Theorem idomrootle 37773

Description: No element of an integral domain can have more than N N-th roots. (Contributed by Stefan O'Rear, 11-Sep-2015.)

Hypotheses

Ref	Expression
idomrootle.b	|- B = ( Base ` R )
idomrootle.e	|- .^ = (.g ` (mulGrp ` R ) )

Assertion

Ref	Expression
idomrootle	|- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( # ` { y e. B | ( N .^ y ) = X } ) <_ N )

Distinct variable groups:   y, B   y, N   y, R   y, X

Allowed substitution hint:   .^ ( y)

Proof of Theorem idomrootle

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- (Poly1 ` R ) = (Poly1 ` R )
2		eqid 2622	. . 3 |- ( Base ` (Poly1 ` R ) ) = ( Base ` (Poly1 ` R ) )
3		eqid 2622	. . 3 |- ( deg1 ` R ) = ( deg1 ` R )
4		eqid 2622	. . 3 |- (eval1 ` R ) = (eval1 ` R )
5		eqid 2622	. . 3 |- ( 0g ` R ) = ( 0g ` R )
6		eqid 2622	. . 3 |- ( 0g ` (Poly1 ` R ) ) = ( 0g ` (Poly1 ` R ) )
7		simp1 1061	. . 3 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> R e. IDomn )
8		isidom 19304	. . . . . . . . 9 |- ( R e. IDomn <-> ( R e. CRing /\ R e. Domn ) )
9	8	simplbi 476	. . . . . . . 8 |- ( R e. IDomn -> R e. CRing )
10	7, 9	syl 17	. . . . . . 7 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> R e. CRing )
11		crngring 18558	. . . . . . 7 |- ( R e. CRing -> R e. Ring )
12	10, 11	syl 17	. . . . . 6 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> R e. Ring )
13	1	ply1ring 19618	. . . . . 6 |- ( R e. Ring -> (Poly1 ` R ) e. Ring )
14	12, 13	syl 17	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (Poly1 ` R ) e. Ring )
15		ringgrp 18552	. . . . 5 |- ( (Poly1 ` R ) e. Ring -> (Poly1 ` R ) e. Grp )
16	14, 15	syl 17	. . . 4 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (Poly1 ` R ) e. Grp )
17		eqid 2622	. . . . . . . 8 |- (mulGrp ` (Poly1 ` R ) ) = (mulGrp ` (Poly1 ` R ) )
18	17	ringmgp 18553	. . . . . . 7 |- ( (Poly1 ` R ) e. Ring -> (mulGrp ` (Poly1 ` R ) ) e. Mnd )
19	14, 18	syl 17	. . . . . 6 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (mulGrp ` (Poly1 ` R ) ) e. Mnd )
20		mndmgm 17300	. . . . . 6 |- ( (mulGrp ` (Poly1 ` R ) ) e. Mnd -> (mulGrp ` (Poly1 ` R ) ) e. Mgm )
21	19, 20	syl 17	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (mulGrp ` (Poly1 ` R ) ) e. Mgm )
22		simp3 1063	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> N e. NN )
23		eqid 2622	. . . . . . 7 |- (var1 ` R ) = (var1 ` R )
24	23, 1, 2	vr1cl 19587	. . . . . 6 |- ( R e. Ring -> (var1 ` R ) e. ( Base ` (Poly1 ` R ) ) )
25	12, 24	syl 17	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (var1 ` R ) e. ( Base ` (Poly1 ` R ) ) )
26	17, 2	mgpbas 18495	. . . . . 6 |- ( Base ` (Poly1 ` R ) ) = ( Base ` (mulGrp ` (Poly1 ` R ) ) )
27		eqid 2622	. . . . . 6 |- (.g ` (mulGrp ` (Poly1 ` R ) ) ) = (.g ` (mulGrp ` (Poly1 ` R ) ) )
28	26, 27	mulgnncl 17556	. . . . 5 |- ( ( (mulGrp ` (Poly1 ` R ) ) e. Mgm /\ N e. NN /\ (var1 ` R ) e. ( Base ` (Poly1 ` R ) ) ) -> ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) e. ( Base ` (Poly1 ` R ) ) )
29	21, 22, 25, 28	syl3anc 1326	. . . 4 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) e. ( Base ` (Poly1 ` R ) ) )
30		eqid 2622	. . . . . . 7 |- (algSc ` (Poly1 ` R ) ) = (algSc ` (Poly1 ` R ) )
31		idomrootle.b	. . . . . . 7 |- B = ( Base ` R )
32	1, 30, 31, 2	ply1sclf 19655	. . . . . 6 |- ( R e. Ring -> (algSc ` (Poly1 ` R ) ) : B --> ( Base ` (Poly1 ` R ) ) )
33	12, 32	syl 17	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (algSc ` (Poly1 ` R ) ) : B --> ( Base ` (Poly1 ` R ) ) )
34		simp2 1062	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> X e. B )
35	33, 34	ffvelrnd 6360	. . . 4 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( (algSc ` (Poly1 ` R ) ) ` X ) e. ( Base ` (Poly1 ` R ) ) )
36		eqid 2622	. . . . 5 |- ( -g ` (Poly1 ` R ) ) = ( -g ` (Poly1 ` R ) )
37	2, 36	grpsubcl 17495	. . . 4 |- ( ( (Poly1 ` R ) e. Grp /\ ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) e. ( Base ` (Poly1 ` R ) ) /\ ( (algSc ` (Poly1 ` R ) ) ` X ) e. ( Base ` (Poly1 ` R ) ) ) -> ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) e. ( Base ` (Poly1 ` R ) ) )
38	16, 29, 35, 37	syl3anc 1326	. . 3 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) e. ( Base ` (Poly1 ` R ) ) )
39	3, 1, 2	deg1xrcl 23842	. . . . . . . . . 10 |- ( ( (algSc ` (Poly1 ` R ) ) ` X ) e. ( Base ` (Poly1 ` R ) ) -> ( ( deg1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) e. RR* )
40	35, 39	syl 17	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) e. RR* )
41		0xr 10086	. . . . . . . . . 10 |- 0 e. RR*
42	41	a1i 11	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> 0 e. RR* )
43		nnre 11027	. . . . . . . . . . 11 |- ( N e. NN -> N e. RR )
44	43	rexrd 10089	. . . . . . . . . 10 |- ( N e. NN -> N e. RR* )
45	44	3ad2ant3 1084	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> N e. RR* )
46	3, 1, 31, 30	deg1sclle 23872	. . . . . . . . . 10 |- ( ( R e. Ring /\ X e. B ) -> ( ( deg1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) <_ 0 )
47	12, 34, 46	syl2anc 693	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) <_ 0 )
48		nngt0 11049	. . . . . . . . . 10 |- ( N e. NN -> 0 < N )
49	48	3ad2ant3 1084	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> 0 < N )
50	40, 42, 45, 47, 49	xrlelttrd 11991	. . . . . . . 8 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) < N )
51	8	simprbi 480	. . . . . . . . . . 11 |- ( R e. IDomn -> R e. Domn )
52		domnnzr 19295	. . . . . . . . . . 11 |- ( R e. Domn -> R e. NzRing )
53	51, 52	syl 17	. . . . . . . . . 10 |- ( R e. IDomn -> R e. NzRing )
54	7, 53	syl 17	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> R e. NzRing )
55		nnnn0 11299	. . . . . . . . . 10 |- ( N e. NN -> N e. NN0 )
56	55	3ad2ant3 1084	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> N e. NN0 )
57	3, 1, 23, 17, 27	deg1pw 23880	. . . . . . . . 9 |- ( ( R e. NzRing /\ N e. NN0 ) -> ( ( deg1 ` R ) ` ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ) = N )
58	54, 56, 57	syl2anc 693	. . . . . . . 8 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ) = N )
59	50, 58	breqtrrd 4681	. . . . . . 7 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) < ( ( deg1 ` R ) ` ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ) )
60	1, 3, 12, 2, 36, 29, 35, 59	deg1sub 23868	. . . . . 6 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) = ( ( deg1 ` R ) ` ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ) )
61	60, 58	eqtrd 2656	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) = N )
62	61, 56	eqeltrd 2701	. . . 4 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) e. NN0 )
63	3, 1, 6, 2	deg1nn0clb 23850	. . . . 5 |- ( ( R e. Ring /\ ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) e. ( Base ` (Poly1 ` R ) ) ) -> ( ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) =/= ( 0g ` (Poly1 ` R ) ) <-> ( ( deg1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) e. NN0 ) )
64	12, 38, 63	syl2anc 693	. . . 4 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) =/= ( 0g ` (Poly1 ` R ) ) <-> ( ( deg1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) e. NN0 ) )
65	62, 64	mpbird 247	. . 3 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) =/= ( 0g ` (Poly1 ` R ) ) )
66	1, 2, 3, 4, 5, 6, 7, 38, 65	fta1g 23927	. 2 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( # ` ( `' ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) " { ( 0g ` R ) } ) ) <_ ( ( deg1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) )
67		eqid 2622	. . . . . . 7 |- ( R ^s B ) = ( R ^s B )
68		eqid 2622	. . . . . . 7 |- ( Base ` ( R ^s B ) ) = ( Base ` ( R ^s B ) )
69		fvex 6201	. . . . . . . . 9 |- ( Base ` R ) e. _V
70	31, 69	eqeltri 2697	. . . . . . . 8 |- B e. _V
71	70	a1i 11	. . . . . . 7 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> B e. _V )
72	4, 1, 67, 31	evl1rhm 19696	. . . . . . . . . 10 |- ( R e. CRing -> (eval1 ` R ) e. ( (Poly1 ` R ) RingHom ( R ^s B ) ) )
73	10, 72	syl 17	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (eval1 ` R ) e. ( (Poly1 ` R ) RingHom ( R ^s B ) ) )
74	2, 68	rhmf 18726	. . . . . . . . 9 |- ( (eval1 ` R ) e. ( (Poly1 ` R ) RingHom ( R ^s B ) ) -> (eval1 ` R ) : ( Base ` (Poly1 ` R ) ) --> ( Base ` ( R ^s B ) ) )
75	73, 74	syl 17	. . . . . . . 8 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (eval1 ` R ) : ( Base ` (Poly1 ` R ) ) --> ( Base ` ( R ^s B ) ) )
76	75, 38	ffvelrnd 6360	. . . . . . 7 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) e. ( Base ` ( R ^s B ) ) )
77	67, 31, 68, 7, 71, 76	pwselbas 16149	. . . . . 6 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) : B --> B )
78		ffn 6045	. . . . . 6 |- ( ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) : B --> B -> ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) Fn B )
79	77, 78	syl 17	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) Fn B )
80		fniniseg2 6340	. . . . 5 |- ( ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) Fn B -> ( `' ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) " { ( 0g ` R ) } ) = { y e. B | ( ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) ` y ) = ( 0g ` R ) } )
81	79, 80	syl 17	. . . 4 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( `' ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) " { ( 0g ` R ) } ) = { y e. B | ( ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) ` y ) = ( 0g ` R ) } )
82	10	adantr 481	. . . . . . . . 9 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> R e. CRing )
83		simpr 477	. . . . . . . . 9 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> y e. B )
84	4, 23, 31, 1, 2, 82, 83	evl1vard 19701	. . . . . . . . . 10 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> ( (var1 ` R ) e. ( Base ` (Poly1 ` R ) ) /\ ( ( (eval1 ` R ) ` (var1 ` R ) ) ` y ) = y ) )
85		idomrootle.e	. . . . . . . . . 10 |- .^ = (.g ` (mulGrp ` R ) )
86		simpl3 1066	. . . . . . . . . . 11 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> N e. NN )
87	86, 55	syl 17	. . . . . . . . . 10 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> N e. NN0 )
88	4, 1, 31, 2, 82, 83, 84, 27, 85, 87	evl1expd 19709	. . . . . . . . 9 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) e. ( Base ` (Poly1 ` R ) ) /\ ( ( (eval1 ` R ) ` ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ) ` y ) = ( N .^ y ) ) )
89		simpl2 1065	. . . . . . . . . 10 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> X e. B )
90	4, 1, 31, 30, 2, 82, 89, 83	evl1scad 19699	. . . . . . . . 9 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> ( ( (algSc ` (Poly1 ` R ) ) ` X ) e. ( Base ` (Poly1 ` R ) ) /\ ( ( (eval1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) ` y ) = X ) )
91		eqid 2622	. . . . . . . . 9 |- ( -g ` R ) = ( -g ` R )
92	4, 1, 31, 2, 82, 83, 88, 90, 36, 91	evl1subd 19706	. . . . . . . 8 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> ( ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) e. ( Base ` (Poly1 ` R ) ) /\ ( ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) ` y ) = ( ( N .^ y ) ( -g ` R ) X ) ) )
93	92	simprd 479	. . . . . . 7 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> ( ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) ` y ) = ( ( N .^ y ) ( -g ` R ) X ) )
94	93	eqeq1d 2624	. . . . . 6 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> ( ( ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) ` y ) = ( 0g ` R ) <-> ( ( N .^ y ) ( -g ` R ) X ) = ( 0g ` R ) ) )
95		ringgrp 18552	. . . . . . . . 9 |- ( R e. Ring -> R e. Grp )
96	12, 95	syl 17	. . . . . . . 8 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> R e. Grp )
97	96	adantr 481	. . . . . . 7 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> R e. Grp )
98		eqid 2622	. . . . . . . . . . . 12 |- (mulGrp ` R ) = (mulGrp ` R )
99	98	ringmgp 18553	. . . . . . . . . . 11 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
100	12, 99	syl 17	. . . . . . . . . 10 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (mulGrp ` R ) e. Mnd )
101	100	adantr 481	. . . . . . . . 9 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> (mulGrp ` R ) e. Mnd )
102		mndmgm 17300	. . . . . . . . 9 |- ( (mulGrp ` R ) e. Mnd -> (mulGrp ` R ) e. Mgm )
103	101, 102	syl 17	. . . . . . . 8 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> (mulGrp ` R ) e. Mgm )
104	98, 31	mgpbas 18495	. . . . . . . . 9 |- B = ( Base ` (mulGrp ` R ) )
105	104, 85	mulgnncl 17556	. . . . . . . 8 |- ( ( (mulGrp ` R ) e. Mgm /\ N e. NN /\ y e. B ) -> ( N .^ y ) e. B )
106	103, 86, 83, 105	syl3anc 1326	. . . . . . 7 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> ( N .^ y ) e. B )
107	31, 5, 91	grpsubeq0 17501	. . . . . . 7 |- ( ( R e. Grp /\ ( N .^ y ) e. B /\ X e. B ) -> ( ( ( N .^ y ) ( -g ` R ) X ) = ( 0g ` R ) <-> ( N .^ y ) = X ) )
108	97, 106, 89, 107	syl3anc 1326	. . . . . 6 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> ( ( ( N .^ y ) ( -g ` R ) X ) = ( 0g ` R ) <-> ( N .^ y ) = X ) )
109	94, 108	bitrd 268	. . . . 5 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> ( ( ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) ` y ) = ( 0g ` R ) <-> ( N .^ y ) = X ) )
110	109	rabbidva 3188	. . . 4 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> { y e. B | ( ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) ` y ) = ( 0g ` R ) } = { y e. B | ( N .^ y ) = X } )
111	81, 110	eqtrd 2656	. . 3 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( `' ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) " { ( 0g ` R ) } ) = { y e. B | ( N .^ y ) = X } )
112	111	fveq2d 6195	. 2 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( # ` ( `' ( (eval1 ` R ) ` ( ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ( -g ` (Poly1 ` R ) ) ( (algSc ` (Poly1 ` R ) ) ` X ) ) ) " { ( 0g ` R ) } ) ) = ( # ` { y e. B | ( N .^ y ) = X } ) )
113	66, 112, 61	3brtr3d 4684	1 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( # ` { y e. B | ( N .^ y ) = X } ) <_ N )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   {csn 4177   class class class wbr 4653   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR*cxr 10073   < clt 10074   <_ cle 10075   NNcn 11020   NN0cn0 11292   #chash 13117   Basecbs 15857   0gc0g 16100   ^_s cpws 16107  Mgmcmgm 17240   Mndcmnd 17294   Grpcgrp 17422   -gcsg 17424  ._gcmg 17540  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548   RingHom crh 18712  NzRingcnzr 19257  Domncdomn 19280  IDomncidom 19281  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547  eval₁ce1 19679   deg₁ cdg1 23814

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-oi 8415  df-card 8765  df-cda 8990  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-xnn0 11364  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-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  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-srg 18506  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-nzr 19258  df-rlreg 19283  df-domn 19284  df-idom 19285  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-evl 19507  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-evl1 19681  df-cnfld 19747  df-mdeg 23815  df-deg1 23816  df-mon1 23890  df-uc1p 23891  df-q1p 23892  df-r1p 23893

This theorem is referenced by:  idomodle  37774