Theorem ply1remlem 21637

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 17346	. . . . . . . 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 17706	. . . . . . 7 |- ( R e. Ring -> P e. Ring )
7	4, 6	syl 16	. . . . . 6 |- ( ph -> P e. Ring )
8		rnggrp 16653	. . . . . 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 17674	. . . . . 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 17741	. . . . . . 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 5847	. . . . 5 |- ( ph -> ( A ` N ) e. B )
20		ply1rem.m	. . . . . 6 |- .- = ( -g ` P )
21	11, 20	grpsubcl 15609	. . . . 5 |- ( ( P e. Grp /\ X e. B /\ ( A ` N ) e. B ) -> ( X .- ( A ` N ) ) e. B )
22	9, 13, 19, 21	syl3anc 1218	. . . 4 |- ( ph -> ( X .- ( A ` N ) ) e. B )
23	1, 22	syl5eqel 2527	. . 3 |- ( ph -> G e. B )
24	1	fveq2i 5697	. . . . . . 7 |- ( D ` G ) = ( D ` ( X .- ( A ` N ) ) )
25		ply1rem.d	. . . . . . . 8 |- D = ( deg1 ` R )
26	25, 5, 11	deg1xrcl 21556	. . . . . . . . . . 11 |- ( ( A ` N ) e. B -> ( D ` ( A ` N ) ) e. RR* )
27	19, 26	syl 16	. . . . . . . . . 10 |- ( ph -> ( D ` ( A ` N ) ) e. RR* )
28		0xr 9433	. . . . . . . . . . 11 |- 0 e. RR*
29	28	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 e. RR* )
30		1re 9388	. . . . . . . . . . 11 |- 1 e. RR
31		rexr 9432	. . . . . . . . . . 11 |- ( 1 e. RR -> 1 e. RR* )
32	30, 31	mp1i 12	. . . . . . . . . 10 |- ( ph -> 1 e. RR* )
33	25, 5, 15, 14	deg1sclle 21587	. . . . . . . . . . 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 9865	. . . . . . . . . . 11 |- 0 < 1
36	35	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 < 1 )
37	27, 29, 32, 34, 36	xrlelttrd 11137	. . . . . . . . 9 |- ( ph -> ( D ` ( A ` N ) ) < 1 )
38		eqid 2443	. . . . . . . . . . . . . 14 |- (mulGrp ` P ) = (mulGrp ` P )
39	38, 11	mgpbas 16600	. . . . . . . . . . . . 13 |- B = ( Base ` (mulGrp ` P ) )
40		eqid 2443	. . . . . . . . . . . . 13 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
41	39, 40	mulg1 15637	. . . . . . . . . . . 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 5698	. . . . . . . . . 10 |- ( ph -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = ( D ` X ) )
44		1nn0 10598	. . . . . . . . . . 11 |- 1 e. NN0
45	25, 5, 10, 38, 40	deg1pw 21595	. . . . . . . . . . 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 2477	. . . . . . . . 9 |- ( ph -> ( D ` X ) = 1 )
48	37, 47	breqtrrd 4321	. . . . . . . 8 |- ( ph -> ( D ` ( A ` N ) ) < ( D ` X ) )
49	5, 25, 4, 11, 20, 13, 19, 48	deg1sub 21583	. . . . . . 7 |- ( ph -> ( D ` ( X .- ( A ` N ) ) ) = ( D ` X ) )
50	24, 49	syl5eq 2487	. . . . . 6 |- ( ph -> ( D ` G ) = ( D ` X ) )
51	50, 47	eqtrd 2475	. . . . 5 |- ( ph -> ( D ` G ) = 1 )
52	51, 44	syl6eqel 2531	. . . 4 |- ( ph -> ( D ` G ) e. NN0 )
53		eqid 2443	. . . . . 6 |- ( 0g ` P ) = ( 0g ` P )
54	25, 5, 53, 11	deg1nn0clb 21564	. . . . 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 5698	. . . 4 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) = ( (coe1 ` G ) ` 1 ) )
58	1	fveq2i 5697	. . . . . 6 |- (coe1 ` G ) = (coe1 ` ( X .- ( A ` N ) ) )
59	58	fveq1i 5695	. . . . 5 |- ( (coe1 ` G ) ` 1 ) = ( (coe1 ` ( X .- ( A ` N ) ) ) ` 1 )
60	44	a1i 11	. . . . . 6 |- ( ph -> 1 e. NN0 )
61		eqid 2443	. . . . . . 7 |- ( -g ` R ) = ( -g ` R )
62	5, 11, 20, 61	coe1subfv 17723	. . . . . 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 1221	. . . . 5 |- ( ph -> ( (coe1 ` ( X .- ( A ` N ) ) ) ` 1 ) = ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) )
64	59, 63	syl5eq 2487	. . . 4 |- ( ph -> ( (coe1 ` G ) ` 1 ) = ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) )
65	42	oveq2d 6110	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) = ( ( 1r ` R ) ( .s ` P ) X ) )
66	5	ply1sca 17711	. . . . . . . . . . . . 13 |- ( R e. NzRing -> R = (Scalar ` P ) )
67	2, 66	syl 16	. . . . . . . . . . . 12 |- ( ph -> R = (Scalar ` P ) )
68	67	fveq2d 5698	. . . . . . . . . . 11 |- ( ph -> ( 1r ` R ) = ( 1r ` (Scalar ` P ) ) )
69	68	oveq1d 6109	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) X ) = ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) )
70	5	ply1lmod 17710	. . . . . . . . . . . 12 |- ( R e. Ring -> P e. LMod )
71	4, 70	syl 16	. . . . . . . . . . 11 |- ( ph -> P e. LMod )
72		eqid 2443	. . . . . . . . . . . 12 |- (Scalar ` P ) = (Scalar ` P )
73		eqid 2443	. . . . . . . . . . . 12 |- ( .s ` P ) = ( .s ` P )
74		eqid 2443	. . . . . . . . . . . 12 |- ( 1r ` (Scalar ` P ) ) = ( 1r ` (Scalar ` P ) )
75	11, 72, 73, 74	lmodvs1 16979	. . . . . . . . . . 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 2479	. . . . . . . . 9 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) = X )
78	77	fveq2d 5698	. . . . . . . 8 |- ( ph -> (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) = (coe1 ` X ) )
79	78	fveq1d 5696	. . . . . . 7 |- ( ph -> ( (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( (coe1 ` X ) ` 1 ) )
80		eqid 2443	. . . . . . . . . 10 |- ( 1r ` R ) = ( 1r ` R )
81	15, 80	rngidcl 16668	. . . . . . . . 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 17730	. . . . . . . 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 1218	. . . . . . 7 |- ( ph -> ( (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( 1r ` R ) )
86	79, 85	eqtr3d 2477	. . . . . 6 |- ( ph -> ( (coe1 ` X ) ` 1 ) = ( 1r ` R ) )
87		eqid 2443	. . . . . . . . . 10 |- ( 0g ` R ) = ( 0g ` R )
88	5, 14, 15, 87	coe1scl 17742	. . . . . . . . 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 5696	. . . . . . 7 |- ( ph -> ( (coe1 ` ( A ` N ) ) ` 1 ) = ( ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) ` 1 ) )
91		ax-1ne0 9354	. . . . . . . . . . 11 |- 1 =/= 0
92		neeq1 2619	. . . . . . . . . . 11 |- ( x = 1 -> ( x =/= 0 <-> 1 =/= 0 ) )
93	91, 92	mpbiri 233	. . . . . . . . . 10 |- ( x = 1 -> x =/= 0 )
94		ifnefalse 3804	. . . . . . . . . 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 2443	. . . . . . . . 9 |- ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) )
97		fvex 5704	. . . . . . . . 9 |- ( 0g ` R ) e. _V
98	95, 96, 97	fvmpt 5777	. . . . . . . 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 2491	. . . . . 6 |- ( ph -> ( (coe1 ` ( A ` N ) ) ` 1 ) = ( 0g ` R ) )
101	86, 100	oveq12d 6112	. . . . 5 |- ( ph -> ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) = ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) )
102		rnggrp 16653	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
103	4, 102	syl 16	. . . . . 6 |- ( ph -> R e. Grp )
104	15, 87, 61	grpsubid1 15614	. . . . . 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 2475	. . . 4 |- ( ph -> ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) = ( 1r ` R ) )
107	57, 64, 106	3eqtrd 2479	. . 3 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) = ( 1r ` R ) )
108		ply1rem.u	. . . 4 |- U = (Monic1p ` R )
109	5, 11, 53, 25, 108, 80	ismon1p 21617	. . 3 |- ( G e. U <-> ( G e. B /\ G =/= ( 0g ` P ) /\ ( (coe1 ` G ) ` ( D ` G ) ) = ( 1r ` R ) ) )
110	23, 56, 107, 109	syl3anbrc 1172	. 2 |- ( ph -> G e. U )
111	1	fveq2i 5697	. . . . . . . . . 10 |- ( O ` G ) = ( O ` ( X .- ( A ` N ) ) )
112	111	fveq1i 5695	. . . . . . . . 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 17774	. . . . . . . . . . 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 17772	. . . . . . . . . . 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 17779	. . . . . . . . . 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 2487	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( ( O ` G ) ` x ) = ( x ( -g ` R ) N ) )
123	122	eqeq1d 2451	. . . . . . 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 15615	. . . . . . . 8 |- ( ( R e. Grp /\ x e. K /\ N e. K ) -> ( ( x ( -g ` R ) N ) = .0. <-> x = N ) )
126	124, 116, 118, 125	syl3anc 1218	. . . . . . 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 3894	. . . . . 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 2443	. . . . . . 7 |- ( R ^s K ) = ( R ^s K )
132		eqid 2443	. . . . . . 7 |- ( Base ` ( R ^s K ) ) = ( Base ` ( R ^s K ) )
133		fvex 5704	. . . . . . . . 9 |- ( Base ` R ) e. _V
134	15, 133	eqeltri 2513	. . . . . . . 8 |- K e. _V
135	134	a1i 11	. . . . . . 7 |- ( ph -> K e. _V )
136	113, 5, 131, 15	evl1rhm 17769	. . . . . . . . . 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 16819	. . . . . . . . 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 5847	. . . . . . 7 |- ( ph -> ( O ` G ) e. ( Base ` ( R ^s K ) ) )
141	131, 15, 132, 2, 135, 140	pwselbas 14430	. . . . . 6 |- ( ph -> ( O ` G ) : K --> K )
142		ffn 5562	. . . . . 6 |- ( ( O ` G ) : K --> K -> ( O ` G ) Fn K )
143	141, 142	syl 16	. . . . 5 |- ( ph -> ( O ` G ) Fn K )
144		fniniseg 5827	. . . . 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 4021	. . . . . 6 |- ( ph -> { N } C_ K )
147	146	sseld 3358	. . . . 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 2441	. 2 |- ( ph -> ( `' ( O ` G ) " { .0. } ) = { N } )
151	110, 51, 150	3jca 1168	1 |- ( ph -> ( G e. U /\ ( D ` G ) = 1 /\ ( `' ( O ` G ) " { .0. } ) = { N } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2609   _Vcvv 2975   ifcif 3794   {csn 3880   class class class wbr 4295   e. cmpt 4353   `'ccnv 4842   "cima 4846   Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6094   RRcr 9284   0cc0 9285   1c1 9286   RR*cxr 9420   < clt 9421   <_ cle 9422   NN0cn0 10582   Basecbs 14177  Scalarcsca 14244   .scvsca 14245   0gc0g 14381   ^_s cpws 14388   Grpcgrp 15413   -gcsg 15416  ._gcmg 15417  mulGrpcmgp 16594   1rcur 16606   Ringcrg 16648   CRingccrg 16649   RingHom crh 16807   LModclmod 16951  NzRingcnzr 17342  algSccascl 17386  var₁cv1 17635  Poly₁cpl1 17636  coe₁cco1 17637  eval₁ce1 17752   deg₁ cdg1 21526  Monic_1pcmn1 21600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364  ax-mulf 9365

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-iin 4177  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-ofr 6324  df-om 6480  df-1st 6580  df-2nd 6581  df-supp 6694  df-tpos 6748  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-ixp 7267  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-fsupp 7624  df-sup 7694  df-oi 7727  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-fz 11441  df-fzo 11552  df-seq 11810  df-hash 12107  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-starv 14256  df-sca 14257  df-vsca 14258  df-ip 14259  df-tset 14260  df-ple 14261  df-ds 14263  df-unif 14264  df-hom 14265  df-cco 14266  df-0g 14383  df-gsum 14384  df-prds 14389  df-pws 14391  df-mre 14527  df-mrc 14528  df-acs 14530  df-mnd 15418  df-mhm 15467  df-submnd 15468  df-grp 15548  df-minusg 15549  df-sbg 15550  df-mulg 15551  df-subg 15681  df-ghm 15748  df-cntz 15838  df-cmn 16282  df-abl 16283  df-mgp 16595  df-ur 16607  df-srg 16611  df-rng 16650  df-cring 16651  df-oppr 16718  df-dvdsr 16736  df-unit 16737  df-invr 16767  df-rnghom 16809  df-subrg 16866  df-lmod 16953  df-lss 17017  df-lsp 17056  df-nzr 17343  df-rlreg 17357  df-assa 17387  df-asp 17388  df-ascl 17389  df-psr 17426  df-mvr 17427  df-mpl 17428  df-opsr 17430  df-evls 17591  df-evl 17592  df-psr1 17639  df-vr1 17640  df-ply1 17641  df-coe1 17642  df-evl1 17754  df-cnfld 17822  df-mdeg 21527  df-deg1 21528  df-mon1 21605

This theorem is referenced by:  ply1rem  21638  facth1  21639  fta1glem1  21640  fta1glem2  21641