Theorem idomsubgmo 29516

Description: The units of an integral domain have at most one subgroup of any single finite cardinality. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Revised by NM, 17-Jun-2017.)

Hypothesis

Ref	Expression
idomsubgmo.g	|- G = ( (mulGrp ` R ) ↾s (Unit ` R ) )

Assertion

Ref	Expression
idomsubgmo	|- ( ( R e. IDomn /\ N e. NN ) -> E* y e. (SubGrp ` G ) ( # ` y ) = N )

Distinct variable groups:   y, G   y, N   y, R

Proof of Theorem idomsubgmo

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

Step	Hyp	Ref	Expression
1		fvex 5696	. . . . . . . . 9 |- ( Base ` G ) e. _V
2	1	rabex 4438	. . . . . . . 8 |- { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } e. _V
3		simp2l 1014	. . . . . . . . . . 11 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> y e. (SubGrp ` G ) )
4		eqid 2438	. . . . . . . . . . . 12 |- ( Base ` G ) = ( Base ` G )
5	4	subgss 15673	. . . . . . . . . . 11 |- ( y e. (SubGrp ` G ) -> y C_ ( Base ` G ) )
6	3, 5	syl 16	. . . . . . . . . 10 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> y C_ ( Base ` G ) )
7		simpl2l 1041	. . . . . . . . . . . 12 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. y ) -> y e. (SubGrp ` G ) )
8		simp3l 1016	. . . . . . . . . . . . . . 15 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) = N )
9		simp1r 1013	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> N e. NN )
10	9	nnnn0d 10628	. . . . . . . . . . . . . . 15 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> N e. NN0 )
11	8, 10	eqeltrd 2512	. . . . . . . . . . . . . 14 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) e. NN0 )
12		vex 2970	. . . . . . . . . . . . . . 15 |- y e. _V
13		hashclb 12120	. . . . . . . . . . . . . . 15 |- ( y e. _V -> ( y e. Fin <-> ( # ` y ) e. NN0 ) )
14	12, 13	ax-mp 5	. . . . . . . . . . . . . 14 |- ( y e. Fin <-> ( # ` y ) e. NN0 )
15	11, 14	sylibr 212	. . . . . . . . . . . . 13 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> y e. Fin )
16	15	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. y ) -> y e. Fin )
17		simpr 461	. . . . . . . . . . . 12 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. y ) -> z e. y )
18		eqid 2438	. . . . . . . . . . . . 13 |- ( od ` G ) = ( od ` G )
19	18	odsubdvds 16061	. . . . . . . . . . . 12 |- ( ( y e. (SubGrp ` G ) /\ y e. Fin /\ z e. y ) -> ( ( od ` G ) ` z ) || ( # ` y ) )
20	7, 16, 17, 19	syl3anc 1218	. . . . . . . . . . 11 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. y ) -> ( ( od ` G ) ` z ) || ( # ` y ) )
21	8	adantr 465	. . . . . . . . . . 11 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. y ) -> ( # ` y ) = N )
22	20, 21	breqtrd 4311	. . . . . . . . . 10 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. y ) -> ( ( od ` G ) ` z ) || N )
23	6, 22	ssrabdv 3426	. . . . . . . . 9 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> y C_ { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } )
24		simp2r 1015	. . . . . . . . . . 11 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> x e. (SubGrp ` G ) )
25	4	subgss 15673	. . . . . . . . . . 11 |- ( x e. (SubGrp ` G ) -> x C_ ( Base ` G ) )
26	24, 25	syl 16	. . . . . . . . . 10 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> x C_ ( Base ` G ) )
27		simpl2r 1042	. . . . . . . . . . . 12 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. x ) -> x e. (SubGrp ` G ) )
28		simp3r 1017	. . . . . . . . . . . . . . 15 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` x ) = N )
29	28, 10	eqeltrd 2512	. . . . . . . . . . . . . 14 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` x ) e. NN0 )
30		vex 2970	. . . . . . . . . . . . . . 15 |- x e. _V
31		hashclb 12120	. . . . . . . . . . . . . . 15 |- ( x e. _V -> ( x e. Fin <-> ( # ` x ) e. NN0 ) )
32	30, 31	ax-mp 5	. . . . . . . . . . . . . 14 |- ( x e. Fin <-> ( # ` x ) e. NN0 )
33	29, 32	sylibr 212	. . . . . . . . . . . . 13 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> x e. Fin )
34	33	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. x ) -> x e. Fin )
35		simpr 461	. . . . . . . . . . . 12 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. x ) -> z e. x )
36	18	odsubdvds 16061	. . . . . . . . . . . 12 |- ( ( x e. (SubGrp ` G ) /\ x e. Fin /\ z e. x ) -> ( ( od ` G ) ` z ) || ( # ` x ) )
37	27, 34, 35, 36	syl3anc 1218	. . . . . . . . . . 11 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. x ) -> ( ( od ` G ) ` z ) || ( # ` x ) )
38	28	adantr 465	. . . . . . . . . . 11 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. x ) -> ( # ` x ) = N )
39	37, 38	breqtrd 4311	. . . . . . . . . 10 |- ( ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) /\ z e. x ) -> ( ( od ` G ) ` z ) || N )
40	26, 39	ssrabdv 3426	. . . . . . . . 9 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> x C_ { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } )
41	23, 40	unssd 3527	. . . . . . . 8 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( y u. x ) C_ { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } )
42		ssdomg 7347	. . . . . . . 8 |- ( { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } e. _V -> ( ( y u. x ) C_ { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } -> ( y u. x ) ~<_ { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ) )
43	2, 41, 42	mpsyl 63	. . . . . . 7 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( y u. x ) ~<_ { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } )
44		idomsubgmo.g	. . . . . . . . . . 11 |- G = ( (mulGrp ` R ) ↾s (Unit ` R ) )
45	44, 4, 18	idomodle 29514	. . . . . . . . . 10 |- ( ( R e. IDomn /\ N e. NN ) -> ( # ` { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ) <_ N )
46	45	3ad2ant1 1009	. . . . . . . . 9 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ) <_ N )
47	46, 8	breqtrrd 4313	. . . . . . . 8 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ) <_ ( # ` y ) )
48	2	a1i 11	. . . . . . . . . 10 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } e. _V )
49		hashbnd 12101	. . . . . . . . . 10 |- ( ( { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } e. _V /\ ( # ` y ) e. NN0 /\ ( # ` { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ) <_ ( # ` y ) ) -> { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } e. Fin )
50	48, 11, 47, 49	syl3anc 1218	. . . . . . . . 9 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } e. Fin )
51		hashdom 12134	. . . . . . . . 9 |- ( ( { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } e. Fin /\ y e. _V ) -> ( ( # ` { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ) <_ ( # ` y ) <-> { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ~<_ y ) )
52	50, 12, 51	sylancl 662	. . . . . . . 8 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( ( # ` { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ) <_ ( # ` y ) <-> { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ~<_ y ) )
53	47, 52	mpbid 210	. . . . . . 7 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ~<_ y )
54		domtr 7354	. . . . . . 7 |- ( ( ( y u. x ) ~<_ { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } /\ { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ~<_ y ) -> ( y u. x ) ~<_ y )
55	43, 53, 54	syl2anc 661	. . . . . 6 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( y u. x ) ~<_ y )
56	12, 30	unex 6373	. . . . . . 7 |- ( y u. x ) e. _V
57		ssun1 3514	. . . . . . 7 |- y C_ ( y u. x )
58		ssdomg 7347	. . . . . . 7 |- ( ( y u. x ) e. _V -> ( y C_ ( y u. x ) -> y ~<_ ( y u. x ) ) )
59	56, 57, 58	mp2 9	. . . . . 6 |- y ~<_ ( y u. x )
60		sbth 7423	. . . . . 6 |- ( ( ( y u. x ) ~<_ y /\ y ~<_ ( y u. x ) ) -> ( y u. x ) ~~ y )
61	55, 59, 60	sylancl 662	. . . . 5 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( y u. x ) ~~ y )
62	8, 28	eqtr4d 2473	. . . . . . 7 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) = ( # ` x ) )
63		hashen 12110	. . . . . . . 8 |- ( ( y e. Fin /\ x e. Fin ) -> ( ( # ` y ) = ( # ` x ) <-> y ~~ x ) )
64	15, 33, 63	syl2anc 661	. . . . . . 7 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( ( # ` y ) = ( # ` x ) <-> y ~~ x ) )
65	62, 64	mpbid 210	. . . . . 6 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> y ~~ x )
66		fiuneneq 29515	. . . . . 6 |- ( ( y ~~ x /\ y e. Fin ) -> ( ( y u. x ) ~~ y <-> y = x ) )
67	65, 15, 66	syl2anc 661	. . . . 5 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( ( y u. x ) ~~ y <-> y = x ) )
68	61, 67	mpbid 210	. . . 4 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> y = x )
69	68	3expia 1189	. . 3 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) ) -> ( ( ( # ` y ) = N /\ ( # ` x ) = N ) -> y = x ) )
70	69	ralrimivva 2803	. 2 |- ( ( R e. IDomn /\ N e. NN ) -> A. y e. (SubGrp ` G ) A. x e. (SubGrp ` G ) ( ( ( # ` y ) = N /\ ( # ` x ) = N ) -> y = x ) )
71		fveq2 5686	. . . 4 |- ( y = x -> ( # ` y ) = ( # ` x ) )
72	71	eqeq1d 2446	. . 3 |- ( y = x -> ( ( # ` y ) = N <-> ( # ` x ) = N ) )
73	72	rmo4 3147	. 2 |- ( E* y e. (SubGrp ` G ) ( # ` y ) = N <-> A. y e. (SubGrp ` G ) A. x e. (SubGrp ` G ) ( ( ( # ` y ) = N /\ ( # ` x ) = N ) -> y = x ) )
74	70, 73	sylibr 212	1 |- ( ( R e. IDomn /\ N e. NN ) -> E* y e. (SubGrp ` G ) ( # ` y ) = N )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   A.wral 2710   E*wrmo 2713   {crab 2714   _Vcvv 2967   u. cun 3321   C_ wss 3323   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   ~~ cen 7299   ~<_ cdom 7300   Fincfn 7302   <_ cle 9411   NNcn 10314   NN0cn0 10571   #chash 12095   || cdivides 13527   Basecbs 14166   ↾_s cress 14167  SubGrpcsubg 15666   odcod 16019  mulGrpcmgp 16579  Unitcui 16719  IDomncidom 17329

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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-ofr 6316  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-ec 7095  df-qs 7099  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-dvds 13528  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-0g 14372  df-gsum 14373  df-prds 14378  df-pws 14380  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-eqg 15671  df-ghm 15736  df-cntz 15826  df-od 16023  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-cring 16636  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-rnghom 16794  df-subrg 16841  df-lmod 16928  df-lss 16991  df-lsp 17030  df-nzr 17317  df-rlreg 17331  df-domn 17332  df-idom 17333  df-assa 17361  df-asp 17362  df-ascl 17363  df-psr 17400  df-mvr 17401  df-mpl 17402  df-opsr 17404  df-evls 17563  df-evl 17564  df-psr1 17611  df-vr1 17612  df-ply1 17613  df-coe1 17614  df-evl1 17726  df-cnfld 17794  df-mdeg 21499  df-deg1 21500  df-mon1 21577  df-uc1p 21578  df-q1p 21579  df-r1p 21580

This theorem is referenced by:  proot1mul  29517