Theorem ply1remlem 21519

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 17265	. . . . . . . 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 17601	. . . . . . 7 |- ( R e. Ring -> P e. Ring )
7	4, 6	syl 16	. . . . . 6 |- ( ph -> P e. Ring )
8		rnggrp 16586	. . . . . 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 17570	. . . . . 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 17636	. . . . . . 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 5832	. . . . 5 |- ( ph -> ( A ` N ) e. B )
20		ply1rem.m	. . . . . 6 |- .- = ( -g ` P )
21	11, 20	grpsubcl 15586	. . . . 5 |- ( ( P e. Grp /\ X e. B /\ ( A ` N ) e. B ) -> ( X .- ( A ` N ) ) e. B )
22	9, 13, 19, 21	syl3anc 1211	. . . 4 |- ( ph -> ( X .- ( A ` N ) ) e. B )
23	1, 22	syl5eqel 2517	. . 3 |- ( ph -> G e. B )
24	1	fveq2i 5682	. . . . . . 7 |- ( D ` G ) = ( D ` ( X .- ( A ` N ) ) )
25		ply1rem.d	. . . . . . . 8 |- D = ( deg1 ` R )
26	25, 5, 11	deg1xrcl 21438	. . . . . . . . . . 11 |- ( ( A ` N ) e. B -> ( D ` ( A ` N ) ) e. RR* )
27	19, 26	syl 16	. . . . . . . . . 10 |- ( ph -> ( D ` ( A ` N ) ) e. RR* )
28		0xr 9418	. . . . . . . . . . 11 |- 0 e. RR*
29	28	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 e. RR* )
30		1re 9373	. . . . . . . . . . 11 |- 1 e. RR
31		rexr 9417	. . . . . . . . . . 11 |- ( 1 e. RR -> 1 e. RR* )
32	30, 31	mp1i 12	. . . . . . . . . 10 |- ( ph -> 1 e. RR* )
33	25, 5, 15, 14	deg1sclle 21469	. . . . . . . . . . 11 |- ( ( R e. Ring /\ N e. K ) -> ( D ` ( A ` N ) ) <_ 0 )
34	4, 18, 33	syl2anc 654	. . . . . . . . . 10 |- ( ph -> ( D ` ( A ` N ) ) <_ 0 )
35		0lt1 9850	. . . . . . . . . . 11 |- 0 < 1
36	35	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 < 1 )
37	27, 29, 32, 34, 36	xrlelttrd 11122	. . . . . . . . 9 |- ( ph -> ( D ` ( A ` N ) ) < 1 )
38		eqid 2433	. . . . . . . . . . . . . 14 |- (mulGrp ` P ) = (mulGrp ` P )
39	38, 11	mgpbas 16571	. . . . . . . . . . . . 13 |- B = ( Base ` (mulGrp ` P ) )
40		eqid 2433	. . . . . . . . . . . . 13 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
41	39, 40	mulg1 15614	. . . . . . . . . . . 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 5683	. . . . . . . . . 10 |- ( ph -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = ( D ` X ) )
44		1nn0 10583	. . . . . . . . . . 11 |- 1 e. NN0
45	25, 5, 10, 38, 40	deg1pw 21477	. . . . . . . . . . 11 |- ( ( R e. NzRing /\ 1 e. NN0 ) -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = 1 )
46	2, 44, 45	sylancl 655	. . . . . . . . . 10 |- ( ph -> ( D ` ( 1 (.g ` (mulGrp ` P ) ) X ) ) = 1 )
47	43, 46	eqtr3d 2467	. . . . . . . . 9 |- ( ph -> ( D ` X ) = 1 )
48	37, 47	breqtrrd 4306	. . . . . . . 8 |- ( ph -> ( D ` ( A ` N ) ) < ( D ` X ) )
49	5, 25, 4, 11, 20, 13, 19, 48	deg1sub 21465	. . . . . . 7 |- ( ph -> ( D ` ( X .- ( A ` N ) ) ) = ( D ` X ) )
50	24, 49	syl5eq 2477	. . . . . 6 |- ( ph -> ( D ` G ) = ( D ` X ) )
51	50, 47	eqtrd 2465	. . . . 5 |- ( ph -> ( D ` G ) = 1 )
52	51, 44	syl6eqel 2521	. . . 4 |- ( ph -> ( D ` G ) e. NN0 )
53		eqid 2433	. . . . . 6 |- ( 0g ` P ) = ( 0g ` P )
54	25, 5, 53, 11	deg1nn0clb 21446	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( G =/= ( 0g ` P ) <-> ( D ` G ) e. NN0 ) )
55	4, 23, 54	syl2anc 654	. . . 4 |- ( ph -> ( G =/= ( 0g ` P ) <-> ( D ` G ) e. NN0 ) )
56	52, 55	mpbird 232	. . 3 |- ( ph -> G =/= ( 0g ` P ) )
57	51	fveq2d 5683	. . . 4 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) = ( (coe1 ` G ) ` 1 ) )
58	1	fveq2i 5682	. . . . . 6 |- (coe1 ` G ) = (coe1 ` ( X .- ( A ` N ) ) )
59	58	fveq1i 5680	. . . . 5 |- ( (coe1 ` G ) ` 1 ) = ( (coe1 ` ( X .- ( A ` N ) ) ) ` 1 )
60	44	a1i 11	. . . . . 6 |- ( ph -> 1 e. NN0 )
61		eqid 2433	. . . . . . 7 |- ( -g ` R ) = ( -g ` R )
62	5, 11, 20, 61	coe1subfv 17618	. . . . . 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 1214	. . . . 5 |- ( ph -> ( (coe1 ` ( X .- ( A ` N ) ) ) ` 1 ) = ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) )
64	59, 63	syl5eq 2477	. . . 4 |- ( ph -> ( (coe1 ` G ) ` 1 ) = ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) )
65	42	oveq2d 6096	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) = ( ( 1r ` R ) ( .s ` P ) X ) )
66	5	ply1sca 17606	. . . . . . . . . . . . 13 |- ( R e. NzRing -> R = (Scalar ` P ) )
67	2, 66	syl 16	. . . . . . . . . . . 12 |- ( ph -> R = (Scalar ` P ) )
68	67	fveq2d 5683	. . . . . . . . . . 11 |- ( ph -> ( 1r ` R ) = ( 1r ` (Scalar ` P ) ) )
69	68	oveq1d 6095	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) X ) = ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) )
70	5	ply1lmod 17605	. . . . . . . . . . . 12 |- ( R e. Ring -> P e. LMod )
71	4, 70	syl 16	. . . . . . . . . . 11 |- ( ph -> P e. LMod )
72		eqid 2433	. . . . . . . . . . . 12 |- (Scalar ` P ) = (Scalar ` P )
73		eqid 2433	. . . . . . . . . . . 12 |- ( .s ` P ) = ( .s ` P )
74		eqid 2433	. . . . . . . . . . . 12 |- ( 1r ` (Scalar ` P ) ) = ( 1r ` (Scalar ` P ) )
75	11, 72, 73, 74	lmodvs1 16900	. . . . . . . . . . 11 |- ( ( P e. LMod /\ X e. B ) -> ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) = X )
76	71, 13, 75	syl2anc 654	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) X ) = X )
77	65, 69, 76	3eqtrd 2469	. . . . . . . . 9 |- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) = X )
78	77	fveq2d 5683	. . . . . . . 8 |- ( ph -> (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) = (coe1 ` X ) )
79	78	fveq1d 5681	. . . . . . 7 |- ( ph -> ( (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( (coe1 ` X ) ` 1 ) )
80		eqid 2433	. . . . . . . . . 10 |- ( 1r ` R ) = ( 1r ` R )
81	15, 80	rngidcl 16601	. . . . . . . . 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 17625	. . . . . . . 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 1211	. . . . . . 7 |- ( ph -> ( (coe1 ` ( ( 1r ` R ) ( .s ` P ) ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( 1r ` R ) )
86	79, 85	eqtr3d 2467	. . . . . 6 |- ( ph -> ( (coe1 ` X ) ` 1 ) = ( 1r ` R ) )
87		eqid 2433	. . . . . . . . . 10 |- ( 0g ` R ) = ( 0g ` R )
88	5, 14, 15, 87	coe1scl 17637	. . . . . . . . 9 |- ( ( R e. Ring /\ N e. K ) -> (coe1 ` ( A ` N ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) )
89	4, 18, 88	syl2anc 654	. . . . . . . 8 |- ( ph -> (coe1 ` ( A ` N ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) )
90	89	fveq1d 5681	. . . . . . 7 |- ( ph -> ( (coe1 ` ( A ` N ) ) ` 1 ) = ( ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) ` 1 ) )
91		ax-1ne0 9339	. . . . . . . . . . 11 |- 1 =/= 0
92		neeq1 2606	. . . . . . . . . . 11 |- ( x = 1 -> ( x =/= 0 <-> 1 =/= 0 ) )
93	91, 92	mpbiri 233	. . . . . . . . . 10 |- ( x = 1 -> x =/= 0 )
94		ifnefalse 3789	. . . . . . . . . 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 2433	. . . . . . . . 9 |- ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) ) = ( x e. NN0 |-> if ( x = 0 , N , ( 0g ` R ) ) )
97		fvex 5689	. . . . . . . . 9 |- ( 0g ` R ) e. _V
98	95, 96, 97	fvmpt 5762	. . . . . . . 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 2481	. . . . . 6 |- ( ph -> ( (coe1 ` ( A ` N ) ) ` 1 ) = ( 0g ` R ) )
101	86, 100	oveq12d 6098	. . . . 5 |- ( ph -> ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) = ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) )
102		rnggrp 16586	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
103	4, 102	syl 16	. . . . . 6 |- ( ph -> R e. Grp )
104	15, 87, 61	grpsubid1 15591	. . . . . 6 |- ( ( R e. Grp /\ ( 1r ` R ) e. K ) -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) = ( 1r ` R ) )
105	103, 82, 104	syl2anc 654	. . . . 5 |- ( ph -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) = ( 1r ` R ) )
106	101, 105	eqtrd 2465	. . . 4 |- ( ph -> ( ( (coe1 ` X ) ` 1 ) ( -g ` R ) ( (coe1 ` ( A ` N ) ) ` 1 ) ) = ( 1r ` R ) )
107	57, 64, 106	3eqtrd 2469	. . 3 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) = ( 1r ` R ) )
108		ply1rem.u	. . . 4 |- U = (Monic1p ` R )
109	5, 11, 53, 25, 108, 80	ismon1p 21499	. . 3 |- ( G e. U <-> ( G e. B /\ G =/= ( 0g ` P ) /\ ( (coe1 ` G ) ` ( D ` G ) ) = ( 1r ` R ) ) )
110	23, 56, 107, 109	syl3anbrc 1165	. 2 |- ( ph -> G e. U )
111	1	fveq2i 5682	. . . . . . . . . 10 |- ( O ` G ) = ( O ` ( X .- ( A ` N ) ) )
112	111	fveq1i 5680	. . . . . . . . 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 462	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> R e. CRing )
116		simpr 458	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> x e. K )
117	113, 10, 15, 5, 11, 115, 116	evl1vard 21384	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> ( X e. B /\ ( ( O ` X ) ` x ) = x ) )
118	18	adantr 462	. . . . . . . . . . . 12 |- ( ( ph /\ x e. K ) -> N e. K )
119	113, 5, 15, 14, 11, 115, 118, 116	evl1scad 21382	. . . . . . . . . . 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 21386	. . . . . . . . . 10 |- ( ( ph /\ x e. K ) -> ( ( X .- ( A ` N ) ) e. B /\ ( ( O ` ( X .- ( A ` N ) ) ) ` x ) = ( x ( -g ` R ) N ) ) )
121	120	simprd 460	. . . . . . . . 9 |- ( ( ph /\ x e. K ) -> ( ( O ` ( X .- ( A ` N ) ) ) ` x ) = ( x ( -g ` R ) N ) )
122	112, 121	syl5eq 2477	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( ( O ` G ) ` x ) = ( x ( -g ` R ) N ) )
123	122	eqeq1d 2441	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( ( ( O ` G ) ` x ) = .0. <-> ( x ( -g ` R ) N ) = .0. ) )
124	103	adantr 462	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> R e. Grp )
125	15, 83, 61	grpsubeq0 15592	. . . . . . . 8 |- ( ( R e. Grp /\ x e. K /\ N e. K ) -> ( ( x ( -g ` R ) N ) = .0. <-> x = N ) )
126	124, 116, 118, 125	syl3anc 1211	. . . . . . 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 3879	. . . . . 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 634	. . . 4 |- ( ph -> ( ( x e. K /\ ( ( O ` G ) ` x ) = .0. ) <-> ( x e. K /\ x e. { N } ) ) )
131		eqid 2433	. . . . . . 7 |- ( R ^s K ) = ( R ^s K )
132		eqid 2433	. . . . . . 7 |- ( Base ` ( R ^s K ) ) = ( Base ` ( R ^s K ) )
133		fvex 5689	. . . . . . . . 9 |- ( Base ` R ) e. _V
134	15, 133	eqeltri 2503	. . . . . . . 8 |- K e. _V
135	134	a1i 11	. . . . . . 7 |- ( ph -> K e. _V )
136	113, 5, 131, 15	evl1rhm 21380	. . . . . . . . . 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 16748	. . . . . . . . 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 5832	. . . . . . 7 |- ( ph -> ( O ` G ) e. ( Base ` ( R ^s K ) ) )
141	131, 15, 132, 2, 135, 140	pwselbas 14410	. . . . . 6 |- ( ph -> ( O ` G ) : K --> K )
142		ffn 5547	. . . . . 6 |- ( ( O ` G ) : K --> K -> ( O ` G ) Fn K )
143	141, 142	syl 16	. . . . 5 |- ( ph -> ( O ` G ) Fn K )
144		fniniseg 5812	. . . . 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 4006	. . . . . 6 |- ( ph -> { N } C_ K )
147	146	sseld 3343	. . . . 5 |- ( ph -> ( x e. { N } -> x e. K ) )
148	147	pm4.71rd 628	. . . 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 2431	. 2 |- ( ph -> ( `' ( O ` G ) " { .0. } ) = { N } )
151	110, 51, 150	3jca 1161	1 |- ( ph -> ( G e. U /\ ( D ` G ) = 1 /\ ( `' ( O ` G ) " { .0. } ) = { N } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 958   = wceq 1362   e. wcel 1755   =/= wne 2596   _Vcvv 2962   ifcif 3779   {csn 3865   class class class wbr 4280   e. cmpt 4338   `'ccnv 4826   "cima 4830   Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080   RRcr 9269   0cc0 9270   1c1 9271   RR*cxr 9405   < clt 9406   <_ cle 9407   NN0cn0 10567   Basecbs 14157  Scalarcsca 14224   .scvsca 14225   0gc0g 14361   ^_s cpws 14368   Grpcgrp 15393   -gcsg 15396  ._gcmg 15397  mulGrpcmgp 16565   Ringcrg 16577   CRingccrg 16578   1rcur 16579   RingHom crh 16738   LModclmod 16872  NzRingcnzr 17261  algSccascl 17305  var₁cv1 17527  Poly₁cpl1 17528  eval₁ce1 17530  coe₁cco1 17531   deg₁ cdg1 21408  Monic_1pcmn1 21482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-ofr 6310  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-tpos 6734  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-fz 11425  df-fzo 11533  df-seq 11791  df-hash 12088  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-0g 14363  df-gsum 14364  df-prds 14369  df-pws 14371  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-mhm 15447  df-submnd 15448  df-grp 15525  df-minusg 15526  df-sbg 15527  df-mulg 15528  df-subg 15658  df-ghm 15725  df-cntz 15815  df-cmn 16259  df-abl 16260  df-mgp 16566  df-rng 16580  df-cring 16581  df-ur 16582  df-oppr 16649  df-dvdsr 16667  df-unit 16668  df-invr 16698  df-rnghom 16740  df-subrg 16787  df-lmod 16874  df-lss 16936  df-lsp 16975  df-nzr 17262  df-rlreg 17276  df-assa 17306  df-asp 17307  df-ascl 17308  df-psr 17351  df-mvr 17352  df-mpl 17353  df-evls 17354  df-evl 17355  df-opsr 17359  df-psr1 17533  df-vr1 17534  df-ply1 17535  df-evl1 17537  df-coe1 17538  df-cnfld 17663  df-mdeg 21409  df-deg1 21410  df-mon1 21487

This theorem is referenced by:  ply1rem  21520  facth1  21521  fta1glem1  21522  fta1glem2  21523