Theorem idomsubgmo 29704

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 5802	. . . . . . . . 9 |- ( Base ` G ) e. _V
2	1	rabex 4544	. . . . . . . 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 2451	. . . . . . . . . . . 12 |- ( Base ` G ) = ( Base ` G )
5	4	subgss 15793	. . . . . . . . . . 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 10740	. . . . . . . . . . . . . . 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 2539	. . . . . . . . . . . . . 14 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) e. NN0 )
12		vex 3074	. . . . . . . . . . . . . . 15 |- y e. _V
13		hashclb 12238	. . . . . . . . . . . . . . 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 2451	. . . . . . . . . . . . 13 |- ( od ` G ) = ( od ` G )
19	18	odsubdvds 16183	. . . . . . . . . . . 12 |- ( ( y e. (SubGrp ` G ) /\ y e. Fin /\ z e. y ) -> ( ( od ` G ) ` z ) || ( # ` y ) )
20	7, 16, 17, 19	syl3anc 1219	. . . . . . . . . . 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 4417	. . . . . . . . . 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 3532	. . . . . . . . 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 15793	. . . . . . . . . . 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 2539	. . . . . . . . . . . . . 14 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` x ) e. NN0 )
30		vex 3074	. . . . . . . . . . . . . . 15 |- x e. _V
31		hashclb 12238	. . . . . . . . . . . . . . 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 16183	. . . . . . . . . . . 12 |- ( ( x e. (SubGrp ` G ) /\ x e. Fin /\ z e. x ) -> ( ( od ` G ) ` z ) || ( # ` x ) )
37	27, 34, 35, 36	syl3anc 1219	. . . . . . . . . . 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 4417	. . . . . . . . . 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 3532	. . . . . . . . 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 3633	. . . . . . . 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 7458	. . . . . . . 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 29702	. . . . . . . . . 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 4419	. . . . . . . 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 12219	. . . . . . . . . 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 1219	. . . . . . . . 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 12253	. . . . . . . . 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 7465	. . . . . . 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 6481	. . . . . . 7 |- ( y u. x ) e. _V
57		ssun1 3620	. . . . . . 7 |- y C_ ( y u. x )
58		ssdomg 7458	. . . . . . 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 7534	. . . . . 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 2495	. . . . . . 7 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) = ( # ` x ) )
63		hashen 12228	. . . . . . . 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 29703	. . . . . 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 1190	. . 3 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) ) -> ( ( ( # ` y ) = N /\ ( # ` x ) = N ) -> y = x ) )
70	69	ralrimivva 2907	. 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 5792	. . . 4 |- ( y = x -> ( # ` y ) = ( # ` x ) )
72	71	eqeq1d 2453	. . 3 |- ( y = x -> ( ( # ` y ) = N <-> ( # ` x ) = N ) )
73	72	rmo4 3252	. 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 1370   e. wcel 1758   A.wral 2795   E*wrmo 2798   {crab 2799   _Vcvv 3071   u. cun 3427   C_ wss 3429   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   ~~ cen 7410   ~<_ cdom 7411   Fincfn 7413   <_ cle 9523   NNcn 10426   NN0cn0 10683   #chash 12213   || cdivides 13646   Basecbs 14285   ↾_s cress 14286  SubGrpcsubg 15786   odcod 16141  mulGrpcmgp 16705  Unitcui 16846  IDomncidom 17467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-disj 4364  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-ofr 6424  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-tpos 6848  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-omul 7028  df-er 7204  df-ec 7206  df-qs 7210  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-sup 7795  df-oi 7828  df-card 8213  df-acn 8216  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-sum 13275  df-dvds 13647  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-0g 14491  df-gsum 14492  df-prds 14497  df-pws 14499  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-minusg 15657  df-sbg 15658  df-mulg 15659  df-subg 15789  df-eqg 15791  df-ghm 15856  df-cntz 15946  df-od 16145  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-srg 16722  df-rng 16762  df-cring 16763  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-rnghom 16921  df-subrg 16978  df-lmod 17065  df-lss 17129  df-lsp 17168  df-nzr 17455  df-rlreg 17469  df-domn 17470  df-idom 17471  df-assa 17499  df-asp 17500  df-ascl 17501  df-psr 17538  df-mvr 17539  df-mpl 17540  df-opsr 17542  df-evls 17704  df-evl 17705  df-psr1 17752  df-vr1 17753  df-ply1 17754  df-coe1 17755  df-evl1 17869  df-cnfld 17937  df-mdeg 21650  df-deg1 21651  df-mon1 21728  df-uc1p 21729  df-q1p 21730  df-r1p 21731

This theorem is referenced by:  proot1mul  29705