Theorem ply1remlem 23922

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 19261	. . . . . . . 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 19618	. . . . . . 7 |- ( R e. Ring -> P e. Ring )
7	4, 6	syl 17	. . . . . 6 |- ( ph -> P e. Ring )
8		ringgrp 18552	. . . . . 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 19587	. . . . . 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 19655	. . . . . . 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 6360	. . . . 5 |- ( ph -> ( A ` N ) e. B )
20		ply1rem.m	. . . . . 6 |- .- = ( -g ` P )
21	11, 20	grpsubcl 17495	. . . . 5 |- ( ( P e. Grp /\ X e. B /\ ( A ` N ) e. B ) -> ( X .- ( A ` N ) ) e. B )
22	9, 13, 19, 21	syl3anc 1326	. . . 4 |- ( ph -> ( X .- ( A ` N ) ) e. B )
23	1, 22	syl5eqel 2705	. . 3 |- ( ph -> G e. B )
24	1	fveq2i 6194	. . . . . . 7 |- ( D ` G ) = ( D ` ( X .- ( A ` N ) ) )
25		ply1rem.d	. . . . . . . 8 |- D = ( deg1 ` R )
26	25, 5, 11	deg1xrcl 23842	. . . . . . . . . . 11 |- ( ( A ` N ) e. B -> ( D ` ( A ` N ) ) e. RR* )
27	19, 26	syl 17	. . . . . . . . . 10 |- ( ph -> ( D ` ( A ` N ) ) e. RR* )
28		0xr 10086	. . . . . . . . . . 11 |- 0 e. RR*
29	28	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 e. RR* )
30		1re 10039	. . . . . . . . . . 11 |- 1 e. RR
31		rexr 10085	. . . . . . . . . . 11 |- ( 1 e. RR -> 1 e. RR* )
32	30, 31	mp1i 13	. . . . . . . . . 10 |- ( ph -> 1 e. RR* )
33	25, 5, 15, 14	deg1sclle 23872	. . . . . . . . . . 11 |- ( ( R e. Ring /\ N e. K ) -> ( D ` ( A ` N ) ) <_ 0 )
34	4, 18, 33	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( D ` ( A ` N ) ) <_ 0 )
35		0lt1 10550	. . . . . . . . . . 11 |- 0 < 1
36	35	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 < 1 )
37	27, 29, 32, 34, 36	xrlelttrd 11991	. . . . . . . . 9 |- ( ph -> ( D ` ( A ` N ) ) < 1 )
38		eqid 2622	. . . . . . . . . . . . . 14 |- (mulGrp ` P ) = (mulGrp ` P )
39	38, 11	mgpbas 18495	. . . . . . . . . . . . 13 |- B = ( Base ` (mulGrp ` P ) )
40		eqid 2622	. . . . . . . . . . . . 13 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
41	39, 40	mulg1 17548	. . . . . . . . . . . 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 6195	. . . . . . . . . 10 |- ( ph -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = ( D ` X ) )
44		1nn0 11308	. . . . . . . . . . 11 |- 1 e. NN0
45	25, 5, 10, 38, 40	deg1pw 23880	. . . . . . . . . . 11 |- ( ( R e. NzRing /\ 1 e. NN0 ) -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = 1 )
46	2, 44, 45	sylancl 694	. . . . . . . . . 10 |- ( ph -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = 1 )
47	43, 46	eqtr3d 2658	. . . . . . . . 9 |- ( ph -> ( D ` X ) = 1 )
48	37, 47	breqtrrd 4681	. . . . . . . 8 |- ( ph -> ( D ` ( A ` N ) ) < ( D ` X ) )
49	5, 25, 4, 11, 20, 13, 19, 48	deg1sub 23868	. . . . . . 7 |- ( ph -> ( D ` ( X .- ( A ` N ) ) ) = ( D ` X ) )
50	24, 49	syl5eq 2668	. . . . . 6 |- ( ph -> ( D ` G ) = ( D ` X ) )
51	50, 47	eqtrd 2656	. . . . 5 |- ( ph -> ( D ` G ) = 1 )
52	51, 44	syl6eqel 2709	. . . 4 |- ( ph -> ( D ` G ) e. NN0 )
53		eqid 2622	. . . . . 6 |- ( 0g ` P ) = ( 0g ` P )
54	25, 5, 53, 11	deg1nn0clb 23850	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( G =/= ( 0g ` P ) <-> ( D ` G ) e. NN0 ) )
55	4, 23, 54	syl2anc 693	. . . 4 |- ( ph -> ( G =/= ( 0g ` P ) <-> ( D ` G ) e. NN0 ) )
56	52, 55	mpbird 247	. . 3 |- ( ph -> G =/= ( 0g ` P ) )
57	51	fveq2d 6195	. . . 4 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) = ( (coe1 ` G ) ` 1 ) )
58	1	fveq2i 6194	. . . . . 6 |- (coe1 ` G ) = (coe1 ` ( X .- ( A ` N ) ) )
59	58	fveq1i 6192	. . . . 5 |- ( (coe1 ` G ) ` 1 ) = ( (coe1 ` ( X .- ( A ` N ) ) ) ` 1 )
60	44	a1i 11	. . . . . 6 |- ( ph -> 1 e. NN0 )
61		eqid 2622	. . . . . . 7 |- ( -g ` R ) = ( -g ` R )
62	5, 11, 20, 61	coe1subfv 19636	. . . . . 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 1329	. . . . 5 |- ( ph -> ( (coe1 ` ( X .- ( A ` N ) ) ) ` 1 ) = ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) )
64	59, 63	syl5eq 2668	. . . 4 |- ( ph -> ( (coe1 ` G ) ` 1 ) = ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) )
65	42	oveq2d 6666	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) = ( ( 1r ` R ) ( .s ` P ) X ) )
66	5	ply1sca 19623	. . . . . . . . . . . . 13 |- ( R e. NzRing -> R = (Scalar ` P ) )
67	2, 66	syl 17	. . . . . . . . . . . 12 |- ( ph -> R = (Scalar ` P ) )
68	67	fveq2d 6195	. . . . . . . . . . 11 |- ( ph -> ( 1r ` R ) = ( 1r ` (Scalar ` P ) ) )
69	68	oveq1d 6665	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) X ) = ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) )
70	5	ply1lmod 19622	. . . . . . . . . . . 12 |- ( R e. Ring -> P e. LMod )
71	4, 70	syl 17	. . . . . . . . . . 11 |- ( ph -> P e. LMod )
72		eqid 2622	. . . . . . . . . . . 12 |- (Scalar ` P ) = (Scalar ` P )
73		eqid 2622	. . . . . . . . . . . 12 |- ( .s ` P ) = ( .s ` P )
74		eqid 2622	. . . . . . . . . . . 12 |- ( 1r ` (Scalar ` P ) ) = ( 1r ` (Scalar ` P ) )
75	11, 72, 73, 74	lmodvs1 18891	. . . . . . . . . . 11 |- ( ( P e. LMod /\ X e. B ) -> ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) = X )
76	71, 13, 75	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) = X )
77	65, 69, 76	3eqtrd 2660	. . . . . . . . 9 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) = X )
78	77	fveq2d 6195	. . . . . . . 8 |- ( ph -> (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) = (coe1 ` X ) )
79	78	fveq1d 6193	. . . . . . 7 |- ( ph -> ( (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( (coe1 ` X ) ` 1 ) )
80		eqid 2622	. . . . . . . . . 10 |- ( 1r ` R ) = ( 1r ` R )
81	15, 80	ringidcl 18568	. . . . . . . . 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 19644	. . . . . . . 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 1326	. . . . . . 7 |- ( ph -> ( (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( 1r ` R ) )
86	79, 85	eqtr3d 2658	. . . . . 6 |- ( ph -> ( (coe1 ` X ) ` 1 ) = ( 1r ` R ) )
87		eqid 2622	. . . . . . . . . 10 |- ( 0g ` R ) = ( 0g ` R )
88	5, 14, 15, 87	coe1scl 19657	. . . . . . . . 9 |- ( ( R e. Ring /\ N e. K ) -> (coe1 ` ( A ` N ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) )
89	4, 18, 88	syl2anc 693	. . . . . . . 8 |- ( ph -> (coe1 ` ( A ` N ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) )
90	89	fveq1d 6193	. . . . . . 7 |- ( ph -> ( (coe1 ` ( A ` N ) ) ` 1 ) = ( ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) ` 1 ) )
91		ax-1ne0 10005	. . . . . . . . . . 11 |- 1 =/= 0
92		neeq1 2856	. . . . . . . . . . 11 |- ( x = 1 -> ( x =/= 0 <-> 1 =/= 0 ) )
93	91, 92	mpbiri 248	. . . . . . . . . 10 |- ( x = 1 -> x =/= 0 )
94		ifnefalse 4098	. . . . . . . . . 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 2622	. . . . . . . . 9 |- ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) )
97		fvex 6201	. . . . . . . . 9 |- ( 0g ` R ) e. _V
98	95, 96, 97	fvmpt 6282	. . . . . . . 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 2672	. . . . . 6 |- ( ph -> ( (coe1 ` ( A ` N ) ) ` 1 ) = ( 0g ` R ) )
101	86, 100	oveq12d 6668	. . . . 5 |- ( ph -> ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) = ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) )
102		ringgrp 18552	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
103	4, 102	syl 17	. . . . . 6 |- ( ph -> R e. Grp )
104	15, 87, 61	grpsubid1 17500	. . . . . 6 |- ( ( R e. Grp /\ ( 1r ` R ) e. K ) -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) = ( 1r ` R ) )
105	103, 82, 104	syl2anc 693	. . . . 5 |- ( ph -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) = ( 1r ` R ) )
106	101, 105	eqtrd 2656	. . . 4 |- ( ph -> ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) = ( 1r ` R ) )
107	57, 64, 106	3eqtrd 2660	. . 3 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) = ( 1r ` R ) )
108		ply1rem.u	. . . 4 |- U = (Monic1p ` R )
109	5, 11, 53, 25, 108, 80	ismon1p 23902	. . 3 |- ( G e. U <-> ( G e. B /\ G =/= ( 0g ` P ) /\ ( (coe1 ` G ) ` ( D ` G ) ) = ( 1r ` R ) ) )
110	23, 56, 107, 109	syl3anbrc 1246	. 2 |- ( ph -> G e. U )
111	1	fveq2i 6194	. . . . . . . . . 10 |- ( O ` G ) = ( O ` ( X .- ( A ` N ) ) )
112	111	fveq1i 6192	. . . . . . . . 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 481	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> R e. CRing )
116		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> x e. K )
117	113, 10, 15, 5, 11, 115, 116	evl1vard 19701	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> ( X e. B /\ ( ( O ` X ) ` x ) = x ) )
118	18	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. K ) -> N e. K )
119	113, 5, 15, 14, 11, 115, 118, 116	evl1scad 19699	. . . . . . . . . . 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 19706	. . . . . . . . . 10 |- ( ( ph /\ x e. K ) -> ( ( X .- ( A ` N ) ) e. B /\ ( ( O ` ( X .- ( A ` N ) ) ) ` x ) = ( x ( -g ` R ) N ) ) )
121	120	simprd 479	. . . . . . . . 9 |- ( ( ph /\ x e. K ) -> ( ( O ` ( X .- ( A ` N ) ) ) ` x ) = ( x ( -g ` R ) N ) )
122	112, 121	syl5eq 2668	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( ( O ` G ) ` x ) = ( x ( -g ` R ) N ) )
123	122	eqeq1d 2624	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( ( ( O ` G ) ` x ) = .0. <-> ( x ( -g ` R ) N ) = .0. ) )
124	103	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> R e. Grp )
125	15, 83, 61	grpsubeq0 17501	. . . . . . . 8 |- ( ( R e. Grp /\ x e. K /\ N e. K ) -> ( ( x ( -g ` R ) N ) = .0. <-> x = N ) )
126	124, 116, 118, 125	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( ( x ( -g ` R ) N ) = .0. <-> x = N ) )
127	123, 126	bitrd 268	. . . . . 6 |- ( ( ph /\ x e. K ) -> ( ( ( O ` G ) ` x ) = .0. <-> x = N ) )
128		velsn 4193	. . . . . 6 |- ( x e. { N } <-> x = N )
129	127, 128	syl6bbr 278	. . . . 5 |- ( ( ph /\ x e. K ) -> ( ( ( O ` G ) ` x ) = .0. <-> x e. { N } ) )
130	129	pm5.32da 673	. . . 4 |- ( ph -> ( ( x e. K /\ ( ( O ` G ) ` x ) = .0. ) <-> ( x e. K /\ x e. { N } ) ) )
131		eqid 2622	. . . . . . 7 |- ( R ^s K ) = ( R ^s K )
132		eqid 2622	. . . . . . 7 |- ( Base ` ( R ^s K ) ) = ( Base ` ( R ^s K ) )
133		fvex 6201	. . . . . . . . 9 |- ( Base ` R ) e. _V
134	15, 133	eqeltri 2697	. . . . . . . 8 |- K e. _V
135	134	a1i 11	. . . . . . 7 |- ( ph -> K e. _V )
136	113, 5, 131, 15	evl1rhm 19696	. . . . . . . . . 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 18726	. . . . . . . . 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 6360	. . . . . . 7 |- ( ph -> ( O ` G ) e. ( Base ` ( R ^s K ) ) )
141	131, 15, 132, 2, 135, 140	pwselbas 16149	. . . . . 6 |- ( ph -> ( O ` G ) : K --> K )
142		ffn 6045	. . . . . 6 |- ( ( O ` G ) : K --> K -> ( O ` G ) Fn K )
143	141, 142	syl 17	. . . . 5 |- ( ph -> ( O ` G ) Fn K )
144		fniniseg 6338	. . . . 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 4340	. . . . . 6 |- ( ph -> { N } C_ K )
147	146	sseld 3602	. . . . 5 |- ( ph -> ( x e. { N } -> x e. K ) )
148	147	pm4.71rd 667	. . . 4 |- ( ph -> ( x e. { N } <-> ( x e. K /\ x e. { N } ) ) )
149	130, 145, 148	3bitr4d 300	. . 3 |- ( ph -> ( x e. ( `' ( O ` G ) " { .0. } ) <-> x e. { N } ) )
150	149	eqrdv 2620	. 2 |- ( ph -> ( `' ( O ` G ) " { .0. } ) = { N } )
151	110, 51, 150	3jca 1242	1 |- ( ph -> ( G e. U /\ ( D ` G ) = 1 /\ ( `' ( O ` G ) " { .0. } ) = { N } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   RR*cxr 10073   < clt 10074   <_ cle 10075   NN0cn0 11292   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   ^_s cpws 16107   Grpcgrp 17422   -gcsg 17424  ._gcmg 17540  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   CRingccrg 18548   RingHom crh 18712   LModclmod 18863  NzRingcnzr 19257  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548  eval₁ce1 19679   deg₁ cdg1 23814  Monic_1pcmn1 23885

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-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-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-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

This theorem is referenced by:  ply1rem  23923  facth1  23924  fta1glem1  23925  fta1glem2  23926