Theorem idomsubgmo 30760

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 5874	. . . . . . . . 9 |- ( Base ` G ) e. _V
2	1	rabex 4598	. . . . . . . 8 |- { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } e. _V
3		simp2l 1022	. . . . . . . . . . 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 2467	. . . . . . . . . . . 12 |- ( Base ` G ) = ( Base ` G )
5	4	subgss 15997	. . . . . . . . . . 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 1049	. . . . . . . . . . . 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 1024	. . . . . . . . . . . . . . 15 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) = N )
9		simp1r 1021	. . . . . . . . . . . . . . . 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 10848	. . . . . . . . . . . . . . 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 2555	. . . . . . . . . . . . . 14 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) e. NN0 )
12		vex 3116	. . . . . . . . . . . . . . 15 |- y e. _V
13		hashclb 12394	. . . . . . . . . . . . . . 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 2467	. . . . . . . . . . . . 13 |- ( od ` G ) = ( od ` G )
19	18	odsubdvds 16387	. . . . . . . . . . . 12 |- ( ( y e. (SubGrp ` G ) /\ y e. Fin /\ z e. y ) -> ( ( od ` G ) ` z ) || ( # ` y ) )
20	7, 16, 17, 19	syl3anc 1228	. . . . . . . . . . 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 4471	. . . . . . . . . 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 3579	. . . . . . . . 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 1023	. . . . . . . . . . 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 15997	. . . . . . . . . . 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 1050	. . . . . . . . . . . 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 1025	. . . . . . . . . . . . . . 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 2555	. . . . . . . . . . . . . 14 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` x ) e. NN0 )
30		vex 3116	. . . . . . . . . . . . . . 15 |- x e. _V
31		hashclb 12394	. . . . . . . . . . . . . . 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 16387	. . . . . . . . . . . 12 |- ( ( x e. (SubGrp ` G ) /\ x e. Fin /\ z e. x ) -> ( ( od ` G ) ` z ) || ( # ` x ) )
37	27, 34, 35, 36	syl3anc 1228	. . . . . . . . . . 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 4471	. . . . . . . . . 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 3579	. . . . . . . . 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 3680	. . . . . . . 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 7558	. . . . . . . 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 30758	. . . . . . . . . 10 |- ( ( R e. IDomn /\ N e. NN ) -> ( # ` { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ) <_ N )
46	45	3ad2ant1 1017	. . . . . . . . 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 4473	. . . . . . . 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 12375	. . . . . . . . . 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 1228	. . . . . . . . 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 12411	. . . . . . . . 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 7565	. . . . . . 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 6580	. . . . . . 7 |- ( y u. x ) e. _V
57		ssun1 3667	. . . . . . 7 |- y C_ ( y u. x )
58		ssdomg 7558	. . . . . . 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 7634	. . . . . 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 2511	. . . . . . 7 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) = ( # ` x ) )
63		hashen 12384	. . . . . . . 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 30759	. . . . . 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 1198	. . 3 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) ) -> ( ( ( # ` y ) = N /\ ( # ` x ) = N ) -> y = x ) )
70	69	ralrimivva 2885	. 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 5864	. . . 4 |- ( y = x -> ( # ` y ) = ( # ` x ) )
72	71	eqeq1d 2469	. . 3 |- ( y = x -> ( ( # ` y ) = N <-> ( # ` x ) = N ) )
73	72	rmo4 3296	. 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 973   = wceq 1379   e. wcel 1767   A.wral 2814   E*wrmo 2817   {crab 2818   _Vcvv 3113   u. cun 3474   C_ wss 3476   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   ~~ cen 7510   ~<_ cdom 7511   Fincfn 7513   <_ cle 9625   NNcn 10532   NN0cn0 10791   #chash 12369   || cdivides 13843   Basecbs 14486   ↾_s cress 14487  SubGrpcsubg 15990   odcod 16345  mulGrpcmgp 16931  Unitcui 17072  IDomncidom 17700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-ofr 6523  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-ec 7310  df-qs 7314  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-dvds 13844  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-0g 14693  df-gsum 14694  df-prds 14699  df-pws 14701  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-eqg 15995  df-ghm 16060  df-cntz 16150  df-od 16349  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-srg 16948  df-rng 16988  df-cring 16989  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-rnghom 17148  df-subrg 17210  df-lmod 17297  df-lss 17362  df-lsp 17401  df-nzr 17688  df-rlreg 17702  df-domn 17703  df-idom 17704  df-assa 17732  df-asp 17733  df-ascl 17734  df-psr 17776  df-mvr 17777  df-mpl 17778  df-opsr 17780  df-evls 17942  df-evl 17943  df-psr1 17990  df-vr1 17991  df-ply1 17992  df-coe1 17993  df-evl1 18124  df-cnfld 18192  df-mdeg 22188  df-deg1 22189  df-mon1 22266  df-uc1p 22267  df-q1p 22268  df-r1p 22269

This theorem is referenced by:  proot1mul  30761