Theorem ig1pdvds 23936

Description: The monic generator of an ideal divides all elements of the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Sep-2020.)

Hypotheses

Ref	Expression
ig1pval.p	|- P = (Poly1 ` R )
ig1pval.g	|- G = (idlGen1p ` R )
ig1pcl.u	|- U = (LIdeal ` P )
ig1pdvds.d	|- .|| = ( ||r ` P )

Assertion

Ref	Expression
ig1pdvds	|- ( ( R e. DivRing /\ I e. U /\ X e. I ) -> ( G ` I ) .|| X )

Proof of Theorem ig1pdvds

Step	Hyp	Ref	Expression
1		drngring 18754	. . . . . . 7 |- ( R e. DivRing -> R e. Ring )
2		ig1pval.p	. . . . . . . 8 |- P = (Poly1 ` R )
3	2	ply1ring 19618	. . . . . . 7 |- ( R e. Ring -> P e. Ring )
4	1, 3	syl 17	. . . . . 6 |- ( R e. DivRing -> P e. Ring )
5	4	3ad2ant1 1082	. . . . 5 |- ( ( R e. DivRing /\ I e. U /\ X e. I ) -> P e. Ring )
6		eqid 2622	. . . . . . . 8 |- ( Base ` P ) = ( Base ` P )
7		ig1pcl.u	. . . . . . . 8 |- U = (LIdeal ` P )
8	6, 7	lidlss 19210	. . . . . . 7 |- ( I e. U -> I C_ ( Base ` P ) )
9	8	3ad2ant2 1083	. . . . . 6 |- ( ( R e. DivRing /\ I e. U /\ X e. I ) -> I C_ ( Base ` P ) )
10		ig1pval.g	. . . . . . . 8 |- G = (idlGen1p ` R )
11	2, 10, 7	ig1pcl 23935	. . . . . . 7 |- ( ( R e. DivRing /\ I e. U ) -> ( G ` I ) e. I )
12	11	3adant3 1081	. . . . . 6 |- ( ( R e. DivRing /\ I e. U /\ X e. I ) -> ( G ` I ) e. I )
13	9, 12	sseldd 3604	. . . . 5 |- ( ( R e. DivRing /\ I e. U /\ X e. I ) -> ( G ` I ) e. ( Base ` P ) )
14		ig1pdvds.d	. . . . . 6 |- .|| = ( ||r ` P )
15		eqid 2622	. . . . . 6 |- ( 0g ` P ) = ( 0g ` P )
16	6, 14, 15	dvdsr01 18655	. . . . 5 |- ( ( P e. Ring /\ ( G ` I ) e. ( Base ` P ) ) -> ( G ` I ) .|| ( 0g ` P ) )
17	5, 13, 16	syl2anc 693	. . . 4 |- ( ( R e. DivRing /\ I e. U /\ X e. I ) -> ( G ` I ) .|| ( 0g ` P ) )
18	17	adantr 481	. . 3 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I = { ( 0g ` P ) } ) -> ( G ` I ) .|| ( 0g ` P ) )
19		eleq2 2690	. . . . . 6 |- ( I = { ( 0g ` P ) } -> ( X e. I <-> X e. { ( 0g ` P ) } ) )
20	19	biimpac 503	. . . . 5 |- ( ( X e. I /\ I = { ( 0g ` P ) } ) -> X e. { ( 0g ` P ) } )
21	20	3ad2antl3 1225	. . . 4 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I = { ( 0g ` P ) } ) -> X e. { ( 0g ` P ) } )
22		elsni 4194	. . . 4 |- ( X e. { ( 0g ` P ) } -> X = ( 0g ` P ) )
23	21, 22	syl 17	. . 3 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I = { ( 0g ` P ) } ) -> X = ( 0g ` P ) )
24	18, 23	breqtrrd 4681	. 2 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I = { ( 0g ` P ) } ) -> ( G ` I ) .|| X )
25		simpl1 1064	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> R e. DivRing )
26	25, 1	syl 17	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> R e. Ring )
27		simpl2 1065	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> I e. U )
28	27, 8	syl 17	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> I C_ ( Base ` P ) )
29		simpl3 1066	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> X e. I )
30	28, 29	sseldd 3604	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> X e. ( Base ` P ) )
31		simpr 477	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> I =/= { ( 0g ` P ) } )
32		eqid 2622	. . . . . . . . . . 11 |- ( deg1 ` R ) = ( deg1 ` R )
33		eqid 2622	. . . . . . . . . . 11 |- (Monic1p ` R ) = (Monic1p ` R )
34	2, 10, 15, 7, 32, 33	ig1pval3 23934	. . . . . . . . . 10 |- ( ( R e. DivRing /\ I e. U /\ I =/= { ( 0g ` P ) } ) -> ( ( G ` I ) e. I /\ ( G ` I ) e. (Monic1p ` R ) /\ ( ( deg1 ` R ) ` ( G ` I ) ) = inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) ) )
35	25, 27, 31, 34	syl3anc 1326	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( G ` I ) e. I /\ ( G ` I ) e. (Monic1p ` R ) /\ ( ( deg1 ` R ) ` ( G ` I ) ) = inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) ) )
36	35	simp2d 1074	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( G ` I ) e. (Monic1p ` R ) )
37		eqid 2622	. . . . . . . . 9 |- (Unic1p ` R ) = (Unic1p ` R )
38	37, 33	mon1puc1p 23910	. . . . . . . 8 |- ( ( R e. Ring /\ ( G ` I ) e. (Monic1p ` R ) ) -> ( G ` I ) e. (Unic1p ` R ) )
39	26, 36, 38	syl2anc 693	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( G ` I ) e. (Unic1p ` R ) )
40		eqid 2622	. . . . . . . 8 |- (rem1p ` R ) = (rem1p ` R )
41	40, 2, 6, 37, 32	r1pdeglt 23918	. . . . . . 7 |- ( ( R e. Ring /\ X e. ( Base ` P ) /\ ( G ` I ) e. (Unic1p ` R ) ) -> ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) < ( ( deg1 ` R ) ` ( G ` I ) ) )
42	26, 30, 39, 41	syl3anc 1326	. . . . . 6 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) < ( ( deg1 ` R ) ` ( G ` I ) ) )
43	35	simp3d 1075	. . . . . 6 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( deg1 ` R ) ` ( G ` I ) ) = inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) )
44	42, 43	breqtrd 4679	. . . . 5 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) < inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) )
45	32, 2, 6	deg1xrf 23841	. . . . . . 7 |- ( deg1 ` R ) : ( Base ` P ) --> RR*
46	35	simp1d 1073	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( G ` I ) e. I )
47	28, 46	sseldd 3604	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( G ` I ) e. ( Base ` P ) )
48		eqid 2622	. . . . . . . . . . 11 |- (quot1p ` R ) = (quot1p ` R )
49		eqid 2622	. . . . . . . . . . 11 |- ( .r ` P ) = ( .r ` P )
50		eqid 2622	. . . . . . . . . . 11 |- ( -g ` P ) = ( -g ` P )
51	40, 2, 6, 48, 49, 50	r1pval 23916	. . . . . . . . . 10 |- ( ( X e. ( Base ` P ) /\ ( G ` I ) e. ( Base ` P ) ) -> ( X (rem1p ` R ) ( G ` I ) ) = ( X ( -g ` P ) ( ( X (quot1p ` R ) ( G ` I ) ) ( .r ` P ) ( G ` I ) ) ) )
52	30, 47, 51	syl2anc 693	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( X (rem1p ` R ) ( G ` I ) ) = ( X ( -g ` P ) ( ( X (quot1p ` R ) ( G ` I ) ) ( .r ` P ) ( G ` I ) ) ) )
53	26, 3	syl 17	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> P e. Ring )
54	48, 2, 6, 37	q1pcl 23915	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ X e. ( Base ` P ) /\ ( G ` I ) e. (Unic1p ` R ) ) -> ( X (quot1p ` R ) ( G ` I ) ) e. ( Base ` P ) )
55	26, 30, 39, 54	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( X (quot1p ` R ) ( G ` I ) ) e. ( Base ` P ) )
56	7, 6, 49	lidlmcl 19217	. . . . . . . . . . 11 |- ( ( ( P e. Ring /\ I e. U ) /\ ( ( X (quot1p ` R ) ( G ` I ) ) e. ( Base ` P ) /\ ( G ` I ) e. I ) ) -> ( ( X (quot1p ` R ) ( G ` I ) ) ( .r ` P ) ( G ` I ) ) e. I )
57	53, 27, 55, 46, 56	syl22anc 1327	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( X (quot1p ` R ) ( G ` I ) ) ( .r ` P ) ( G ` I ) ) e. I )
58	7, 50	lidlsubcl 19216	. . . . . . . . . 10 |- ( ( ( P e. Ring /\ I e. U ) /\ ( X e. I /\ ( ( X (quot1p ` R ) ( G ` I ) ) ( .r ` P ) ( G ` I ) ) e. I ) ) -> ( X ( -g ` P ) ( ( X (quot1p ` R ) ( G ` I ) ) ( .r ` P ) ( G ` I ) ) ) e. I )
59	53, 27, 29, 57, 58	syl22anc 1327	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( X ( -g ` P ) ( ( X (quot1p ` R ) ( G ` I ) ) ( .r ` P ) ( G ` I ) ) ) e. I )
60	52, 59	eqeltrd 2701	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( X (rem1p ` R ) ( G ` I ) ) e. I )
61	28, 60	sseldd 3604	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( X (rem1p ` R ) ( G ` I ) ) e. ( Base ` P ) )
62		ffvelrn 6357	. . . . . . 7 |- ( ( ( deg1 ` R ) : ( Base ` P ) --> RR* /\ ( X (rem1p ` R ) ( G ` I ) ) e. ( Base ` P ) ) -> ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) e. RR* )
63	45, 61, 62	sylancr 695	. . . . . 6 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) e. RR* )
64	28	ssdifd 3746	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( I \ { ( 0g ` P ) } ) C_ ( ( Base ` P ) \ { ( 0g ` P ) } ) )
65		imass2 5501	. . . . . . . . . 10 |- ( ( I \ { ( 0g ` P ) } ) C_ ( ( Base ` P ) \ { ( 0g ` P ) } ) -> ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) C_ ( ( deg1 ` R ) " ( ( Base ` P ) \ { ( 0g ` P ) } ) ) )
66	64, 65	syl 17	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) C_ ( ( deg1 ` R ) " ( ( Base ` P ) \ { ( 0g ` P ) } ) ) )
67	32, 2, 15, 6	deg1n0ima 23849	. . . . . . . . . . 11 |- ( R e. Ring -> ( ( deg1 ` R ) " ( ( Base ` P ) \ { ( 0g ` P ) } ) ) C_ NN0 )
68	26, 67	syl 17	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( deg1 ` R ) " ( ( Base ` P ) \ { ( 0g ` P ) } ) ) C_ NN0 )
69		nn0uz 11722	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
70	68, 69	syl6sseq 3651	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( deg1 ` R ) " ( ( Base ` P ) \ { ( 0g ` P ) } ) ) C_ ( ZZ>= ` 0 ) )
71	66, 70	sstrd 3613	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) C_ ( ZZ>= ` 0 ) )
72		uzssz 11707	. . . . . . . . 9 |- ( ZZ>= ` 0 ) C_ ZZ
73		zssre 11384	. . . . . . . . . 10 |- ZZ C_ RR
74		ressxr 10083	. . . . . . . . . 10 |- RR C_ RR*
75	73, 74	sstri 3612	. . . . . . . . 9 |- ZZ C_ RR*
76	72, 75	sstri 3612	. . . . . . . 8 |- ( ZZ>= ` 0 ) C_ RR*
77	71, 76	syl6ss 3615	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) C_ RR* )
78	7, 15	lidl0cl 19212	. . . . . . . . . . . 12 |- ( ( P e. Ring /\ I e. U ) -> ( 0g ` P ) e. I )
79	53, 27, 78	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( 0g ` P ) e. I )
80	79	snssd 4340	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> { ( 0g ` P ) } C_ I )
81	31	necomd 2849	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> { ( 0g ` P ) } =/= I )
82		pssdifn0 3944	. . . . . . . . . 10 |- ( ( { ( 0g ` P ) } C_ I /\ { ( 0g ` P ) } =/= I ) -> ( I \ { ( 0g ` P ) } ) =/= (/) )
83	80, 81, 82	syl2anc 693	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( I \ { ( 0g ` P ) } ) =/= (/) )
84		ffn 6045	. . . . . . . . . . . 12 |- ( ( deg1 ` R ) : ( Base ` P ) --> RR* -> ( deg1 ` R ) Fn ( Base ` P ) )
85	45, 84	ax-mp 5	. . . . . . . . . . 11 |- ( deg1 ` R ) Fn ( Base ` P )
86	28	ssdifssd 3748	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( I \ { ( 0g ` P ) } ) C_ ( Base ` P ) )
87		fnimaeq0 6013	. . . . . . . . . . 11 |- ( ( ( deg1 ` R ) Fn ( Base ` P ) /\ ( I \ { ( 0g ` P ) } ) C_ ( Base ` P ) ) -> ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) = (/) <-> ( I \ { ( 0g ` P ) } ) = (/) ) )
88	85, 86, 87	sylancr 695	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) = (/) <-> ( I \ { ( 0g ` P ) } ) = (/) ) )
89	88	necon3bid 2838	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) =/= (/) <-> ( I \ { ( 0g ` P ) } ) =/= (/) ) )
90	83, 89	mpbird 247	. . . . . . . 8 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) =/= (/) )
91		infssuzcl 11772	. . . . . . . 8 |- ( ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) C_ ( ZZ>= ` 0 ) /\ ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) =/= (/) ) -> inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) e. ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) )
92	71, 90, 91	syl2anc 693	. . . . . . 7 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) e. ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) )
93	77, 92	sseldd 3604	. . . . . 6 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) e. RR* )
94		xrltnle 10105	. . . . . 6 |- ( ( ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) e. RR* /\ inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) e. RR* ) -> ( ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) < inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) <-> -. inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) <_ ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) ) )
95	63, 93, 94	syl2anc 693	. . . . 5 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) < inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) <-> -. inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) <_ ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) ) )
96	44, 95	mpbid 222	. . . 4 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> -. inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) <_ ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) )
97	71	adantr 481	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) /\ ( X (rem1p ` R ) ( G ` I ) ) =/= ( 0g ` P ) ) -> ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) C_ ( ZZ>= ` 0 ) )
98	85	a1i 11	. . . . . . . 8 |- ( ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) /\ ( X (rem1p ` R ) ( G ` I ) ) =/= ( 0g ` P ) ) -> ( deg1 ` R ) Fn ( Base ` P ) )
99	86	adantr 481	. . . . . . . 8 |- ( ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) /\ ( X (rem1p ` R ) ( G ` I ) ) =/= ( 0g ` P ) ) -> ( I \ { ( 0g ` P ) } ) C_ ( Base ` P ) )
100	60	adantr 481	. . . . . . . . 9 |- ( ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) /\ ( X (rem1p ` R ) ( G ` I ) ) =/= ( 0g ` P ) ) -> ( X (rem1p ` R ) ( G ` I ) ) e. I )
101		simpr 477	. . . . . . . . 9 |- ( ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) /\ ( X (rem1p ` R ) ( G ` I ) ) =/= ( 0g ` P ) ) -> ( X (rem1p ` R ) ( G ` I ) ) =/= ( 0g ` P ) )
102		eldifsn 4317	. . . . . . . . 9 |- ( ( X (rem1p ` R ) ( G ` I ) ) e. ( I \ { ( 0g ` P ) } ) <-> ( ( X (rem1p ` R ) ( G ` I ) ) e. I /\ ( X (rem1p ` R ) ( G ` I ) ) =/= ( 0g ` P ) ) )
103	100, 101, 102	sylanbrc 698	. . . . . . . 8 |- ( ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) /\ ( X (rem1p ` R ) ( G ` I ) ) =/= ( 0g ` P ) ) -> ( X (rem1p ` R ) ( G ` I ) ) e. ( I \ { ( 0g ` P ) } ) )
104		fnfvima 6496	. . . . . . . 8 |- ( ( ( deg1 ` R ) Fn ( Base ` P ) /\ ( I \ { ( 0g ` P ) } ) C_ ( Base ` P ) /\ ( X (rem1p ` R ) ( G ` I ) ) e. ( I \ { ( 0g ` P ) } ) ) -> ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) e. ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) )
105	98, 99, 103, 104	syl3anc 1326	. . . . . . 7 |- ( ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) /\ ( X (rem1p ` R ) ( G ` I ) ) =/= ( 0g ` P ) ) -> ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) e. ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) )
106		infssuzle 11771	. . . . . . 7 |- ( ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) C_ ( ZZ>= ` 0 ) /\ ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) e. ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) ) -> inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) <_ ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) )
107	97, 105, 106	syl2anc 693	. . . . . 6 |- ( ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) /\ ( X (rem1p ` R ) ( G ` I ) ) =/= ( 0g ` P ) ) -> inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) <_ ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) )
108	107	ex 450	. . . . 5 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( X (rem1p ` R ) ( G ` I ) ) =/= ( 0g ` P ) -> inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) <_ ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) ) )
109	108	necon1bd 2812	. . . 4 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( -. inf ( ( ( deg1 ` R ) " ( I \ { ( 0g ` P ) } ) ) , RR , < ) <_ ( ( deg1 ` R ) ` ( X (rem1p ` R ) ( G ` I ) ) ) -> ( X (rem1p ` R ) ( G ` I ) ) = ( 0g ` P ) ) )
110	96, 109	mpd 15	. . 3 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( X (rem1p ` R ) ( G ` I ) ) = ( 0g ` P ) )
111	2, 14, 6, 37, 15, 40	dvdsr1p 23921	. . . 4 |- ( ( R e. Ring /\ X e. ( Base ` P ) /\ ( G ` I ) e. (Unic1p ` R ) ) -> ( ( G ` I ) .|| X <-> ( X (rem1p ` R ) ( G ` I ) ) = ( 0g ` P ) ) )
112	26, 30, 39, 111	syl3anc 1326	. . 3 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( ( G ` I ) .|| X <-> ( X (rem1p ` R ) ( G ` I ) ) = ( 0g ` P ) ) )
113	110, 112	mpbird 247	. 2 |- ( ( ( R e. DivRing /\ I e. U /\ X e. I ) /\ I =/= { ( 0g ` P ) } ) -> ( G ` I ) .|| X )
114	24, 113	pm2.61dane 2881	1 |- ( ( R e. DivRing /\ I e. U /\ X e. I ) -> ( G ` I ) .|| X )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   "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   ZZcz 11377   ZZ>=cuz 11687   Basecbs 15857   .rcmulr 15942   0gc0g 16100   -gcsg 17424   Ringcrg 18547   ||_rcdsr 18638   DivRingcdr 18747  LIdealclidl 19170  Poly₁cpl1 19547   deg₁ cdg1 23814  Monic_1pcmn1 23885  Unic_1pcuc1p 23886  quot_1pcq1p 23887  rem_1pcr1p 23888  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-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  df-q1p 23892  df-r1p 23893  df-ig1p 23894

This theorem is referenced by:  ig1prsp  23937