Theorem ply1remlem 22298

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		nzrrng 17691	. . . . . . . 8 |- ( R e. NzRing -> R e. Ring )
4	2, 3	syl 16	. . . . . . 7 |- ( ph -> R e. Ring )
5		ply1rem.p	. . . . . . . 8 |- P = (Poly1 ` R )
6	5	ply1rng 18060	. . . . . . 7 |- ( R e. Ring -> P e. Ring )
7	4, 6	syl 16	. . . . . 6 |- ( ph -> P e. Ring )
8		rnggrp 16991	. . . . . 6 |- ( P e. Ring -> P e. Grp )
9	7, 8	syl 16	. . . . 5 |- ( ph -> P e. Grp )
10		ply1rem.x	. . . . . . 7 |- X = (var1 ` R )
11		ply1rem.b	. . . . . . 7 |- B = ( Base ` P )
12	10, 5, 11	vr1cl 18029	. . . . . 6 |- ( R e. Ring -> X e. B )
13	4, 12	syl 16	. . . . 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 18097	. . . . . . 7 |- ( R e. Ring -> A : K --> B )
17	4, 16	syl 16	. . . . . 6 |- ( ph -> A : K --> B )
18		ply1rem.3	. . . . . 6 |- ( ph -> N e. K )
19	17, 18	ffvelrnd 6020	. . . . 5 |- ( ph -> ( A ` N ) e. B )
20		ply1rem.m	. . . . . 6 |- .- = ( -g ` P )
21	11, 20	grpsubcl 15919	. . . . 5 |- ( ( P e. Grp /\ X e. B /\ ( A ` N ) e. B ) -> ( X .- ( A ` N ) ) e. B )
22	9, 13, 19, 21	syl3anc 1228	. . . 4 |- ( ph -> ( X .- ( A ` N ) ) e. B )
23	1, 22	syl5eqel 2559	. . 3 |- ( ph -> G e. B )
24	1	fveq2i 5867	. . . . . . 7 |- ( D ` G ) = ( D ` ( X .- ( A ` N ) ) )
25		ply1rem.d	. . . . . . . 8 |- D = ( deg1 ` R )
26	25, 5, 11	deg1xrcl 22217	. . . . . . . . . . 11 |- ( ( A ` N ) e. B -> ( D ` ( A ` N ) ) e. RR* )
27	19, 26	syl 16	. . . . . . . . . 10 |- ( ph -> ( D ` ( A ` N ) ) e. RR* )
28		0xr 9636	. . . . . . . . . . 11 |- 0 e. RR*
29	28	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 e. RR* )
30		1re 9591	. . . . . . . . . . 11 |- 1 e. RR
31		rexr 9635	. . . . . . . . . . 11 |- ( 1 e. RR -> 1 e. RR* )
32	30, 31	mp1i 12	. . . . . . . . . 10 |- ( ph -> 1 e. RR* )
33	25, 5, 15, 14	deg1sclle 22248	. . . . . . . . . . 11 |- ( ( R e. Ring /\ N e. K ) -> ( D ` ( A ` N ) ) <_ 0 )
34	4, 18, 33	syl2anc 661	. . . . . . . . . 10 |- ( ph -> ( D ` ( A ` N ) ) <_ 0 )
35		0lt1 10071	. . . . . . . . . . 11 |- 0 < 1
36	35	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 < 1 )
37	27, 29, 32, 34, 36	xrlelttrd 11359	. . . . . . . . 9 |- ( ph -> ( D ` ( A ` N ) ) < 1 )
38		eqid 2467	. . . . . . . . . . . . . 14 |- (mulGrp ` P ) = (mulGrp ` P )
39	38, 11	mgpbas 16937	. . . . . . . . . . . . 13 |- B = ( Base ` (mulGrp ` P ) )
40		eqid 2467	. . . . . . . . . . . . 13 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
41	39, 40	mulg1 15949	. . . . . . . . . . . 12 |- ( X e. B -> ( 1 (.g ` (mulGrp ` P ) ) X ) = X )
42	13, 41	syl 16	. . . . . . . . . . 11 |- ( ph -> ( 1 (.g ` (mulGrp ` P ) ) X ) = X )
43	42	fveq2d 5868	. . . . . . . . . 10 |- ( ph -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = ( D ` X ) )
44		1nn0 10807	. . . . . . . . . . 11 |- 1 e. NN0
45	25, 5, 10, 38, 40	deg1pw 22256	. . . . . . . . . . 11 |- ( ( R e. NzRing /\ 1 e. NN0 ) -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = 1 )
46	2, 44, 45	sylancl 662	. . . . . . . . . 10 |- ( ph -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = 1 )
47	43, 46	eqtr3d 2510	. . . . . . . . 9 |- ( ph -> ( D ` X ) = 1 )
48	37, 47	breqtrrd 4473	. . . . . . . 8 |- ( ph -> ( D ` ( A ` N ) ) < ( D ` X ) )
49	5, 25, 4, 11, 20, 13, 19, 48	deg1sub 22244	. . . . . . 7 |- ( ph -> ( D ` ( X .- ( A ` N ) ) ) = ( D ` X ) )
50	24, 49	syl5eq 2520	. . . . . 6 |- ( ph -> ( D ` G ) = ( D ` X ) )
51	50, 47	eqtrd 2508	. . . . 5 |- ( ph -> ( D ` G ) = 1 )
52	51, 44	syl6eqel 2563	. . . 4 |- ( ph -> ( D ` G ) e. NN0 )
53		eqid 2467	. . . . . 6 |- ( 0g ` P ) = ( 0g ` P )
54	25, 5, 53, 11	deg1nn0clb 22225	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( G =/= ( 0g ` P ) <-> ( D ` G ) e. NN0 ) )
55	4, 23, 54	syl2anc 661	. . . 4 |- ( ph -> ( G =/= ( 0g ` P ) <-> ( D ` G ) e. NN0 ) )
56	52, 55	mpbird 232	. . 3 |- ( ph -> G =/= ( 0g ` P ) )
57	51	fveq2d 5868	. . . 4 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) = ( (coe1 ` G ) ` 1 ) )
58	1	fveq2i 5867	. . . . . 6 |- (coe1 ` G ) = (coe1 ` ( X .- ( A ` N ) ) )
59	58	fveq1i 5865	. . . . 5 |- ( (coe1 ` G ) ` 1 ) = ( (coe1 ` ( X .- ( A ` N ) ) ) ` 1 )
60	44	a1i 11	. . . . . 6 |- ( ph -> 1 e. NN0 )
61		eqid 2467	. . . . . . 7 |- ( -g ` R ) = ( -g ` R )
62	5, 11, 20, 61	coe1subfv 18078	. . . . . 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 1231	. . . . 5 |- ( ph -> ( (coe1 ` ( X .- ( A ` N ) ) ) ` 1 ) = ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) )
64	59, 63	syl5eq 2520	. . . 4 |- ( ph -> ( (coe1 ` G ) ` 1 ) = ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) )
65	42	oveq2d 6298	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) = ( ( 1r ` R ) ( .s ` P ) X ) )
66	5	ply1sca 18065	. . . . . . . . . . . . 13 |- ( R e. NzRing -> R = (Scalar ` P ) )
67	2, 66	syl 16	. . . . . . . . . . . 12 |- ( ph -> R = (Scalar ` P ) )
68	67	fveq2d 5868	. . . . . . . . . . 11 |- ( ph -> ( 1r ` R ) = ( 1r ` (Scalar ` P ) ) )
69	68	oveq1d 6297	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) X ) = ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) )
70	5	ply1lmod 18064	. . . . . . . . . . . 12 |- ( R e. Ring -> P e. LMod )
71	4, 70	syl 16	. . . . . . . . . . 11 |- ( ph -> P e. LMod )
72		eqid 2467	. . . . . . . . . . . 12 |- (Scalar ` P ) = (Scalar ` P )
73		eqid 2467	. . . . . . . . . . . 12 |- ( .s ` P ) = ( .s ` P )
74		eqid 2467	. . . . . . . . . . . 12 |- ( 1r ` (Scalar ` P ) ) = ( 1r ` (Scalar ` P ) )
75	11, 72, 73, 74	lmodvs1 17323	. . . . . . . . . . 11 |- ( ( P e. LMod /\ X e. B ) -> ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) = X )
76	71, 13, 75	syl2anc 661	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) = X )
77	65, 69, 76	3eqtrd 2512	. . . . . . . . 9 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) = X )
78	77	fveq2d 5868	. . . . . . . 8 |- ( ph -> (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) = (coe1 ` X ) )
79	78	fveq1d 5866	. . . . . . 7 |- ( ph -> ( (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( (coe1 ` X ) ` 1 ) )
80		eqid 2467	. . . . . . . . . 10 |- ( 1r ` R ) = ( 1r ` R )
81	15, 80	rngidcl 17006	. . . . . . . . 9 |- ( R e. Ring -> ( 1r ` R ) e. K )
82	4, 81	syl 16	. . . . . . . 8 |- ( ph -> ( 1r ` R ) e. K )
83		ply1rem.z	. . . . . . . . 9 |- .0. = ( 0g ` R )
84	83, 15, 5, 10, 73, 38, 40	coe1tmfv1 18086	. . . . . . . 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 1228	. . . . . . 7 |- ( ph -> ( (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( 1r ` R ) )
86	79, 85	eqtr3d 2510	. . . . . 6 |- ( ph -> ( (coe1 ` X ) ` 1 ) = ( 1r ` R ) )
87		eqid 2467	. . . . . . . . . 10 |- ( 0g ` R ) = ( 0g ` R )
88	5, 14, 15, 87	coe1scl 18099	. . . . . . . . 9 |- ( ( R e. Ring /\ N e. K ) -> (coe1 ` ( A ` N ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) )
89	4, 18, 88	syl2anc 661	. . . . . . . 8 |- ( ph -> (coe1 ` ( A ` N ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) )
90	89	fveq1d 5866	. . . . . . 7 |- ( ph -> ( (coe1 ` ( A ` N ) ) ` 1 ) = ( ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) ` 1 ) )
91		ax-1ne0 9557	. . . . . . . . . . 11 |- 1 =/= 0
92		neeq1 2748	. . . . . . . . . . 11 |- ( x = 1 -> ( x =/= 0 <-> 1 =/= 0 ) )
93	91, 92	mpbiri 233	. . . . . . . . . 10 |- ( x = 1 -> x =/= 0 )
94		ifnefalse 3951	. . . . . . . . . 10 |- ( x =/= 0 -> if ( x = 0 , N , ( 0g ` R ) ) = ( 0g ` R ) )
95	93, 94	syl 16	. . . . . . . . 9 |- ( x = 1 -> if ( x = 0 , N , ( 0g ` R ) ) = ( 0g ` R ) )
96		eqid 2467	. . . . . . . . 9 |- ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) )
97		fvex 5874	. . . . . . . . 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 2524	. . . . . 6 |- ( ph -> ( (coe1 ` ( A ` N ) ) ` 1 ) = ( 0g ` R ) )
101	86, 100	oveq12d 6300	. . . . 5 |- ( ph -> ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) = ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) )
102		rnggrp 16991	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
103	4, 102	syl 16	. . . . . 6 |- ( ph -> R e. Grp )
104	15, 87, 61	grpsubid1 15924	. . . . . 6 |- ( ( R e. Grp /\ ( 1r ` R ) e. K ) -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) = ( 1r ` R ) )
105	103, 82, 104	syl2anc 661	. . . . 5 |- ( ph -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) = ( 1r ` R ) )
106	101, 105	eqtrd 2508	. . . 4 |- ( ph -> ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) = ( 1r ` R ) )
107	57, 64, 106	3eqtrd 2512	. . 3 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) = ( 1r ` R ) )
108		ply1rem.u	. . . 4 |- U = (Monic1p ` R )
109	5, 11, 53, 25, 108, 80	ismon1p 22278	. . 3 |- ( G e. U <-> ( G e. B /\ G =/= ( 0g ` P ) /\ ( (coe1 ` G ) ` ( D ` G ) ) = ( 1r ` R ) ) )
110	23, 56, 107, 109	syl3anbrc 1180	. 2 |- ( ph -> G e. U )
111	1	fveq2i 5867	. . . . . . . . . 10 |- ( O ` G ) = ( O ` ( X .- ( A ` N ) ) )
112	111	fveq1i 5865	. . . . . . . . 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 465	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> R e. CRing )
116		simpr 461	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> x e. K )
117	113, 10, 15, 5, 11, 115, 116	evl1vard 18144	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> ( X e. B /\ ( ( O ` X ) ` x ) = x ) )
118	18	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ x e. K ) -> N e. K )
119	113, 5, 15, 14, 11, 115, 118, 116	evl1scad 18142	. . . . . . . . . . 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 18149	. . . . . . . . . 10 |- ( ( ph /\ x e. K ) -> ( ( X .- ( A ` N ) ) e. B /\ ( ( O ` ( X .- ( A ` N ) ) ) ` x ) = ( x ( -g ` R ) N ) ) )
121	120	simprd 463	. . . . . . . . 9 |- ( ( ph /\ x e. K ) -> ( ( O ` ( X .- ( A ` N ) ) ) ` x ) = ( x ( -g ` R ) N ) )
122	112, 121	syl5eq 2520	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( ( O ` G ) ` x ) = ( x ( -g ` R ) N ) )
123	122	eqeq1d 2469	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( ( ( O ` G ) ` x ) = .0. <-> ( x ( -g ` R ) N ) = .0. ) )
124	103	adantr 465	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> R e. Grp )
125	15, 83, 61	grpsubeq0 15925	. . . . . . . 8 |- ( ( R e. Grp /\ x e. K /\ N e. K ) -> ( ( x ( -g ` R ) N ) = .0. <-> x = N ) )
126	124, 116, 118, 125	syl3anc 1228	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( ( x ( -g ` R ) N ) = .0. <-> x = N ) )
127	123, 126	bitrd 253	. . . . . 6 |- ( ( ph /\ x e. K ) -> ( ( ( O ` G ) ` x ) = .0. <-> x = N ) )
128		elsn 4041	. . . . . 6 |- ( x e. { N } <-> x = N )
129	127, 128	syl6bbr 263	. . . . 5 |- ( ( ph /\ x e. K ) -> ( ( ( O ` G ) ` x ) = .0. <-> x e. { N } ) )
130	129	pm5.32da 641	. . . 4 |- ( ph -> ( ( x e. K /\ ( ( O ` G ) ` x ) = .0. ) <-> ( x e. K /\ x e. { N } ) ) )
131		eqid 2467	. . . . . . 7 |- ( R ^s K ) = ( R ^s K )
132		eqid 2467	. . . . . . 7 |- ( Base ` ( R ^s K ) ) = ( Base ` ( R ^s K ) )
133		fvex 5874	. . . . . . . . 9 |- ( Base ` R ) e. _V
134	15, 133	eqeltri 2551	. . . . . . . 8 |- K e. _V
135	134	a1i 11	. . . . . . 7 |- ( ph -> K e. _V )
136	113, 5, 131, 15	evl1rhm 18139	. . . . . . . . . 10 |- ( R e. CRing -> O e. ( P RingHom ( R ^s K ) ) )
137	114, 136	syl 16	. . . . . . . . 9 |- ( ph -> O e. ( P RingHom ( R ^s K ) ) )
138	11, 132	rhmf 17159	. . . . . . . . 9 |- ( O e. ( P RingHom ( R ^s K ) ) -> O : B --> ( Base ` ( R ^s K ) ) )
139	137, 138	syl 16	. . . . . . . 8 |- ( ph -> O : B --> ( Base ` ( R ^s K ) ) )
140	139, 23	ffvelrnd 6020	. . . . . . 7 |- ( ph -> ( O ` G ) e. ( Base ` ( R ^s K ) ) )
141	131, 15, 132, 2, 135, 140	pwselbas 14740	. . . . . 6 |- ( ph -> ( O ` G ) : K --> K )
142		ffn 5729	. . . . . 6 |- ( ( O ` G ) : K --> K -> ( O ` G ) Fn K )
143	141, 142	syl 16	. . . . 5 |- ( ph -> ( O ` G ) Fn K )
144		fniniseg 6000	. . . . 5 |- ( ( O ` G ) Fn K -> ( x e. ( `' ( O ` G ) " { .0. } ) <-> ( x e. K /\ ( ( O ` G ) ` x ) = .0. ) ) )
145	143, 144	syl 16	. . . 4 |- ( ph -> ( x e. ( `' ( O ` G ) " { .0. } ) <-> ( x e. K /\ ( ( O ` G ) ` x ) = .0. ) ) )
146	18	snssd 4172	. . . . . 6 |- ( ph -> { N } C_ K )
147	146	sseld 3503	. . . . 5 |- ( ph -> ( x e. { N } -> x e. K ) )
148	147	pm4.71rd 635	. . . 4 |- ( ph -> ( x e. { N } <-> ( x e. K /\ x e. { N } ) ) )
149	130, 145, 148	3bitr4d 285	. . 3 |- ( ph -> ( x e. ( `' ( O ` G ) " { .0. } ) <-> x e. { N } ) )
150	149	eqrdv 2464	. 2 |- ( ph -> ( `' ( O ` G ) " { .0. } ) = { N } )
151	110, 51, 150	3jca 1176	1 |- ( ph -> ( G e. U /\ ( D ` G ) = 1 /\ ( `' ( O ` G ) " { .0. } ) = { N } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   _Vcvv 3113   ifcif 3939   {csn 4027   class class class wbr 4447   |-> cmpt 4505   `'ccnv 4998   "cima 5002   Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   RRcr 9487   0cc0 9488   1c1 9489   RR*cxr 9623   < clt 9624   <_ cle 9625   NN0cn0 10791   Basecbs 14486  Scalarcsca 14554   .scvsca 14555   0gc0g 14691   ^_s cpws 14698   Grpcgrp 15723   -gcsg 15726  ._gcmg 15727  mulGrpcmgp 16931   1rcur 16943   Ringcrg 16986   CRingccrg 16987   RingHom crh 17145   LModclmod 17295  NzRingcnzr 17687  algSccascl 17731  var₁cv1 17986  Poly₁cpl1 17987  coe₁cco1 17988  eval₁ce1 18122   deg₁ cdg1 22187  Monic_1pcmn1 22261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-ofr 6523  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-0g 14693  df-gsum 14694  df-prds 14699  df-pws 14701  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-ghm 16060  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-srg 16948  df-rng 16988  df-cring 16989  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-rnghom 17148  df-subrg 17210  df-lmod 17297  df-lss 17362  df-lsp 17401  df-nzr 17688  df-rlreg 17702  df-assa 17732  df-asp 17733  df-ascl 17734  df-psr 17776  df-mvr 17777  df-mpl 17778  df-opsr 17780  df-evls 17942  df-evl 17943  df-psr1 17990  df-vr1 17991  df-ply1 17992  df-coe1 17993  df-evl1 18124  df-cnfld 18192  df-mdeg 22188  df-deg1 22189  df-mon1 22266

This theorem is referenced by:  ply1rem  22299  facth1  22300  fta1glem1  22301  fta1glem2  22302