Theorem ply1remlem 23113

Description: A term of the form x - N is linear, monic, and has exactly one zero. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
ply1rem.p	|- P = (Poly1 ` R )
ply1rem.b	|- B = ( Base ` P )
ply1rem.k	|- K = ( Base ` R )
ply1rem.x	|- X = (var1 ` R )
ply1rem.m	|- .- = ( -g ` P )
ply1rem.a	|- A = (algSc ` P )
ply1rem.g	|- G = ( X .- ( A ` N ) )
ply1rem.o	|- O = (eval1 ` R )
ply1rem.1	|- ( ph -> R e. NzRing )
ply1rem.2	|- ( ph -> R e. CRing )
ply1rem.3	|- ( ph -> N e. K )
ply1rem.u	|- U = (Monic1p ` R )
ply1rem.d	|- D = ( deg1 ` R )
ply1rem.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
ply1remlem	|- ( ph -> ( G e. U /\ ( D ` G ) = 1 /\ ( `' ( O ` G ) " { .0. } ) = { N } ) )

Proof of Theorem ply1remlem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ply1rem.g	. . . 4 |- G = ( X .- ( A ` N ) )
2		ply1rem.1	. . . . . . . 8 |- ( ph -> R e. NzRing )
3		nzrring 18485	. . . . . . . 8 |- ( R e. NzRing -> R e. Ring )
4	2, 3	syl 17	. . . . . . 7 |- ( ph -> R e. Ring )
5		ply1rem.p	. . . . . . . 8 |- P = (Poly1 ` R )
6	5	ply1ring 18841	. . . . . . 7 |- ( R e. Ring -> P e. Ring )
7	4, 6	syl 17	. . . . . 6 |- ( ph -> P e. Ring )
8		ringgrp 17785	. . . . . 6 |- ( P e. Ring -> P e. Grp )
9	7, 8	syl 17	. . . . 5 |- ( ph -> P e. Grp )
10		ply1rem.x	. . . . . . 7 |- X = (var1 ` R )
11		ply1rem.b	. . . . . . 7 |- B = ( Base ` P )
12	10, 5, 11	vr1cl 18810	. . . . . 6 |- ( R e. Ring -> X e. B )
13	4, 12	syl 17	. . . . 5 |- ( ph -> X e. B )
14		ply1rem.a	. . . . . . . 8 |- A = (algSc ` P )
15		ply1rem.k	. . . . . . . 8 |- K = ( Base ` R )
16	5, 14, 15, 11	ply1sclf 18878	. . . . . . 7 |- ( R e. Ring -> A : K --> B )
17	4, 16	syl 17	. . . . . 6 |- ( ph -> A : K --> B )
18		ply1rem.3	. . . . . 6 |- ( ph -> N e. K )
19	17, 18	ffvelrnd 6023	. . . . 5 |- ( ph -> ( A ` N ) e. B )
20		ply1rem.m	. . . . . 6 |- .- = ( -g ` P )
21	11, 20	grpsubcl 16734	. . . . 5 |- ( ( P e. Grp /\ X e. B /\ ( A ` N ) e. B ) -> ( X .- ( A ` N ) ) e. B )
22	9, 13, 19, 21	syl3anc 1268	. . . 4 |- ( ph -> ( X .- ( A ` N ) ) e. B )
23	1, 22	syl5eqel 2533	. . 3 |- ( ph -> G e. B )
24	1	fveq2i 5868	. . . . . . 7 |- ( D ` G ) = ( D ` ( X .- ( A ` N ) ) )
25		ply1rem.d	. . . . . . . 8 |- D = ( deg1 ` R )
26	25, 5, 11	deg1xrcl 23031	. . . . . . . . . . 11 |- ( ( A ` N ) e. B -> ( D ` ( A ` N ) ) e. RR* )
27	19, 26	syl 17	. . . . . . . . . 10 |- ( ph -> ( D ` ( A ` N ) ) e. RR* )
28		0xr 9687	. . . . . . . . . . 11 |- 0 e. RR*
29	28	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 e. RR* )
30		1re 9642	. . . . . . . . . . 11 |- 1 e. RR
31		rexr 9686	. . . . . . . . . . 11 |- ( 1 e. RR -> 1 e. RR* )
32	30, 31	mp1i 13	. . . . . . . . . 10 |- ( ph -> 1 e. RR* )
33	25, 5, 15, 14	deg1sclle 23061	. . . . . . . . . . 11 |- ( ( R e. Ring /\ N e. K ) -> ( D ` ( A ` N ) ) <_ 0 )
34	4, 18, 33	syl2anc 667	. . . . . . . . . 10 |- ( ph -> ( D ` ( A ` N ) ) <_ 0 )
35		0lt1 10136	. . . . . . . . . . 11 |- 0 < 1
36	35	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 < 1 )
37	27, 29, 32, 34, 36	xrlelttrd 11457	. . . . . . . . 9 |- ( ph -> ( D ` ( A ` N ) ) < 1 )
38		eqid 2451	. . . . . . . . . . . . . 14 |- (mulGrp ` P ) = (mulGrp ` P )
39	38, 11	mgpbas 17729	. . . . . . . . . . . . 13 |- B = ( Base ` (mulGrp ` P ) )
40		eqid 2451	. . . . . . . . . . . . 13 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
41	39, 40	mulg1 16765	. . . . . . . . . . . 12 |- ( X e. B -> ( 1 (.g ` (mulGrp ` P ) ) X ) = X )
42	13, 41	syl 17	. . . . . . . . . . 11 |- ( ph -> ( 1 (.g ` (mulGrp ` P ) ) X ) = X )
43	42	fveq2d 5869	. . . . . . . . . 10 |- ( ph -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = ( D ` X ) )
44		1nn0 10885	. . . . . . . . . . 11 |- 1 e. NN0
45	25, 5, 10, 38, 40	deg1pw 23069	. . . . . . . . . . 11 |- ( ( R e. NzRing /\ 1 e. NN0 ) -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = 1 )
46	2, 44, 45	sylancl 668	. . . . . . . . . 10 |- ( ph -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = 1 )
47	43, 46	eqtr3d 2487	. . . . . . . . 9 |- ( ph -> ( D ` X ) = 1 )
48	37, 47	breqtrrd 4429	. . . . . . . 8 |- ( ph -> ( D ` ( A ` N ) ) < ( D ` X ) )
49	5, 25, 4, 11, 20, 13, 19, 48	deg1sub 23057	. . . . . . 7 |- ( ph -> ( D ` ( X .- ( A ` N ) ) ) = ( D ` X ) )
50	24, 49	syl5eq 2497	. . . . . 6 |- ( ph -> ( D ` G ) = ( D ` X ) )
51	50, 47	eqtrd 2485	. . . . 5 |- ( ph -> ( D ` G ) = 1 )
52	51, 44	syl6eqel 2537	. . . 4 |- ( ph -> ( D ` G ) e. NN0 )
53		eqid 2451	. . . . . 6 |- ( 0g ` P ) = ( 0g ` P )
54	25, 5, 53, 11	deg1nn0clb 23039	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( G =/= ( 0g ` P ) <-> ( D ` G ) e. NN0 ) )
55	4, 23, 54	syl2anc 667	. . . 4 |- ( ph -> ( G =/= ( 0g ` P ) <-> ( D ` G ) e. NN0 ) )
56	52, 55	mpbird 236	. . 3 |- ( ph -> G =/= ( 0g ` P ) )
57	51	fveq2d 5869	. . . 4 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) = ( (coe1 ` G ) ` 1 ) )
58	1	fveq2i 5868	. . . . . 6 |- (coe1 ` G ) = (coe1 ` ( X .- ( A ` N ) ) )
59	58	fveq1i 5866	. . . . 5 |- ( (coe1 ` G ) ` 1 ) = ( (coe1 ` ( X .- ( A ` N ) ) ) ` 1 )
60	44	a1i 11	. . . . . 6 |- ( ph -> 1 e. NN0 )
61		eqid 2451	. . . . . . 7 |- ( -g ` R ) = ( -g ` R )
62	5, 11, 20, 61	coe1subfv 18859	. . . . . 6 |- ( ( ( R e. Ring /\ X e. B /\ ( A ` N ) e. B ) /\ 1 e. NN0 ) -> ( (coe1 ` ( X .- ( A ` N ) ) ) ` 1 ) = ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) )
63	4, 13, 19, 60, 62	syl31anc 1271	. . . . 5 |- ( ph -> ( (coe1 ` ( X .- ( A ` N ) ) ) ` 1 ) = ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) )
64	59, 63	syl5eq 2497	. . . 4 |- ( ph -> ( (coe1 ` G ) ` 1 ) = ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) )
65	42	oveq2d 6306	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) = ( ( 1r ` R ) ( .s ` P ) X ) )
66	5	ply1sca 18846	. . . . . . . . . . . . 13 |- ( R e. NzRing -> R = (Scalar ` P ) )
67	2, 66	syl 17	. . . . . . . . . . . 12 |- ( ph -> R = (Scalar ` P ) )
68	67	fveq2d 5869	. . . . . . . . . . 11 |- ( ph -> ( 1r ` R ) = ( 1r ` (Scalar ` P ) ) )
69	68	oveq1d 6305	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) X ) = ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) )
70	5	ply1lmod 18845	. . . . . . . . . . . 12 |- ( R e. Ring -> P e. LMod )
71	4, 70	syl 17	. . . . . . . . . . 11 |- ( ph -> P e. LMod )
72		eqid 2451	. . . . . . . . . . . 12 |- (Scalar ` P ) = (Scalar ` P )
73		eqid 2451	. . . . . . . . . . . 12 |- ( .s ` P ) = ( .s ` P )
74		eqid 2451	. . . . . . . . . . . 12 |- ( 1r ` (Scalar ` P ) ) = ( 1r ` (Scalar ` P ) )
75	11, 72, 73, 74	lmodvs1 18119	. . . . . . . . . . 11 |- ( ( P e. LMod /\ X e. B ) -> ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) = X )
76	71, 13, 75	syl2anc 667	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) = X )
77	65, 69, 76	3eqtrd 2489	. . . . . . . . 9 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) = X )
78	77	fveq2d 5869	. . . . . . . 8 |- ( ph -> (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) = (coe1 ` X ) )
79	78	fveq1d 5867	. . . . . . 7 |- ( ph -> ( (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( (coe1 ` X ) ` 1 ) )
80		eqid 2451	. . . . . . . . . 10 |- ( 1r ` R ) = ( 1r ` R )
81	15, 80	ringidcl 17801	. . . . . . . . 9 |- ( R e. Ring -> ( 1r ` R ) e. K )
82	4, 81	syl 17	. . . . . . . 8 |- ( ph -> ( 1r ` R ) e. K )
83		ply1rem.z	. . . . . . . . 9 |- .0. = ( 0g ` R )
84	83, 15, 5, 10, 73, 38, 40	coe1tmfv1 18867	. . . . . . . 8 |- ( ( R e. Ring /\ ( 1r ` R ) e. K /\ 1 e. NN0 ) -> ( (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( 1r ` R ) )
85	4, 82, 60, 84	syl3anc 1268	. . . . . . 7 |- ( ph -> ( (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( 1r ` R ) )
86	79, 85	eqtr3d 2487	. . . . . 6 |- ( ph -> ( (coe1 ` X ) ` 1 ) = ( 1r ` R ) )
87		eqid 2451	. . . . . . . . . 10 |- ( 0g ` R ) = ( 0g ` R )
88	5, 14, 15, 87	coe1scl 18880	. . . . . . . . 9 |- ( ( R e. Ring /\ N e. K ) -> (coe1 ` ( A ` N ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) )
89	4, 18, 88	syl2anc 667	. . . . . . . 8 |- ( ph -> (coe1 ` ( A ` N ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) )
90	89	fveq1d 5867	. . . . . . 7 |- ( ph -> ( (coe1 ` ( A ` N ) ) ` 1 ) = ( ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) ` 1 ) )
91		ax-1ne0 9608	. . . . . . . . . . 11 |- 1 =/= 0
92		neeq1 2686	. . . . . . . . . . 11 |- ( x = 1 -> ( x =/= 0 <-> 1 =/= 0 ) )
93	91, 92	mpbiri 237	. . . . . . . . . 10 |- ( x = 1 -> x =/= 0 )
94		ifnefalse 3893	. . . . . . . . . 10 |- ( x =/= 0 -> if ( x = 0 , N , ( 0g ` R ) ) = ( 0g ` R ) )
95	93, 94	syl 17	. . . . . . . . 9 |- ( x = 1 -> if ( x = 0 , N , ( 0g ` R ) ) = ( 0g ` R ) )
96		eqid 2451	. . . . . . . . 9 |- ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) )
97		fvex 5875	. . . . . . . . 9 |- ( 0g ` R ) e. _V
98	95, 96, 97	fvmpt 5948	. . . . . . . 8 |- ( 1 e. NN0 -> ( ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) ` 1 ) = ( 0g ` R ) )
99	44, 98	ax-mp 5	. . . . . . 7 |- ( ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) ` 1 ) = ( 0g ` R )
100	90, 99	syl6eq 2501	. . . . . 6 |- ( ph -> ( (coe1 ` ( A ` N ) ) ` 1 ) = ( 0g ` R ) )
101	86, 100	oveq12d 6308	. . . . 5 |- ( ph -> ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) = ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) )
102		ringgrp 17785	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
103	4, 102	syl 17	. . . . . 6 |- ( ph -> R e. Grp )
104	15, 87, 61	grpsubid1 16739	. . . . . 6 |- ( ( R e. Grp /\ ( 1r ` R ) e. K ) -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) = ( 1r ` R ) )
105	103, 82, 104	syl2anc 667	. . . . 5 |- ( ph -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) = ( 1r ` R ) )
106	101, 105	eqtrd 2485	. . . 4 |- ( ph -> ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) = ( 1r ` R ) )
107	57, 64, 106	3eqtrd 2489	. . 3 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) = ( 1r ` R ) )
108		ply1rem.u	. . . 4 |- U = (Monic1p ` R )
109	5, 11, 53, 25, 108, 80	ismon1p 23093	. . 3 |- ( G e. U <-> ( G e. B /\ G =/= ( 0g ` P ) /\ ( (coe1 ` G ) ` ( D ` G ) ) = ( 1r ` R ) ) )
110	23, 56, 107, 109	syl3anbrc 1192	. 2 |- ( ph -> G e. U )
111	1	fveq2i 5868	. . . . . . . . . 10 |- ( O ` G ) = ( O ` ( X .- ( A ` N ) ) )
112	111	fveq1i 5866	. . . . . . . . 9 |- ( ( O ` G ) ` x ) = ( ( O ` ( X .- ( A ` N ) ) ) ` x )
113		ply1rem.o	. . . . . . . . . . 11 |- O = (eval1 ` R )
114		ply1rem.2	. . . . . . . . . . . 12 |- ( ph -> R e. CRing )
115	114	adantr 467	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> R e. CRing )
116		simpr 463	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> x e. K )
117	113, 10, 15, 5, 11, 115, 116	evl1vard 18925	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> ( X e. B /\ ( ( O ` X ) ` x ) = x ) )
118	18	adantr 467	. . . . . . . . . . . 12 |- ( ( ph /\ x e. K ) -> N e. K )
119	113, 5, 15, 14, 11, 115, 118, 116	evl1scad 18923	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> ( ( A ` N ) e. B /\ ( ( O ` ( A ` N ) ) ` x ) = N ) )
120	113, 5, 15, 11, 115, 116, 117, 119, 20, 61	evl1subd 18930	. . . . . . . . . 10 |- ( ( ph /\ x e. K ) -> ( ( X .- ( A ` N ) ) e. B /\ ( ( O ` ( X .- ( A ` N ) ) ) ` x ) = ( x ( -g ` R ) N ) ) )
121	120	simprd 465	. . . . . . . . 9 |- ( ( ph /\ x e. K ) -> ( ( O ` ( X .- ( A ` N ) ) ) ` x ) = ( x ( -g ` R ) N ) )
122	112, 121	syl5eq 2497	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( ( O ` G ) ` x ) = ( x ( -g ` R ) N ) )
123	122	eqeq1d 2453	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( ( ( O ` G ) ` x ) = .0. <-> ( x ( -g ` R ) N ) = .0. ) )
124	103	adantr 467	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> R e. Grp )
125	15, 83, 61	grpsubeq0 16740	. . . . . . . 8 |- ( ( R e. Grp /\ x e. K /\ N e. K ) -> ( ( x ( -g ` R ) N ) = .0. <-> x = N ) )
126	124, 116, 118, 125	syl3anc 1268	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( ( x ( -g ` R ) N ) = .0. <-> x = N ) )
127	123, 126	bitrd 257	. . . . . 6 |- ( ( ph /\ x e. K ) -> ( ( ( O ` G ) ` x ) = .0. <-> x = N ) )
128		elsn 3982	. . . . . 6 |- ( x e. { N } <-> x = N )
129	127, 128	syl6bbr 267	. . . . 5 |- ( ( ph /\ x e. K ) -> ( ( ( O ` G ) ` x ) = .0. <-> x e. { N } ) )
130	129	pm5.32da 647	. . . 4 |- ( ph -> ( ( x e. K /\ ( ( O ` G ) ` x ) = .0. ) <-> ( x e. K /\ x e. { N } ) ) )
131		eqid 2451	. . . . . . 7 |- ( R ^s K ) = ( R ^s K )
132		eqid 2451	. . . . . . 7 |- ( Base ` ( R ^s K ) ) = ( Base ` ( R ^s K ) )
133		fvex 5875	. . . . . . . . 9 |- ( Base ` R ) e. _V
134	15, 133	eqeltri 2525	. . . . . . . 8 |- K e. _V
135	134	a1i 11	. . . . . . 7 |- ( ph -> K e. _V )
136	113, 5, 131, 15	evl1rhm 18920	. . . . . . . . . 10 |- ( R e. CRing -> O e. ( P RingHom ( R ^s K ) ) )
137	114, 136	syl 17	. . . . . . . . 9 |- ( ph -> O e. ( P RingHom ( R ^s K ) ) )
138	11, 132	rhmf 17954	. . . . . . . . 9 |- ( O e. ( P RingHom ( R ^s K ) ) -> O : B --> ( Base ` ( R ^s K ) ) )
139	137, 138	syl 17	. . . . . . . 8 |- ( ph -> O : B --> ( Base ` ( R ^s K ) ) )
140	139, 23	ffvelrnd 6023	. . . . . . 7 |- ( ph -> ( O ` G ) e. ( Base ` ( R ^s K ) ) )
141	131, 15, 132, 2, 135, 140	pwselbas 15387	. . . . . 6 |- ( ph -> ( O ` G ) : K --> K )
142		ffn 5728	. . . . . 6 |- ( ( O ` G ) : K --> K -> ( O ` G ) Fn K )
143	141, 142	syl 17	. . . . 5 |- ( ph -> ( O ` G ) Fn K )
144		fniniseg 6003	. . . . 5 |- ( ( O ` G ) Fn K -> ( x e. ( `' ( O ` G ) " { .0. } ) <-> ( x e. K /\ ( ( O ` G ) ` x ) = .0. ) ) )
145	143, 144	syl 17	. . . 4 |- ( ph -> ( x e. ( `' ( O ` G ) " { .0. } ) <-> ( x e. K /\ ( ( O ` G ) ` x ) = .0. ) ) )
146	18	snssd 4117	. . . . . 6 |- ( ph -> { N } C_ K )
147	146	sseld 3431	. . . . 5 |- ( ph -> ( x e. { N } -> x e. K ) )
148	147	pm4.71rd 641	. . . 4 |- ( ph -> ( x e. { N } <-> ( x e. K /\ x e. { N } ) ) )
149	130, 145, 148	3bitr4d 289	. . 3 |- ( ph -> ( x e. ( `' ( O ` G ) " { .0. } ) <-> x e. { N } ) )
150	149	eqrdv 2449	. 2 |- ( ph -> ( `' ( O ` G ) " { .0. } ) = { N } )
151	110, 51, 150	3jca 1188	1 |- ( ph -> ( G e. U /\ ( D ` G ) = 1 /\ ( `' ( O ` G ) " { .0. } ) = { N } ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   _Vcvv 3045   ifcif 3881   {csn 3968   class class class wbr 4402   |-> cmpt 4461   `'ccnv 4833   "cima 4837   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   RR*cxr 9674   < clt 9675   <_ cle 9676   NN0cn0 10869   Basecbs 15121  Scalarcsca 15193   .scvsca 15194   0gc0g 15338   ^_s cpws 15345   Grpcgrp 16669   -gcsg 16671  ._gcmg 16672  mulGrpcmgp 17723   1rcur 17735   Ringcrg 17780   CRingccrg 17781   RingHom crh 17940   LModclmod 18091  NzRingcnzr 18481  algSccascl 18535  var₁cv1 18769  Poly₁cpl1 18770  coe₁cco1 18771  eval₁ce1 18903   deg₁ cdg1 23003  Monic_1pcmn1 23074

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-hom 15214  df-cco 15215  df-0g 15340  df-gsum 15341  df-prds 15346  df-pws 15348  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-srg 17740  df-ring 17782  df-cring 17783  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-rnghom 17943  df-subrg 18006  df-lmod 18093  df-lss 18156  df-lsp 18195  df-nzr 18482  df-rlreg 18507  df-assa 18536  df-asp 18537  df-ascl 18538  df-psr 18580  df-mvr 18581  df-mpl 18582  df-opsr 18584  df-evls 18729  df-evl 18730  df-psr1 18773  df-vr1 18774  df-ply1 18775  df-coe1 18776  df-evl1 18905  df-cnfld 18971  df-mdeg 23004  df-deg1 23005  df-mon1 23080

This theorem is referenced by:  ply1rem  23114  facth1  23115  fta1glem1  23116  fta1glem2  23117