( Base ` P )
2		ig1peu.u	. . . . . . . . . . 11 |- U = (LIdeal ` P )
3	1, 2	lidlss 18433	. . . . . . . . . 10 |- ( I e. U -> I C_ ( Base ` P ) )
4	3	3ad2ant2 1030	. . . . . . . . 9 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> I C_ ( Base ` P ) )
5	4	ssdifd 3569	. . . . . . . 8 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( I \ { .0. } ) C_ ( ( Base ` P ) \ { .0. } ) )
6		imass2 5204	. . . . . . . 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 17982	. . . . . . . . 9 |- ( R e. DivRing -> R e. Ring )
9	8	3ad2ant1 1029	. . . . . . . 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 23038	. . . . . . . 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 3442	. . . . . 6 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( D " ( I \ { .0. } ) ) C_ NN0 )
16		nn0uz 11193	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
17	15, 16	syl6sseq 3478	. . . . 5 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( D " ( I \ { .0. } ) ) C_ ( ZZ>= ` 0 ) )
18	11	ply1ring 18841	. . . . . . . . . 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 1009	. . . . . . . . 9 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> I e. U )
21	2, 12	lidl0cl 18435	. . . . . . . . 9 |- ( ( P e. Ring /\ I e. U ) -> .0. e. I )
22	19, 20, 21	syl2anc 667	. . . . . . . 8 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> .0. e. I )
23	22	snssd 4117	. . . . . . 7 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> { .0. } C_ I )
24		simp3 1010	. . . . . . . 8 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> I =/= { .0. } )
25	24	necomd 2679	. . . . . . 7 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> { .0. } =/= I )
26		pssdifn0 3827	. . . . . . 7 |- ( ( { .0. } C_ I /\ { .0. } =/= I ) -> ( I \ { .0. } ) =/= (/) )
27	23, 25, 26	syl2anc 667	. . . . . 6 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( I \ { .0. } ) =/= (/) )
28	10, 11, 1	deg1xrf 23030	. . . . . . . . . 10 |- D : ( Base ` P ) --> RR*
29		ffn 5728	. . . . . . . . . 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 3571	. . . . . . . 8 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( I \ { .0. } ) C_ ( Base ` P ) )
33		fnimaeq0 5697	. . . . . . . 8 |- ( ( D Fn ( Base ` P ) /\ ( I \ { .0. } ) C_ ( Base ` P ) ) -> ( ( D " ( I \ { .0. } ) ) = (/) <-> ( I \ { .0. } ) = (/) ) )
34	31, 32, 33	syl2anc 667	. . . . . . 7 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( ( D " ( I \ { .0. } ) ) = (/) <-> ( I \ { .0. } ) = (/) ) )
35	34	necon3bid 2668	. . . . . 6 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( ( D " ( I \ { .0. } ) ) =/= (/) <-> ( I \ { .0. } ) =/= (/) ) )
36	27, 35	mpbird 236	. . . . 5 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( D " ( I \ { .0. } ) ) =/= (/) )
37		infmssuzclOLD 11247	. . . . 5 |- ( ( ( D " ( I \ { .0. } ) ) C_ ( ZZ>= ` 0 ) /\ ( D " ( I \ { .0. } ) ) =/= (/) ) -> sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) e. ( D " ( I \ { .0. } ) ) )
38	17, 36, 37	syl2anc 667	. . . 4 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) e. ( D " ( I \ { .0. } ) ) )
39		fvelimab 5921	. . . . 5 |- ( ( D Fn ( Base ` P ) /\ ( I \ { .0. } ) C_ ( Base ` P ) ) -> ( sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) e. ( D " ( I \ { .0. } ) ) <-> E. h e. ( I \ { .0. } ) ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
40	31, 32, 39	syl2anc 667	. . . 4 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) e. ( D " ( I \ { .0. } ) ) <-> E. h e. ( I \ { .0. } ) ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
41	38, 40	mpbid 214	. . 3 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> E. h e. ( I \ { .0. } ) ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) )
42	19	adantr 467	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> P e. Ring )
43		simpl2 1012	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> I e. U )
44	9	adantr 467	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> R e. Ring )
45		eqid 2451	. . . . . . . . . . 11 |- (algSc ` P ) = (algSc ` P )
46		eqid 2451	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
47	11, 45, 46, 1	ply1sclf 18878	. . . . . . . . . 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 1011	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> R e. DivRing )
50	32	sselda 3432	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> h e. ( Base ` P ) )
51		eldifsni 4098	. . . . . . . . . . . . . 14 |- ( h e. ( I \ { .0. } ) -> h =/= .0. )
52	51	adantl 468	. . . . . . . . . . . . 13 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> h =/= .0. )
53		eqid 2451	. . . . . . . . . . . . . 14 |- (Unic1p ` R ) = (Unic1p ` R )
54	11, 1, 12, 53	drnguc1p 23121	. . . . . . . . . . . . 13 |- ( ( R e. DivRing /\ h e. ( Base ` P ) /\ h =/= .0. ) -> h e. (Unic1p ` R ) )
55	49, 50, 52, 54	syl3anc 1268	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> h e. (Unic1p ` R ) )
56		eqid 2451	. . . . . . . . . . . . 13 |- (Unit ` R ) = (Unit ` R )
57	10, 56, 53	uc1pldg 23099	. . . . . . . . . . . 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 2451	. . . . . . . . . . . 12 |- ( invr ` R ) = ( invr ` R )
60	56, 59	unitinvcl 17902	. . . . . . . . . . 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 667	. . . . . . . . . 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 17887	. . . . . . . . . 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 6023	. . . . . . . 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 3555	. . . . . . . . 9 |- ( h e. ( I \ { .0. } ) -> h e. I )
66	65	adantl 468	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> h e. I )
67		eqid 2451	. . . . . . . . 9 |- ( .r ` P ) = ( .r ` P )
68	2, 1, 67	lidlmcl 18441	. . . . . . . 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 1269	. . . . . . 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 23102	. . . . . . . 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 667	. . . . . . 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 3618	. . . . . 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 2451	. . . . . . . . . 10 |- (RLReg ` R ) = (RLReg ` R )
75	74, 56	unitrrg 18517	. . . . . . . . 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 3433	. . . . . . 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 23064	. . . . . . 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 1268	. . . . . 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 5865	. . . . . . . 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 2453	. . . . . . 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 3150	. . . . . 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 667	. . . . 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 2462	. . . . . 6 |- ( ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) -> ( ( D ` g ) = ( D ` h ) <-> ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
85	84	rexbidv 2901	. . . . 5 |- ( ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) -> ( E. g e. ( I i^i M ) ( D ` g ) = ( D ` h ) <-> E. g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
86	83, 85	syl5ibcom 224	. . . 4 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ h e. ( I \ { .0. } ) ) -> ( ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) -> E. g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
87	86	rexlimdva 2879	. . 3 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( E. h e. ( I \ { .0. } ) ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) -> E. g e. ( I i^i M ) ( D ` g ) = sup ( ( 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 ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) )
89		eqid 2451	. . . . . . 7 |- ( -g ` P ) = ( -g ` P )
90	9	ad2antrr 732	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) ) -> R e. Ring )
91		inss2 3653	. . . . . . . . 9 |- ( I i^i M ) C_ M
92		simprl 764	. . . . . . . . 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 3430	. . . . . . . 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 467	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) ) -> g e. M )
95		simprl 764	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) ) -> ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) )
96		simprr 766	. . . . . . . . 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 3430	. . . . . . . 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 467	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) ) -> h e. M )
99		simprr 766	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) ) -> ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) )
100	10, 70, 11, 89, 90, 94, 95, 98, 99	deg1submon1p 23103	. . . . . 6 |- ( ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) /\ ( ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) ) -> ( D ` ( g ( -g ` P ) h ) ) < sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) )
101	100	ex 436	. . . . 5 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( ( ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) -> ( D ` ( g ( -g ` P ) h ) ) < sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
102	17	ad2antrr 732	. . . . . . . . 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 732	. . . . . . . . . 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 467	. . . . . . . . . . . . 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 1012	. . . . . . . . . . . . 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 3652	. . . . . . . . . . . . . 14 |- ( I i^i M ) C_ I
108	107, 92	sseldi 3430	. . . . . . . . . . . . 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 3430	. . . . . . . . . . . . 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 18439	. . . . . . . . . . . . 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 1269	. . . . . . . . . . . 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 467	. . . . . . . . . . 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 463	. . . . . . . . . . 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 4097	. . . . . . . . . . 11 |- ( ( g ( -g ` P ) h ) e. ( I \ { .0. } ) <-> ( ( g ( -g ` P ) h ) e. I /\ ( g ( -g ` P ) h ) =/= .0. ) )
115	112, 113, 114	sylanbrc 670	. . . . . . . . . 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 6143	. . . . . . . . . 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 1268	. . . . . . . . 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		infmssuzleOLD 11246	. . . . . . . . 9 |- ( ( ( D " ( I \ { .0. } ) ) C_ ( ZZ>= ` 0 ) /\ ( D ` ( g ( -g ` P ) h ) ) e. ( D " ( I \ { .0. } ) ) ) -> sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) <_ ( D ` ( g ( -g ` P ) h ) ) )
119	102, 117, 118	syl2anc 667	. . . . . . . 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. ) -> sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) <_ ( D ` ( g ( -g ` P ) h ) ) )
120	119	ex 436	. . . . . . 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. -> sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) <_ ( D ` ( g ( -g ` P ) h ) ) ) )
121		imassrn 5179	. . . . . . . . . . 11 |- ( D " ( I \ { .0. } ) ) C_ ran D
122		frn 5735	. . . . . . . . . . . 12 |- ( D : ( Base ` P ) --> RR* -> ran D C_ RR* )
123	28, 122	ax-mp 5	. . . . . . . . . . 11 |- ran D C_ RR*
124	121, 123	sstri 3441	. . . . . . . . . 10 |- ( D " ( I \ { .0. } ) ) C_ RR*
125	124, 38	sseldi 3430	. . . . . . . . 9 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) e. RR* )
126	125	adantr 467	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) e. RR* )
127		ringgrp 17785	. . . . . . . . . . . 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 467	. . . . . . . . . 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 3443	. . . . . . . . . . . 12 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( I i^i M ) C_ ( Base ` P ) )
131	130	adantr 467	. . . . . . . . . . 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 3433	. . . . . . . . . 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 3433	. . . . . . . . . 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 16734	. . . . . . . . . 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 1268	. . . . . . . . 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 23031	. . . . . . . . 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 9699	. . . . . . . 8 |- ( ( sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) e. RR* /\ ( D ` ( g ( -g ` P ) h ) ) e. RR* ) -> ( sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) <_ ( D ` ( g ( -g ` P ) h ) ) <-> -. ( D ` ( g ( -g ` P ) h ) ) < sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
139	126, 137, 138	syl2anc 667	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) <_ ( D ` ( g ( -g ` P ) h ) ) <-> -. ( D ` ( g ( -g ` P ) h ) ) < sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
140	120, 139	sylibd 218	. . . . . 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 ) ) < sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
141	140	necon4ad 2643	. . . . 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 ) ) < sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) -> ( g ( -g ` P ) h ) = .0. ) )
142	101, 141	syld 45	. . . 4 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( ( ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) -> ( g ( -g ` P ) h ) = .0. ) )
143	1, 12, 89	grpsubeq0 16740	. . . . 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 1268	. . . 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 218	. . 3 |- ( ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) /\ ( g e. ( I i^i M ) /\ h e. ( I i^i M ) ) ) -> ( ( ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) -> g = h ) )
146	145	ralrimivva 2809	. 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 ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) -> g = h ) )
147		fveq2 5865	. . . 4 |- ( g = h -> ( D ` g ) = ( D ` h ) )
148	147	eqeq1d 2453	. . 3 |- ( g = h -> ( ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) <-> ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) )
149	148	reu4 3232	. 2 |- ( E! g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) <-> ( E. g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ A. g e. ( I i^i M ) A. h e. ( I i^i M ) ( ( ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) /\ ( D ` h ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) ) -> g = h ) ) )
150	88, 146, 149	sylanbrc 670	1 |- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> E! g e. ( I i^i M ) ( D ` g ) = sup ( ( D " ( I \ { .0. } ) ) , RR , `' < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   E!wreu 2739   \ cdif 3401   i^i cin 3403   C_ wss 3404   (/)c0 3731   {csn 3968   class class class wbr 4402   `'ccnv 4833   ran crn 4835   "cima 4837   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   supcsup 7954   RRcr 9538   0cc0 9539   RR*cxr 9674   < clt 9675   <_ cle 9676   NN0cn0 10869   ZZ>=cuz 11159   Basecbs 15121   .rcmulr 15191   0gc0g 15338   Grpcgrp 16669   -gcsg 16671   Ringcrg 17780  Unitcui 17867   invrcinvr 17899   DivRingcdr 17975  LIdealclidl 18393  RLRegcrlreg 18503  algSccascl 18535  Poly₁cpl1 18770  coe₁cco1 18771   deg₁ cdg1 23003  Monic_1pcmn1 23074  Unic_1pcuc1p 23075

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-8 1889  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-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  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-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  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-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  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-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  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-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-ofr 6532  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-0g 15340  df-gsum 15341  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-ghm 16881  df-cntz 16971  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-drng 17977  df-subrg 18006  df-lmod 18093  df-lss 18156  df-sra 18395  df-rgmod 18396  df-lidl 18397  df-rlreg 18507  df-ascl 18538  df-psr 18580  df-mvr 18581  df-mpl 18582  df-opsr 18584  df-psr1 18773  df-vr1 18774  df-ply1 18775  df-coe1 18776  df-cnfld 18971  df-mdeg 23004  df-deg1 23005  df-mon1 23080  df-uc1p 23081

This theorem is referenced by:  ig1pval3OLD  23132