Theorem deg1mhm 37785

Description: Homomorphic property of the polynomial degree. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Hypotheses

Ref	Expression
deg1mhm.d	|- D = ( deg1 ` R )
deg1mhm.b	|- B = ( Base ` P )
deg1mhm.p	|- P = (Poly1 ` R )
deg1mhm.z	|- .0. = ( 0g ` P )
deg1mhm.y	|- Y = ( (mulGrp ` P ) ↾s ( B \ { .0. } ) )
deg1mhm.n	|- N = (ℂfld ↾s NN0 )

Assertion

Ref	Expression
deg1mhm	|- ( R e. Domn -> ( D |` ( B \ { .0. } ) ) e. ( Y MndHom N ) )

Proof of Theorem deg1mhm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		deg1mhm.p	. . . . . 6 |- P = (Poly1 ` R )
2	1	ply1domn 23883	. . . . 5 |- ( R e. Domn -> P e. Domn )
3		deg1mhm.b	. . . . . . 7 |- B = ( Base ` P )
4		deg1mhm.z	. . . . . . 7 |- .0. = ( 0g ` P )
5		eqid 2622	. . . . . . 7 |- (mulGrp ` P ) = (mulGrp ` P )
6	3, 4, 5	isdomn3 37782	. . . . . 6 |- ( P e. Domn <-> ( P e. Ring /\ ( B \ { .0. } ) e. (SubMnd ` (mulGrp ` P ) ) ) )
7	6	simprbi 480	. . . . 5 |- ( P e. Domn -> ( B \ { .0. } ) e. (SubMnd ` (mulGrp ` P ) ) )
8	2, 7	syl 17	. . . 4 |- ( R e. Domn -> ( B \ { .0. } ) e. (SubMnd ` (mulGrp ` P ) ) )
9		deg1mhm.y	. . . . 5 |- Y = ( (mulGrp ` P ) ↾s ( B \ { .0. } ) )
10	9	submmnd 17354	. . . 4 |- ( ( B \ { .0. } ) e. (SubMnd ` (mulGrp ` P ) ) -> Y e. Mnd )
11	8, 10	syl 17	. . 3 |- ( R e. Domn -> Y e. Mnd )
12		nn0subm 19801	. . . 4 |- NN0 e. (SubMnd ` ℂfld )
13		deg1mhm.n	. . . . 5 |- N = (ℂfld ↾s NN0 )
14	13	submmnd 17354	. . . 4 |- ( NN0 e. (SubMnd ` ℂfld ) -> N e. Mnd )
15	12, 14	mp1i 13	. . 3 |- ( R e. Domn -> N e. Mnd )
16	11, 15	jca 554	. 2 |- ( R e. Domn -> ( Y e. Mnd /\ N e. Mnd ) )
17		deg1mhm.d	. . . . . . . 8 |- D = ( deg1 ` R )
18	17, 1, 3	deg1xrf 23841	. . . . . . 7 |- D : B --> RR*
19		ffn 6045	. . . . . . 7 |- ( D : B --> RR* -> D Fn B )
20	18, 19	ax-mp 5	. . . . . 6 |- D Fn B
21		difss 3737	. . . . . 6 |- ( B \ { .0. } ) C_ B
22		fnssres 6004	. . . . . 6 |- ( ( D Fn B /\ ( B \ { .0. } ) C_ B ) -> ( D |` ( B \ { .0. } ) ) Fn ( B \ { .0. } ) )
23	20, 21, 22	mp2an 708	. . . . 5 |- ( D |` ( B \ { .0. } ) ) Fn ( B \ { .0. } )
24	23	a1i 11	. . . 4 |- ( R e. Domn -> ( D |` ( B \ { .0. } ) ) Fn ( B \ { .0. } ) )
25		fvres 6207	. . . . . . 7 |- ( x e. ( B \ { .0. } ) -> ( ( D |` ( B \ { .0. } ) ) ` x ) = ( D ` x ) )
26	25	adantl 482	. . . . . 6 |- ( ( R e. Domn /\ x e. ( B \ { .0. } ) ) -> ( ( D |` ( B \ { .0. } ) ) ` x ) = ( D ` x ) )
27		domnring 19296	. . . . . . . 8 |- ( R e. Domn -> R e. Ring )
28	27	adantr 481	. . . . . . 7 |- ( ( R e. Domn /\ x e. ( B \ { .0. } ) ) -> R e. Ring )
29		eldifi 3732	. . . . . . . 8 |- ( x e. ( B \ { .0. } ) -> x e. B )
30	29	adantl 482	. . . . . . 7 |- ( ( R e. Domn /\ x e. ( B \ { .0. } ) ) -> x e. B )
31		eldifsni 4320	. . . . . . . 8 |- ( x e. ( B \ { .0. } ) -> x =/= .0. )
32	31	adantl 482	. . . . . . 7 |- ( ( R e. Domn /\ x e. ( B \ { .0. } ) ) -> x =/= .0. )
33	17, 1, 4, 3	deg1nn0cl 23848	. . . . . . 7 |- ( ( R e. Ring /\ x e. B /\ x =/= .0. ) -> ( D ` x ) e. NN0 )
34	28, 30, 32, 33	syl3anc 1326	. . . . . 6 |- ( ( R e. Domn /\ x e. ( B \ { .0. } ) ) -> ( D ` x ) e. NN0 )
35	26, 34	eqeltrd 2701	. . . . 5 |- ( ( R e. Domn /\ x e. ( B \ { .0. } ) ) -> ( ( D |` ( B \ { .0. } ) ) ` x ) e. NN0 )
36	35	ralrimiva 2966	. . . 4 |- ( R e. Domn -> A. x e. ( B \ { .0. } ) ( ( D |` ( B \ { .0. } ) ) ` x ) e. NN0 )
37		ffnfv 6388	. . . 4 |- ( ( D |` ( B \ { .0. } ) ) : ( B \ { .0. } ) --> NN0 <-> ( ( D |` ( B \ { .0. } ) ) Fn ( B \ { .0. } ) /\ A. x e. ( B \ { .0. } ) ( ( D |` ( B \ { .0. } ) ) ` x ) e. NN0 ) )
38	24, 36, 37	sylanbrc 698	. . 3 |- ( R e. Domn -> ( D |` ( B \ { .0. } ) ) : ( B \ { .0. } ) --> NN0 )
39		eqid 2622	. . . . . 6 |- (RLReg ` R ) = (RLReg ` R )
40		eqid 2622	. . . . . 6 |- ( .r ` P ) = ( .r ` P )
41	27	adantr 481	. . . . . 6 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> R e. Ring )
42	29	ad2antrl 764	. . . . . 6 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> x e. B )
43	31	ad2antrl 764	. . . . . 6 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> x =/= .0. )
44		simpl 473	. . . . . . 7 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> R e. Domn )
45		eqid 2622	. . . . . . . 8 |- (coe1 ` x ) = (coe1 ` x )
46	17, 1, 4, 3, 39, 45	deg1ldgdomn 23854	. . . . . . 7 |- ( ( R e. Domn /\ x e. B /\ x =/= .0. ) -> ( (coe1 ` x ) ` ( D ` x ) ) e. (RLReg ` R ) )
47	44, 42, 43, 46	syl3anc 1326	. . . . . 6 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> ( (coe1 ` x ) ` ( D ` x ) ) e. (RLReg ` R ) )
48		eldifi 3732	. . . . . . 7 |- ( y e. ( B \ { .0. } ) -> y e. B )
49	48	ad2antll 765	. . . . . 6 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> y e. B )
50		eldifsni 4320	. . . . . . 7 |- ( y e. ( B \ { .0. } ) -> y =/= .0. )
51	50	ad2antll 765	. . . . . 6 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> y =/= .0. )
52	17, 1, 39, 3, 40, 4, 41, 42, 43, 47, 49, 51	deg1mul2 23874	. . . . 5 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> ( D ` ( x ( .r ` P ) y ) ) = ( ( D ` x ) + ( D ` y ) ) )
53		domnring 19296	. . . . . . . . . 10 |- ( P e. Domn -> P e. Ring )
54	2, 53	syl 17	. . . . . . . . 9 |- ( R e. Domn -> P e. Ring )
55	54	adantr 481	. . . . . . . 8 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> P e. Ring )
56	3, 40	ringcl 18561	. . . . . . . 8 |- ( ( P e. Ring /\ x e. B /\ y e. B ) -> ( x ( .r ` P ) y ) e. B )
57	55, 42, 49, 56	syl3anc 1326	. . . . . . 7 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> ( x ( .r ` P ) y ) e. B )
58	2	adantr 481	. . . . . . . 8 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> P e. Domn )
59	3, 40, 4	domnmuln0 19298	. . . . . . . 8 |- ( ( P e. Domn /\ ( x e. B /\ x =/= .0. ) /\ ( y e. B /\ y =/= .0. ) ) -> ( x ( .r ` P ) y ) =/= .0. )
60	58, 42, 43, 49, 51, 59	syl122anc 1335	. . . . . . 7 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> ( x ( .r ` P ) y ) =/= .0. )
61		eldifsn 4317	. . . . . . 7 |- ( ( x ( .r ` P ) y ) e. ( B \ { .0. } ) <-> ( ( x ( .r ` P ) y ) e. B /\ ( x ( .r ` P ) y ) =/= .0. ) )
62	57, 60, 61	sylanbrc 698	. . . . . 6 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> ( x ( .r ` P ) y ) e. ( B \ { .0. } ) )
63		fvres 6207	. . . . . 6 |- ( ( x ( .r ` P ) y ) e. ( B \ { .0. } ) -> ( ( D |` ( B \ { .0. } ) ) ` ( x ( .r ` P ) y ) ) = ( D ` ( x ( .r ` P ) y ) ) )
64	62, 63	syl 17	. . . . 5 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> ( ( D |` ( B \ { .0. } ) ) ` ( x ( .r ` P ) y ) ) = ( D ` ( x ( .r ` P ) y ) ) )
65		fvres 6207	. . . . . . 7 |- ( y e. ( B \ { .0. } ) -> ( ( D |` ( B \ { .0. } ) ) ` y ) = ( D ` y ) )
66	25, 65	oveqan12d 6669	. . . . . 6 |- ( ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) -> ( ( ( D |` ( B \ { .0. } ) ) ` x ) + ( ( D |` ( B \ { .0. } ) ) ` y ) ) = ( ( D ` x ) + ( D ` y ) ) )
67	66	adantl 482	. . . . 5 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> ( ( ( D |` ( B \ { .0. } ) ) ` x ) + ( ( D |` ( B \ { .0. } ) ) ` y ) ) = ( ( D ` x ) + ( D ` y ) ) )
68	52, 64, 67	3eqtr4d 2666	. . . 4 |- ( ( R e. Domn /\ ( x e. ( B \ { .0. } ) /\ y e. ( B \ { .0. } ) ) ) -> ( ( D |` ( B \ { .0. } ) ) ` ( x ( .r ` P ) y ) ) = ( ( ( D |` ( B \ { .0. } ) ) ` x ) + ( ( D |` ( B \ { .0. } ) ) ` y ) ) )
69	68	ralrimivva 2971	. . 3 |- ( R e. Domn -> A. x e. ( B \ { .0. } ) A. y e. ( B \ { .0. } ) ( ( D |` ( B \ { .0. } ) ) ` ( x ( .r ` P ) y ) ) = ( ( ( D |` ( B \ { .0. } ) ) ` x ) + ( ( D |` ( B \ { .0. } ) ) ` y ) ) )
70		eqid 2622	. . . . . . . 8 |- ( 1r ` P ) = ( 1r ` P )
71	3, 70	ringidcl 18568	. . . . . . 7 |- ( P e. Ring -> ( 1r ` P ) e. B )
72	54, 71	syl 17	. . . . . 6 |- ( R e. Domn -> ( 1r ` P ) e. B )
73		domnnzr 19295	. . . . . . 7 |- ( P e. Domn -> P e. NzRing )
74	70, 4	nzrnz 19260	. . . . . . 7 |- ( P e. NzRing -> ( 1r ` P ) =/= .0. )
75	2, 73, 74	3syl 18	. . . . . 6 |- ( R e. Domn -> ( 1r ` P ) =/= .0. )
76		eldifsn 4317	. . . . . 6 |- ( ( 1r ` P ) e. ( B \ { .0. } ) <-> ( ( 1r ` P ) e. B /\ ( 1r ` P ) =/= .0. ) )
77	72, 75, 76	sylanbrc 698	. . . . 5 |- ( R e. Domn -> ( 1r ` P ) e. ( B \ { .0. } ) )
78		fvres 6207	. . . . 5 |- ( ( 1r ` P ) e. ( B \ { .0. } ) -> ( ( D |` ( B \ { .0. } ) ) ` ( 1r ` P ) ) = ( D ` ( 1r ` P ) ) )
79	77, 78	syl 17	. . . 4 |- ( R e. Domn -> ( ( D |` ( B \ { .0. } ) ) ` ( 1r ` P ) ) = ( D ` ( 1r ` P ) ) )
80	5, 70	ringidval 18503	. . . . . . 7 |- ( 1r ` P ) = ( 0g ` (mulGrp ` P ) )
81	9, 80	subm0 17356	. . . . . 6 |- ( ( B \ { .0. } ) e. (SubMnd ` (mulGrp ` P ) ) -> ( 1r ` P ) = ( 0g ` Y ) )
82	8, 81	syl 17	. . . . 5 |- ( R e. Domn -> ( 1r ` P ) = ( 0g ` Y ) )
83	82	fveq2d 6195	. . . 4 |- ( R e. Domn -> ( ( D |` ( B \ { .0. } ) ) ` ( 1r ` P ) ) = ( ( D |` ( B \ { .0. } ) ) ` ( 0g ` Y ) ) )
84		domnnzr 19295	. . . . 5 |- ( R e. Domn -> R e. NzRing )
85		eqid 2622	. . . . . . 7 |- (Monic1p ` R ) = (Monic1p ` R )
86	1, 70, 85, 17	mon1pid 37783	. . . . . 6 |- ( R e. NzRing -> ( ( 1r ` P ) e. (Monic1p ` R ) /\ ( D ` ( 1r ` P ) ) = 0 ) )
87	86	simprd 479	. . . . 5 |- ( R e. NzRing -> ( D ` ( 1r ` P ) ) = 0 )
88	84, 87	syl 17	. . . 4 |- ( R e. Domn -> ( D ` ( 1r ` P ) ) = 0 )
89	79, 83, 88	3eqtr3d 2664	. . 3 |- ( R e. Domn -> ( ( D |` ( B \ { .0. } ) ) ` ( 0g ` Y ) ) = 0 )
90	38, 69, 89	3jca 1242	. 2 |- ( R e. Domn -> ( ( D |` ( B \ { .0. } ) ) : ( B \ { .0. } ) --> NN0 /\ A. x e. ( B \ { .0. } ) A. y e. ( B \ { .0. } ) ( ( D |` ( B \ { .0. } ) ) ` ( x ( .r ` P ) y ) ) = ( ( ( D |` ( B \ { .0. } ) ) ` x ) + ( ( D |` ( B \ { .0. } ) ) ` y ) ) /\ ( ( D |` ( B \ { .0. } ) ) ` ( 0g ` Y ) ) = 0 ) )
91	5, 3	mgpbas 18495	. . . . 5 |- B = ( Base ` (mulGrp ` P ) )
92	9, 91	ressbas2 15931	. . . 4 |- ( ( B \ { .0. } ) C_ B -> ( B \ { .0. } ) = ( Base ` Y ) )
93	21, 92	ax-mp 5	. . 3 |- ( B \ { .0. } ) = ( Base ` Y )
94		nn0sscn 11297	. . . 4 |- NN0 C_ CC
95		cnfldbas 19750	. . . . 5 |- CC = ( Base ` ℂfld )
96	13, 95	ressbas2 15931	. . . 4 |- ( NN0 C_ CC -> NN0 = ( Base ` N ) )
97	94, 96	ax-mp 5	. . 3 |- NN0 = ( Base ` N )
98		fvex 6201	. . . . . 6 |- ( Base ` P ) e. _V
99	3, 98	eqeltri 2697	. . . . 5 |- B e. _V
100		difexg 4808	. . . . 5 |- ( B e. _V -> ( B \ { .0. } ) e. _V )
101	99, 100	ax-mp 5	. . . 4 |- ( B \ { .0. } ) e. _V
102	5, 40	mgpplusg 18493	. . . . 5 |- ( .r ` P ) = ( +g ` (mulGrp ` P ) )
103	9, 102	ressplusg 15993	. . . 4 |- ( ( B \ { .0. } ) e. _V -> ( .r ` P ) = ( +g ` Y ) )
104	101, 103	ax-mp 5	. . 3 |- ( .r ` P ) = ( +g ` Y )
105		nn0ex 11298	. . . 4 |- NN0 e. _V
106		cnfldadd 19751	. . . . 5 |- + = ( +g ` ℂfld )
107	13, 106	ressplusg 15993	. . . 4 |- ( NN0 e. _V -> + = ( +g ` N ) )
108	105, 107	ax-mp 5	. . 3 |- + = ( +g ` N )
109		eqid 2622	. . 3 |- ( 0g ` Y ) = ( 0g ` Y )
110		cnfld0 19770	. . . . 5 |- 0 = ( 0g ` ℂfld )
111	13, 110	subm0 17356	. . . 4 |- ( NN0 e. (SubMnd ` ℂfld ) -> 0 = ( 0g ` N ) )
112	12, 111	ax-mp 5	. . 3 |- 0 = ( 0g ` N )
113	93, 97, 104, 108, 109, 112	ismhm 17337	. 2 |- ( ( D |` ( B \ { .0. } ) ) e. ( Y MndHom N ) <-> ( ( Y e. Mnd /\ N e. Mnd ) /\ ( ( D |` ( B \ { .0. } ) ) : ( B \ { .0. } ) --> NN0 /\ A. x e. ( B \ { .0. } ) A. y e. ( B \ { .0. } ) ( ( D |` ( B \ { .0. } ) ) ` ( x ( .r ` P ) y ) ) = ( ( ( D |` ( B \ { .0. } ) ) ` x ) + ( ( D |` ( B \ { .0. } ) ) ` y ) ) /\ ( ( D |` ( B \ { .0. } ) ) ` ( 0g ` Y ) ) = 0 ) ) )
114	16, 90, 113	sylanbrc 698	1 |- ( R e. Domn -> ( D |` ( B \ { .0. } ) ) e. ( Y MndHom N ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   + caddc 9939   RR*cxr 10073   NN0cn0 11292   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   Mndcmnd 17294   MndHom cmhm 17333  SubMndcsubmnd 17334  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547  NzRingcnzr 19257  RLRegcrlreg 19279  Domncdomn 19280  Poly₁cpl1 19547  coe₁cco1 19548  ℂ_fldccnfld 19746   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-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-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  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-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-lss 18933  df-nzr 19258  df-rlreg 19283  df-domn 19284  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-cnfld 19747  df-mdeg 23815  df-deg1 23816  df-mon1 23890

This theorem is referenced by: (None)