Theorem idomrootle 27379

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 2404	. . 3 |- (Poly1 ` R ) = (Poly1 ` R )
2		eqid 2404	. . 3 |- ( Base ` (Poly1 ` R ) ) = ( Base ` (Poly1 ` R ) )
3		eqid 2404	. . 3 |- ( deg1 ` R ) = ( deg1 ` R )
4		eqid 2404	. . 3 |- (eval1 ` R ) = (eval1 ` R )
5		eqid 2404	. . 3 |- ( 0g ` R ) = ( 0g ` R )
6		eqid 2404	. . 3 |- ( 0g ` (Poly1 ` R ) ) = ( 0g ` (Poly1 ` R ) )
7		simp1 957	. . 3 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> R e. IDomn )
8		isidom 16319	. . . . . . . . 9 |- ( R e. IDomn <-> ( R e. CRing /\ R e. Domn ) )
9	8	simplbi 447	. . . . . . . 8 |- ( R e. IDomn -> R e. CRing )
10	7, 9	syl 16	. . . . . . 7 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> R e. CRing )
11		crngrng 15629	. . . . . . 7 |- ( R e. CRing -> R e. Ring )
12	10, 11	syl 16	. . . . . 6 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> R e. Ring )
13	1	ply1rng 16597	. . . . . 6 |- ( R e. Ring -> (Poly1 ` R ) e. Ring )
14	12, 13	syl 16	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (Poly1 ` R ) e. Ring )
15		rnggrp 15624	. . . . 5 |- ( (Poly1 ` R ) e. Ring -> (Poly1 ` R ) e. Grp )
16	14, 15	syl 16	. . . 4 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (Poly1 ` R ) e. Grp )
17		eqid 2404	. . . . . . 7 |- (mulGrp ` (Poly1 ` R ) ) = (mulGrp ` (Poly1 ` R ) )
18	17	rngmgp 15625	. . . . . 6 |- ( (Poly1 ` R ) e. Ring -> (mulGrp ` (Poly1 ` R ) ) e. Mnd )
19	14, 18	syl 16	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (mulGrp ` (Poly1 ` R ) ) e. Mnd )
20		simp3 959	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> N e. NN )
21		eqid 2404	. . . . . . 7 |- (var1 ` R ) = (var1 ` R )
22	21, 1, 2	vr1cl 16566	. . . . . 6 |- ( R e. Ring -> (var1 ` R ) e. ( Base ` (Poly1 ` R ) ) )
23	12, 22	syl 16	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (var1 ` R ) e. ( Base ` (Poly1 ` R ) ) )
24	17, 2	mgpbas 15609	. . . . . 6 |- ( Base ` (Poly1 ` R ) ) = ( Base ` (mulGrp ` (Poly1 ` R ) ) )
25		eqid 2404	. . . . . 6 |- (.g ` (mulGrp ` (Poly1 ` R ) ) ) = (.g ` (mulGrp ` (Poly1 ` R ) ) )
26	24, 25	mulgnncl 14860	. . . . 5 |- ( ( (mulGrp ` (Poly1 ` R ) ) e. Mnd /\ N e. NN /\ (var1 ` R ) e. ( Base ` (Poly1 ` R ) ) ) -> ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) e. ( Base ` (Poly1 ` R ) ) )
27	19, 20, 23, 26	syl3anc 1184	. . . 4 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) e. ( Base ` (Poly1 ` R ) ) )
28		eqid 2404	. . . . . . 7 |- (algSc ` (Poly1 ` R ) ) = (algSc ` (Poly1 ` R ) )
29		idomrootle.b	. . . . . . 7 |- B = ( Base ` R )
30	1, 28, 29, 2	ply1sclf 16632	. . . . . 6 |- ( R e. Ring -> (algSc ` (Poly1 ` R ) ) : B --> ( Base ` (Poly1 ` R ) ) )
31	12, 30	syl 16	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (algSc ` (Poly1 ` R ) ) : B --> ( Base ` (Poly1 ` R ) ) )
32		simp2 958	. . . . 5 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> X e. B )
33	31, 32	ffvelrnd 5830	. . . 4 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( (algSc ` (Poly1 ` R ) ) ` X ) e. ( Base ` (Poly1 ` R ) ) )
34		eqid 2404	. . . . 5 |- ( -g ` (Poly1 ` R ) ) = ( -g ` (Poly1 ` R ) )
35	2, 34	grpsubcl 14824	. . . 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 ) ) )
36	16, 27, 33, 35	syl3anc 1184	. . 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 ) ) )
37	3, 1, 2	deg1xrcl 19958	. . . . . . . . . 10 |- ( ( (algSc ` (Poly1 ` R ) ) ` X ) e. ( Base ` (Poly1 ` R ) ) -> ( ( deg1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) e. RR* )
38	33, 37	syl 16	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) e. RR* )
39		0xr 9087	. . . . . . . . . 10 |- 0 e. RR*
40	39	a1i 11	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> 0 e. RR* )
41		nnre 9963	. . . . . . . . . . 11 |- ( N e. NN -> N e. RR )
42	41	rexrd 9090	. . . . . . . . . 10 |- ( N e. NN -> N e. RR* )
43	42	3ad2ant3 980	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> N e. RR* )
44	3, 1, 29, 28	deg1sclle 19988	. . . . . . . . . 10 |- ( ( R e. Ring /\ X e. B ) -> ( ( deg1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) <_ 0 )
45	12, 32, 44	syl2anc 643	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) <_ 0 )
46		nngt0 9985	. . . . . . . . . 10 |- ( N e. NN -> 0 < N )
47	46	3ad2ant3 980	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> 0 < N )
48	38, 40, 43, 45, 47	xrlelttrd 10706	. . . . . . . 8 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( (algSc ` (Poly1 ` R ) ) ` X ) ) < N )
49	8	simprbi 451	. . . . . . . . . . 11 |- ( R e. IDomn -> R e. Domn )
50		domnnzr 16310	. . . . . . . . . . 11 |- ( R e. Domn -> R e. NzRing )
51	49, 50	syl 16	. . . . . . . . . 10 |- ( R e. IDomn -> R e. NzRing )
52	7, 51	syl 16	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> R e. NzRing )
53		nnnn0 10184	. . . . . . . . . 10 |- ( N e. NN -> N e. NN0 )
54	53	3ad2ant3 980	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> N e. NN0 )
55	3, 1, 21, 17, 25	deg1pw 19996	. . . . . . . . 9 |- ( ( R e. NzRing /\ N e. NN0 ) -> ( ( deg1 ` R ) ` ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ) = N )
56	52, 54, 55	syl2anc 643	. . . . . . . 8 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( ( deg1 ` R ) ` ( N (.g ` (mulGrp ` (Poly1 ` R ) ) ) (var1 ` R ) ) ) = N )
57	48, 56	breqtrrd 4198	. . . . . . 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 ) ) ) )
58	1, 3, 12, 2, 34, 27, 33, 57	deg1sub 19984	. . . . . 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 ) ) ) )
59	58, 56	eqtrd 2436	. . . . 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 )
60	59, 54	eqeltrd 2478	. . . 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 )
61	3, 1, 6, 2	deg1nn0clb 19966	. . . . 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 ) )
62	12, 36, 61	syl2anc 643	. . . 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 ) )
63	60, 62	mpbird 224	. . 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 ) ) )
64	1, 2, 3, 4, 5, 6, 7, 36, 63	fta1g 20043	. 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 ) ) ) )
65		eqid 2404	. . . . . . 7 |- ( R ^s B ) = ( R ^s B )
66		eqid 2404	. . . . . . 7 |- ( Base ` ( R ^s B ) ) = ( Base ` ( R ^s B ) )
67		fvex 5701	. . . . . . . . 9 |- ( Base ` R ) e. _V
68	29, 67	eqeltri 2474	. . . . . . . 8 |- B e. _V
69	68	a1i 11	. . . . . . 7 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> B e. _V )
70	4, 1, 65, 29	evl1rhm 19902	. . . . . . . . . 10 |- ( R e. CRing -> (eval1 ` R ) e. ( (Poly1 ` R ) RingHom ( R ^s B ) ) )
71	10, 70	syl 16	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (eval1 ` R ) e. ( (Poly1 ` R ) RingHom ( R ^s B ) ) )
72	2, 66	rhmf 15782	. . . . . . . . 9 |- ( (eval1 ` R ) e. ( (Poly1 ` R ) RingHom ( R ^s B ) ) -> (eval1 ` R ) : ( Base ` (Poly1 ` R ) ) --> ( Base ` ( R ^s B ) ) )
73	71, 72	syl 16	. . . . . . . 8 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (eval1 ` R ) : ( Base ` (Poly1 ` R ) ) --> ( Base ` ( R ^s B ) ) )
74	73, 36	ffvelrnd 5830	. . . . . . 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 ) ) )
75	65, 29, 66, 7, 69, 74	pwselbas 13666	. . . . . 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 )
76		ffn 5550	. . . . . 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 )
77	75, 76	syl 16	. . . . 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 )
78		fniniseg2 5812	. . . . 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 ) } )
79	77, 78	syl 16	. . . 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 ) } )
80	10	adantr 452	. . . . . . . . 9 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> R e. CRing )
81		simpr 448	. . . . . . . . 9 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> y e. B )
82	4, 21, 29, 1, 2, 80, 81	evl1vard 19906	. . . . . . . . . 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 ) )
83		idomrootle.e	. . . . . . . . . 10 |- .^ = (.g ` (mulGrp ` R ) )
84		simpl3 962	. . . . . . . . . . 11 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> N e. NN )
85	84, 53	syl 16	. . . . . . . . . 10 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> N e. NN0 )
86	4, 1, 29, 2, 80, 81, 82, 25, 83, 85	evl1expd 19911	. . . . . . . . 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 ) ) )
87		simpl2 961	. . . . . . . . . 10 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> X e. B )
88	4, 1, 29, 28, 2, 80, 87, 81	evl1scad 19904	. . . . . . . . 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 ) )
89		eqid 2404	. . . . . . . . 9 |- ( -g ` R ) = ( -g ` R )
90	4, 1, 29, 2, 80, 81, 86, 88, 34, 89	evl1subd 19908	. . . . . . . 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 ) ) )
91	90	simprd 450	. . . . . . 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 ) )
92	91	eqeq1d 2412	. . . . . 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 ) ) )
93		rnggrp 15624	. . . . . . . . 9 |- ( R e. Ring -> R e. Grp )
94	12, 93	syl 16	. . . . . . . 8 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> R e. Grp )
95	94	adantr 452	. . . . . . 7 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> R e. Grp )
96		eqid 2404	. . . . . . . . . . 11 |- (mulGrp ` R ) = (mulGrp ` R )
97	96	rngmgp 15625	. . . . . . . . . 10 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
98	12, 97	syl 16	. . . . . . . . 9 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> (mulGrp ` R ) e. Mnd )
99	98	adantr 452	. . . . . . . 8 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> (mulGrp ` R ) e. Mnd )
100	96, 29	mgpbas 15609	. . . . . . . . 9 |- B = ( Base ` (mulGrp ` R ) )
101	100, 83	mulgnncl 14860	. . . . . . . 8 |- ( ( (mulGrp ` R ) e. Mnd /\ N e. NN /\ y e. B ) -> ( N .^ y ) e. B )
102	99, 84, 81, 101	syl3anc 1184	. . . . . . 7 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> ( N .^ y ) e. B )
103	29, 5, 89	grpsubeq0 14830	. . . . . . 7 |- ( ( R e. Grp /\ ( N .^ y ) e. B /\ X e. B ) -> ( ( ( N .^ y ) ( -g ` R ) X ) = ( 0g ` R ) <-> ( N .^ y ) = X ) )
104	95, 102, 87, 103	syl3anc 1184	. . . . . 6 |- ( ( ( R e. IDomn /\ X e. B /\ N e. NN ) /\ y e. B ) -> ( ( ( N .^ y ) ( -g ` R ) X ) = ( 0g ` R ) <-> ( N .^ y ) = X ) )
105	92, 104	bitrd 245	. . . . 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 ) )
106	105	rabbidva 2907	. . . 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 } )
107	79, 106	eqtrd 2436	. . 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 } )
108	107	fveq2d 5691	. 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 } ) )
109	64, 108, 59	3brtr3d 4201	1 |- ( ( R e. IDomn /\ X e. B /\ N e. NN ) -> ( # ` { y e. B | ( N .^ y ) = X } ) <_ N )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   {crab 2670   _Vcvv 2916   {csn 3774   class class class wbr 4172   `'ccnv 4836   "cima 4840   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946   RR*cxr 9075   < clt 9076   <_ cle 9077   NNcn 9956   NN0cn0 10177   #chash 11573   Basecbs 13424   ^_s cpws 13625   0gc0g 13678   Mndcmnd 14639   Grpcgrp 14640   -gcsg 14643  ._gcmg 14644  mulGrpcmgp 15603   Ringcrg 15615   CRingccrg 15616   RingHom crh 15772  NzRingcnzr 16283  Domncdomn 16295  IDomncidom 16296  algSccascl 16326  var₁cv1 16525  Poly₁cpl1 16526  eval₁ce1 16528   deg₁ cdg1 19930

This theorem is referenced by:  idomodle  27380

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-ofr 6265  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-prds 13626  df-pws 13628  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-rnghom 15774  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-nzr 16284  df-rlreg 16298  df-domn 16299  df-idom 16300  df-assa 16327  df-asp 16328  df-ascl 16329  df-psr 16372  df-mvr 16373  df-mpl 16374  df-evls 16375  df-evl 16376  df-opsr 16380  df-psr1 16531  df-vr1 16532  df-ply1 16533  df-evl1 16535  df-coe1 16536  df-cnfld 16659  df-mdeg 19931  df-deg1 19932  df-mon1 20006  df-uc1p 20007  df-q1p 20008  df-r1p 20009