Theorem frgp0 18173

Description: The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
frgp0.m	|- G = (freeGrp ` I )
frgp0.r	|- .~ = ( ~FG ` I )

Assertion

Ref	Expression
frgp0	|- ( I e. V -> ( G e. Grp /\ [ (/) ] .~ = ( 0g ` G ) ) )

Proof of Theorem frgp0

Dummy variables a b c d x y z n v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgp0.m	. . 3 |- G = (freeGrp ` I )
2		eqid 2622	. . 3 |- (freeMnd ` ( I X. 2o ) ) = (freeMnd ` ( I X. 2o ) )
3		frgp0.r	. . 3 |- .~ = ( ~FG ` I )
4	1, 2, 3	frgpval 18171	. 2 |- ( I e. V -> G = ( (freeMnd ` ( I X. 2o ) ) /.s .~ ) )
5		2on 7568	. . . . 5 |- 2o e. On
6		xpexg 6960	. . . . 5 |- ( ( I e. V /\ 2o e. On ) -> ( I X. 2o ) e. _V )
7	5, 6	mpan2 707	. . . 4 |- ( I e. V -> ( I X. 2o ) e. _V )
8		eqid 2622	. . . . 5 |- ( Base ` (freeMnd ` ( I X. 2o ) ) ) = ( Base ` (freeMnd ` ( I X. 2o ) ) )
9	2, 8	frmdbas 17389	. . . 4 |- ( ( I X. 2o ) e. _V -> ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) )
10	7, 9	syl 17	. . 3 |- ( I e. V -> ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) )
11	10	eqcomd 2628	. 2 |- ( I e. V -> Word ( I X. 2o ) = ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
12		eqidd 2623	. 2 |- ( I e. V -> ( +g ` (freeMnd ` ( I X. 2o ) ) ) = ( +g ` (freeMnd ` ( I X. 2o ) ) ) )
13		eqid 2622	. . . 4 |- ( _I ` Word ( I X. 2o ) ) = ( _I ` Word ( I X. 2o ) )
14	13, 3	efger 18131	. . 3 |- .~ Er ( _I ` Word ( I X. 2o ) )
15		wrdexg 13315	. . . . 5 |- ( ( I X. 2o ) e. _V -> Word ( I X. 2o ) e. _V )
16		fvi 6255	. . . . 5 |- (Word ( I X. 2o ) e. _V -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
17	7, 15, 16	3syl 18	. . . 4 |- ( I e. V -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
18		ereq2 7750	. . . 4 |- ( ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) -> ( .~ Er ( _I ` Word ( I X. 2o ) ) <-> .~ Er Word ( I X. 2o ) ) )
19	17, 18	syl 17	. . 3 |- ( I e. V -> ( .~ Er ( _I ` Word ( I X. 2o ) ) <-> .~ Er Word ( I X. 2o ) ) )
20	14, 19	mpbii 223	. 2 |- ( I e. V -> .~ Er Word ( I X. 2o ) )
21		fvexd 6203	. 2 |- ( I e. V -> (freeMnd ` ( I X. 2o ) ) e. _V )
22		eqid 2622	. . . 4 |- ( +g ` (freeMnd ` ( I X. 2o ) ) ) = ( +g ` (freeMnd ` ( I X. 2o ) ) )
23	1, 2, 3, 22	frgpcpbl 18172	. . 3 |- ( ( a .~ b /\ c .~ d ) -> ( a ( +g ` (freeMnd ` ( I X. 2o ) ) ) c ) .~ ( b ( +g ` (freeMnd ` ( I X. 2o ) ) ) d ) )
24	23	a1i 11	. 2 |- ( I e. V -> ( ( a .~ b /\ c .~ d ) -> ( a ( +g ` (freeMnd ` ( I X. 2o ) ) ) c ) .~ ( b ( +g ` (freeMnd ` ( I X. 2o ) ) ) d ) ) )
25	2	frmdmnd 17396	. . . . . 6 |- ( ( I X. 2o ) e. _V -> (freeMnd ` ( I X. 2o ) ) e. Mnd )
26	7, 25	syl 17	. . . . 5 |- ( I e. V -> (freeMnd ` ( I X. 2o ) ) e. Mnd )
27	26	3ad2ant1 1082	. . . 4 |- ( ( I e. V /\ x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) ) -> (freeMnd ` ( I X. 2o ) ) e. Mnd )
28		simp2 1062	. . . . 5 |- ( ( I e. V /\ x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) ) -> x e. Word ( I X. 2o ) )
29	11	3ad2ant1 1082	. . . . 5 |- ( ( I e. V /\ x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) ) -> Word ( I X. 2o ) = ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
30	28, 29	eleqtrd 2703	. . . 4 |- ( ( I e. V /\ x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) ) -> x e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
31		simp3 1063	. . . . 5 |- ( ( I e. V /\ x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) ) -> y e. Word ( I X. 2o ) )
32	31, 29	eleqtrd 2703	. . . 4 |- ( ( I e. V /\ x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) ) -> y e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
33	8, 22	mndcl 17301	. . . 4 |- ( ( (freeMnd ` ( I X. 2o ) ) e. Mnd /\ x e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) /\ y e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) ) -> ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
34	27, 30, 32, 33	syl3anc 1326	. . 3 |- ( ( I e. V /\ x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) ) -> ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
35	34, 29	eleqtrrd 2704	. 2 |- ( ( I e. V /\ x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) ) -> ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) e. Word ( I X. 2o ) )
36	20	adantr 481	. . . 4 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> .~ Er Word ( I X. 2o ) )
37	26	adantr 481	. . . . . 6 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> (freeMnd ` ( I X. 2o ) ) e. Mnd )
38	34	3adant3r3 1276	. . . . . 6 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
39		simpr3 1069	. . . . . . 7 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> z e. Word ( I X. 2o ) )
40	11	adantr 481	. . . . . . 7 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> Word ( I X. 2o ) = ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
41	39, 40	eleqtrd 2703	. . . . . 6 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> z e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
42	8, 22	mndcl 17301	. . . . . 6 |- ( ( (freeMnd ` ( I X. 2o ) ) e. Mnd /\ ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) /\ z e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) ) -> ( ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) ( +g ` (freeMnd ` ( I X. 2o ) ) ) z ) e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
43	37, 38, 41, 42	syl3anc 1326	. . . . 5 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> ( ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) ( +g ` (freeMnd ` ( I X. 2o ) ) ) z ) e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
44	43, 40	eleqtrrd 2704	. . . 4 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> ( ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) ( +g ` (freeMnd ` ( I X. 2o ) ) ) z ) e. Word ( I X. 2o ) )
45	36, 44	erref 7762	. . 3 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> ( ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) ( +g ` (freeMnd ` ( I X. 2o ) ) ) z ) .~ ( ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) ( +g ` (freeMnd ` ( I X. 2o ) ) ) z ) )
46	30	3adant3r3 1276	. . . 4 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> x e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
47	32	3adant3r3 1276	. . . 4 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> y e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
48	8, 22	mndass 17302	. . . 4 |- ( ( (freeMnd ` ( I X. 2o ) ) e. Mnd /\ ( x e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) /\ y e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) /\ z e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) ) ) -> ( ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) ( +g ` (freeMnd ` ( I X. 2o ) ) ) z ) = ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) ( y ( +g ` (freeMnd ` ( I X. 2o ) ) ) z ) ) )
49	37, 46, 47, 41, 48	syl13anc 1328	. . 3 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> ( ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) ( +g ` (freeMnd ` ( I X. 2o ) ) ) z ) = ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) ( y ( +g ` (freeMnd ` ( I X. 2o ) ) ) z ) ) )
50	45, 49	breqtrd 4679	. 2 |- ( ( I e. V /\ ( x e. Word ( I X. 2o ) /\ y e. Word ( I X. 2o ) /\ z e. Word ( I X. 2o ) ) ) -> ( ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) y ) ( +g ` (freeMnd ` ( I X. 2o ) ) ) z ) .~ ( x ( +g ` (freeMnd ` ( I X. 2o ) ) ) ( y ( +g ` (freeMnd ` ( I X. 2o ) ) ) z ) ) )
51		wrd0 13330	. . 3 |- (/) e. Word ( I X. 2o )
52	51	a1i 11	. 2 |- ( I e. V -> (/) e. Word ( I X. 2o ) )
53	51, 11	syl5eleq 2707	. . . . . 6 |- ( I e. V -> (/) e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
54	53	adantr 481	. . . . 5 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> (/) e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
55	11	eleq2d 2687	. . . . . 6 |- ( I e. V -> ( x e. Word ( I X. 2o ) <-> x e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) ) )
56	55	biimpa 501	. . . . 5 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> x e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
57	2, 8, 22	frmdadd 17392	. . . . 5 |- ( ( (/) e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) /\ x e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) ) -> ( (/) ( +g ` (freeMnd ` ( I X. 2o ) ) ) x ) = ( (/) ++ x ) )
58	54, 56, 57	syl2anc 693	. . . 4 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> ( (/) ( +g ` (freeMnd ` ( I X. 2o ) ) ) x ) = ( (/) ++ x ) )
59		ccatlid 13369	. . . . 5 |- ( x e. Word ( I X. 2o ) -> ( (/) ++ x ) = x )
60	59	adantl 482	. . . 4 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> ( (/) ++ x ) = x )
61	58, 60	eqtrd 2656	. . 3 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> ( (/) ( +g ` (freeMnd ` ( I X. 2o ) ) ) x ) = x )
62	20	adantr 481	. . . 4 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> .~ Er Word ( I X. 2o ) )
63		simpr 477	. . . 4 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> x e. Word ( I X. 2o ) )
64	62, 63	erref 7762	. . 3 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> x .~ x )
65	61, 64	eqbrtrd 4675	. 2 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> ( (/) ( +g ` (freeMnd ` ( I X. 2o ) ) ) x ) .~ x )
66		revcl 13510	. . . 4 |- ( x e. Word ( I X. 2o ) -> (reverse ` x ) e. Word ( I X. 2o ) )
67	66	adantl 482	. . 3 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> (reverse ` x ) e. Word ( I X. 2o ) )
68		eqid 2622	. . . . 5 |- ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
69	68	efgmf 18126	. . . 4 |- ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) : ( I X. 2o ) --> ( I X. 2o )
70	69	a1i 11	. . 3 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) : ( I X. 2o ) --> ( I X. 2o ) )
71		wrdco 13577	. . 3 |- ( ( (reverse ` x ) e. Word ( I X. 2o ) /\ ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) : ( I X. 2o ) --> ( I X. 2o ) ) -> ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) o. (reverse ` x ) ) e. Word ( I X. 2o ) )
72	67, 70, 71	syl2anc 693	. 2 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) o. (reverse ` x ) ) e. Word ( I X. 2o ) )
73	11	adantr 481	. . . . 5 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> Word ( I X. 2o ) = ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
74	72, 73	eleqtrd 2703	. . . 4 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) o. (reverse ` x ) ) e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
75	2, 8, 22	frmdadd 17392	. . . 4 |- ( ( ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) o. (reverse ` x ) ) e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) /\ x e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) ) -> ( ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) o. (reverse ` x ) ) ( +g ` (freeMnd ` ( I X. 2o ) ) ) x ) = ( ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) o. (reverse ` x ) ) ++ x ) )
76	74, 56, 75	syl2anc 693	. . 3 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> ( ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) o. (reverse ` x ) ) ( +g ` (freeMnd ` ( I X. 2o ) ) ) x ) = ( ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) o. (reverse ` x ) ) ++ x ) )
77	17	eleq2d 2687	. . . . 5 |- ( I e. V -> ( x e. ( _I ` Word ( I X. 2o ) ) <-> x e. Word ( I X. 2o ) ) )
78	77	biimpar 502	. . . 4 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> x e. ( _I ` Word ( I X. 2o ) ) )
79		eqid 2622	. . . . 5 |- ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) = ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) )
80	13, 3, 68, 79	efginvrel1 18141	. . . 4 |- ( x e. ( _I ` Word ( I X. 2o ) ) -> ( ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) o. (reverse ` x ) ) ++ x ) .~ (/) )
81	78, 80	syl 17	. . 3 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> ( ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) o. (reverse ` x ) ) ++ x ) .~ (/) )
82	76, 81	eqbrtrd 4675	. 2 |- ( ( I e. V /\ x e. Word ( I X. 2o ) ) -> ( ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) o. (reverse ` x ) ) ( +g ` (freeMnd ` ( I X. 2o ) ) ) x ) .~ (/) )
83	4, 11, 12, 20, 21, 24, 35, 50, 52, 65, 72, 82	qusgrp2 17533	1 |- ( I e. V -> ( G e. Grp /\ [ (/) ] .~ = ( 0g ` G ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   (/)c0 3915   <.cop 4183   <.cotp 4185   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   o. ccom 5118   Oncon0 5723   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   Er wer 7739   [cec 7740   0cc0 9936   ...cfz 12326   #chash 13117  Word cword 13291   ++ cconcat 13293   splice csplice 13296  reversecreverse 13297   <"cs2 13586   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294  freeMndcfrmd 17384   Grpcgrp 17422   ~_FG cefg 18119  freeGrpcfrgp 18120

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  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-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-0g 16102  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-frmd 17386  df-grp 17425  df-efg 18122  df-frgp 18123

This theorem is referenced by:  frgpgrp  18175  frgpinv  18177  frgpmhm  18178