Theorem idomsubgmo 35499

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 5858	. . . . . . . . 9 |- ( Base ` G ) e. _V
2	1	rabex 4544	. . . . . . . 8 |- { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } e. _V
3		simp2l 1023	. . . . . . . . . . 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 2402	. . . . . . . . . . . 12 |- ( Base ` G ) = ( Base ` G )
5	4	subgss 16524	. . . . . . . . . . 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 1050	. . . . . . . . . . . 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 1025	. . . . . . . . . . . . . . 15 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) = N )
9		simp1r 1022	. . . . . . . . . . . . . . . 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 10892	. . . . . . . . . . . . . . 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 2490	. . . . . . . . . . . . . 14 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) e. NN0 )
12		vex 3061	. . . . . . . . . . . . . . 15 |- y e. _V
13		hashclb 12475	. . . . . . . . . . . . . . 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 463	. . . . . . . . . . . 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 459	. . . . . . . . . . . 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 2402	. . . . . . . . . . . . 13 |- ( od ` G ) = ( od ` G )
19	18	odsubdvds 16913	. . . . . . . . . . . 12 |- ( ( y e. (SubGrp ` G ) /\ y e. Fin /\ z e. y ) -> ( ( od ` G ) ` z ) || ( # ` y ) )
20	7, 16, 17, 19	syl3anc 1230	. . . . . . . . . . 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 463	. . . . . . . . . . 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 4418	. . . . . . . . . 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 3517	. . . . . . . . 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 1024	. . . . . . . . . . 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 16524	. . . . . . . . . . 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 1051	. . . . . . . . . . . 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 1026	. . . . . . . . . . . . . . 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 2490	. . . . . . . . . . . . . 14 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` x ) e. NN0 )
30		vex 3061	. . . . . . . . . . . . . . 15 |- x e. _V
31		hashclb 12475	. . . . . . . . . . . . . . 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 463	. . . . . . . . . . . 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 459	. . . . . . . . . . . 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 16913	. . . . . . . . . . . 12 |- ( ( x e. (SubGrp ` G ) /\ x e. Fin /\ z e. x ) -> ( ( od ` G ) ` z ) || ( # ` x ) )
37	27, 34, 35, 36	syl3anc 1230	. . . . . . . . . . 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 463	. . . . . . . . . . 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 4418	. . . . . . . . . 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 3517	. . . . . . . . 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 3618	. . . . . . . 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 7598	. . . . . . . 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 62	. . . . . . 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 35497	. . . . . . . . . 10 |- ( ( R e. IDomn /\ N e. NN ) -> ( # ` { z e. ( Base ` G ) | ( ( od ` G ) ` z ) || N } ) <_ N )
46	45	3ad2ant1 1018	. . . . . . . . 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 4420	. . . . . . . 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 12456	. . . . . . . . . 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 1230	. . . . . . . . 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 12493	. . . . . . . . 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 660	. . . . . . . 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 7605	. . . . . . 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 659	. . . . . 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 6579	. . . . . . 7 |- ( y u. x ) e. _V
57		ssun1 3605	. . . . . . 7 |- y C_ ( y u. x )
58		ssdomg 7598	. . . . . . 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 7674	. . . . . 6 |- ( ( ( y u. x ) ~<_ y /\ y ~<_ ( y u. x ) ) -> ( y u. x ) ~~ y )
61	55, 59, 60	sylancl 660	. . . . 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 2446	. . . . . . 7 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) /\ ( ( # ` y ) = N /\ ( # ` x ) = N ) ) -> ( # ` y ) = ( # ` x ) )
63		hashen 12465	. . . . . . . 8 |- ( ( y e. Fin /\ x e. Fin ) -> ( ( # ` y ) = ( # ` x ) <-> y ~~ x ) )
64	15, 33, 63	syl2anc 659	. . . . . . 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 35498	. . . . . 6 |- ( ( y ~~ x /\ y e. Fin ) -> ( ( y u. x ) ~~ y <-> y = x ) )
67	65, 15, 66	syl2anc 659	. . . . 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 1199	. . 3 |- ( ( ( R e. IDomn /\ N e. NN ) /\ ( y e. (SubGrp ` G ) /\ x e. (SubGrp ` G ) ) ) -> ( ( ( # ` y ) = N /\ ( # ` x ) = N ) -> y = x ) )
70	69	ralrimivva 2824	. 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 5848	. . . 4 |- ( y = x -> ( # ` y ) = ( # ` x ) )
72	71	eqeq1d 2404	. . 3 |- ( y = x -> ( ( # ` y ) = N <-> ( # ` x ) = N ) )
73	72	rmo4 3241	. 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 367   /\ w3a 974   = wceq 1405   e. wcel 1842   A.wral 2753   E*wrmo 2756   {crab 2757   _Vcvv 3058   u. cun 3411   C_ wss 3413   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   ~~ cen 7550   ~<_ cdom 7551   Fincfn 7553   <_ cle 9658   NNcn 10575   NN0cn0 10835   #chash 12450   || cdvds 14193   Basecbs 14839   ↾_s cress 14840  SubGrpcsubg 16517   odcod 16871  mulGrpcmgp 17459  Unitcui 17606  IDomncidom 18247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600  ax-mulf 9601

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-disj 4366  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-ofr 6521  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-tpos 6957  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-omul 7171  df-er 7347  df-ec 7349  df-qs 7353  df-map 7458  df-pm 7459  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-sup 7934  df-oi 7968  df-card 8351  df-acn 8354  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-rp 11265  df-fz 11725  df-fzo 11853  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-sum 13656  df-dvds 14194  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-hom 14931  df-cco 14932  df-0g 15054  df-gsum 15055  df-prds 15060  df-pws 15062  df-mre 15198  df-mrc 15199  df-acs 15201  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-mhm 16288  df-submnd 16289  df-grp 16379  df-minusg 16380  df-sbg 16381  df-mulg 16382  df-subg 16520  df-eqg 16522  df-ghm 16587  df-cntz 16677  df-od 16875  df-cmn 17122  df-abl 17123  df-mgp 17460  df-ur 17472  df-srg 17476  df-ring 17518  df-cring 17519  df-oppr 17590  df-dvdsr 17608  df-unit 17609  df-invr 17639  df-rnghom 17682  df-subrg 17745  df-lmod 17832  df-lss 17897  df-lsp 17936  df-nzr 18224  df-rlreg 18249  df-domn 18250  df-idom 18251  df-assa 18279  df-asp 18280  df-ascl 18281  df-psr 18323  df-mvr 18324  df-mpl 18325  df-opsr 18327  df-evls 18489  df-evl 18490  df-psr1 18537  df-vr1 18538  df-ply1 18539  df-coe1 18540  df-evl1 18671  df-cnfld 18739  df-mdeg 22743  df-deg1 22744  df-mon1 22821  df-uc1p 22822  df-q1p 22823  df-r1p 22824

This theorem is referenced by:  proot1mul  35500