Theorem idomsubgmo 37776

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 6201	. . . . . . . . 9 |- ( Base ` G ) e. _V
2	1	rabex 4813	. . . . . . . 8 |- { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } e. _V
3		simp2l 1087	. . . . . . . . . . 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 2622	. . . . . . . . . . . 12 |- ( Base ` G ) = ( Base ` G )
5	4	subgss 17595	. . . . . . . . . . 11 |- ( y e. (SubGrp ` G ) -> y C_ ( Base ` G ) )
6	3, 5	syl 17	. . . . . . . . . 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 1114	. . . . . . . . . . . 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 1089	. . . . . . . . . . . . . . 15 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) = N )
9		simp1r 1086	. . . . . . . . . . . . . . . 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 11351	. . . . . . . . . . . . . . 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 2701	. . . . . . . . . . . . . 14 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) e. NN0 )
12		vex 3203	. . . . . . . . . . . . . . 15 |- y e. _V
13		hashclb 13149	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 477	. . . . . . . . . . . 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 2622	. . . . . . . . . . . . 13 |- ( od ` G ) = ( od ` G )
19	18	odsubdvds 17986	. . . . . . . . . . . 12 |- ( ( y e. (SubGrp ` G ) /\ y e. Fin /\ z e. y ) -> ( ( od ` G ) ` z ) || ( # ` y ) )
20	7, 16, 17, 19	syl3anc 1326	. . . . . . . . . . 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 481	. . . . . . . . . . 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 4679	. . . . . . . . . 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 3681	. . . . . . . . 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 1088	. . . . . . . . . . 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 17595	. . . . . . . . . . 11 |- ( x e. (SubGrp ` G ) -> x C_ ( Base ` G ) )
26	24, 25	syl 17	. . . . . . . . . 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 1115	. . . . . . . . . . . 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 1090	. . . . . . . . . . . . . . 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 2701	. . . . . . . . . . . . . 14 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` x ) e. NN0 )
30		vex 3203	. . . . . . . . . . . . . . 15 |- x e. _V
31		hashclb 13149	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 477	. . . . . . . . . . . 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 17986	. . . . . . . . . . . 12 |- ( ( x e. (SubGrp ` G ) /\ x e. Fin /\ z e. x ) -> ( ( od ` G ) ` z ) || ( # ` x ) )
37	27, 34, 35, 36	syl3anc 1326	. . . . . . . . . . 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 481	. . . . . . . . . . 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 4679	. . . . . . . . . 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 3681	. . . . . . . . 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 3789	. . . . . . . 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 8001	. . . . . . . 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 68	. . . . . . 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 37774	. . . . . . . . . 10 |- ( ( R e. IDomn /\ N e. NN ) -> ( # ` { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ) <_ N )
46	45	3ad2ant1 1082	. . . . . . . . 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 4681	. . . . . . . 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 13123	. . . . . . . . . 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 1326	. . . . . . . . 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 13168	. . . . . . . . 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 694	. . . . . . . 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 222	. . . . . . 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 8009	. . . . . . 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 693	. . . . . 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 6956	. . . . . . 7 |- ( y u. x ) e. _V
57		ssun1 3776	. . . . . . 7 |- y C_ ( y u. x )
58		ssdomg 8001	. . . . . . 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 8080	. . . . . 6 |- ( ( ( y u. x ) ~<_ y /\ y ~<_ ( y u. x ) ) -> ( y u. x ) ~~ y )
61	55, 59, 60	sylancl 694	. . . . 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 2659	. . . . . . 7 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) = ( # ` x ) )
63		hashen 13135	. . . . . . . 8 |- ( ( y e. Fin /\ x e. Fin ) -> ( ( # ` y ) = ( # ` x ) <-> y ~~ x ) )
64	15, 33, 63	syl2anc 693	. . . . . . 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 222	. . . . . 6 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> y ~~ x )
66		fiuneneq 37775	. . . . . 6 |- ( ( y ~~ x /\ y e. Fin ) -> ( ( y u. x ) ~~ y <-> y = x ) )
67	65, 15, 66	syl2anc 693	. . . . 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 222	. . . 4 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> y = x )
69	68	3expia 1267	. . 3 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) ) -> ( ( ( # ` y ) = N /\ ( # ` x ) = N ) -> y = x ) )
70	69	ralrimivva 2971	. 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 6191	. . . 4 |- ( y = x -> ( # ` y ) = ( # ` x ) )
72	71	eqeq1d 2624	. . 3 |- ( y = x -> ( ( # ` y ) = N <-> ( # ` x ) = N ) )
73	72	rmo4 3399	. 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 224	1 |- ( ( R e. IDomn /\ N e. NN ) -> E* y e. (SubGrp ` G ) ( # ` y ) = N )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E*wrmo 2915   {crab 2916   _Vcvv 3200   u. cun 3572   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955   <_ cle 10075   NNcn 11020   NN0cn0 11292   #chash 13117   || cdvds 14983   Basecbs 15857   ↾_s cress 15858  SubGrpcsubg 17588   odcod 17944  mulGrpcmgp 18489  Unitcui 18639  IDomncidom 19281

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-fal 1489  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-disj 4621  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-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  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-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  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-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  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-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  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-eqg 17593  df-ghm 17658  df-cntz 17750  df-od 17948  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-nzr 19258  df-rlreg 19283  df-domn 19284  df-idom 19285  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-evl 19507  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-evl1 19681  df-cnfld 19747  df-mdeg 23815  df-deg1 23816  df-mon1 23890  df-uc1p 23891  df-q1p 23892  df-r1p 23893

This theorem is referenced by:  proot1mul  37777