Theorem frgpup1 18188

Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

Hypotheses

Ref	Expression
frgpup.b	|- B = ( Base ` H )
frgpup.n	|- N = ( invg ` H )
frgpup.t	|- T = ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) )
frgpup.h	|- ( ph -> H e. Grp )
frgpup.i	|- ( ph -> I e. V )
frgpup.a	|- ( ph -> F : I --> B )
frgpup.w	|- W = ( _I ` Word ( I X. 2o ) )
frgpup.r	|- .~ = ( ~FG ` I )
frgpup.g	|- G = (freeGrp ` I )
frgpup.x	|- X = ( Base ` G )
frgpup.e	|- E = ran ( g e. W |-> <. [ g ] .~ , ( H gsumg ( T o. g ) ) >. )

Assertion

Ref	Expression
frgpup1	|- ( ph -> E e. ( G GrpHom H ) )

Distinct variable groups:   y, g, z   g, H   y, F, z   y, N, z   B, g, y, z   T, g   .~ , g   ph, g, y, z   y, I, z   g, W

Allowed substitution hints:   .~ ( y, z)   T( y, z)   E( y, z, g)   F( g)   G( y, z, g)   H( y, z)   I( g)   N( g)   V( y, z, g)   W( y, z)   X( y, z, g)

Proof of Theorem frgpup1

Dummy variables a u c t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgpup.x	. 2 |- X = ( Base ` G )
2		frgpup.b	. 2 |- B = ( Base ` H )
3		eqid 2622	. 2 |- ( +g ` G ) = ( +g ` G )
4		eqid 2622	. 2 |- ( +g ` H ) = ( +g ` H )
5		frgpup.i	. . 3 |- ( ph -> I e. V )
6		frgpup.g	. . . 4 |- G = (freeGrp ` I )
7	6	frgpgrp 18175	. . 3 |- ( I e. V -> G e. Grp )
8	5, 7	syl 17	. 2 |- ( ph -> G e. Grp )
9		frgpup.h	. 2 |- ( ph -> H e. Grp )
10		frgpup.n	. . 3 |- N = ( invg ` H )
11		frgpup.t	. . 3 |- T = ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) )
12		frgpup.a	. . 3 |- ( ph -> F : I --> B )
13		frgpup.w	. . 3 |- W = ( _I ` Word ( I X. 2o ) )
14		frgpup.r	. . 3 |- .~ = ( ~FG ` I )
15		frgpup.e	. . 3 |- E = ran ( g e. W |-> <. [ g ] .~ , ( H gsumg ( T o. g ) ) >. )
16	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupf 18186	. 2 |- ( ph -> E : X --> B )
17		eqid 2622	. . . . . . . . . . 11 |- (freeMnd ` ( I X. 2o ) ) = (freeMnd ` ( I X. 2o ) )
18	6, 17, 14	frgpval 18171	. . . . . . . . . 10 |- ( I e. V -> G = ( (freeMnd ` ( I X. 2o ) ) /.s .~ ) )
19	5, 18	syl 17	. . . . . . . . 9 |- ( ph -> G = ( (freeMnd ` ( I X. 2o ) ) /.s .~ ) )
20		2on 7568	. . . . . . . . . . . . 13 |- 2o e. On
21		xpexg 6960	. . . . . . . . . . . . 13 |- ( ( I e. V /\ 2o e. On ) -> ( I X. 2o ) e. _V )
22	5, 20, 21	sylancl 694	. . . . . . . . . . . 12 |- ( ph -> ( I X. 2o ) e. _V )
23		wrdexg 13315	. . . . . . . . . . . 12 |- ( ( I X. 2o ) e. _V -> Word ( I X. 2o ) e. _V )
24		fvi 6255	. . . . . . . . . . . 12 |- (Word ( I X. 2o ) e. _V -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
25	22, 23, 24	3syl 18	. . . . . . . . . . 11 |- ( ph -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
26	13, 25	syl5eq 2668	. . . . . . . . . 10 |- ( ph -> W = Word ( I X. 2o ) )
27		eqid 2622	. . . . . . . . . . . 12 |- ( Base ` (freeMnd ` ( I X. 2o ) ) ) = ( Base ` (freeMnd ` ( I X. 2o ) ) )
28	17, 27	frmdbas 17389	. . . . . . . . . . 11 |- ( ( I X. 2o ) e. _V -> ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) )
29	22, 28	syl 17	. . . . . . . . . 10 |- ( ph -> ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) )
30	26, 29	eqtr4d 2659	. . . . . . . . 9 |- ( ph -> W = ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
31		fvex 6201	. . . . . . . . . . 11 |- ( ~FG ` I ) e. _V
32	14, 31	eqeltri 2697	. . . . . . . . . 10 |- .~ e. _V
33	32	a1i 11	. . . . . . . . 9 |- ( ph -> .~ e. _V )
34		fvexd 6203	. . . . . . . . 9 |- ( ph -> (freeMnd ` ( I X. 2o ) ) e. _V )
35	19, 30, 33, 34	qusbas 16205	. . . . . . . 8 |- ( ph -> ( W /. .~ ) = ( Base ` G ) )
36	35, 1	syl6reqr 2675	. . . . . . 7 |- ( ph -> X = ( W /. .~ ) )
37		eqimss 3657	. . . . . . 7 |- ( X = ( W /. .~ ) -> X C_ ( W /. .~ ) )
38	36, 37	syl 17	. . . . . 6 |- ( ph -> X C_ ( W /. .~ ) )
39	38	adantr 481	. . . . 5 |- ( ( ph /\ a e. X ) -> X C_ ( W /. .~ ) )
40	39	sselda 3603	. . . 4 |- ( ( ( ph /\ a e. X ) /\ c e. X ) -> c e. ( W /. .~ ) )
41		eqid 2622	. . . . 5 |- ( W /. .~ ) = ( W /. .~ )
42		oveq2 6658	. . . . . . 7 |- ( [ u ] .~ = c -> ( a ( +g ` G ) [ u ] .~ ) = ( a ( +g ` G ) c ) )
43	42	fveq2d 6195	. . . . . 6 |- ( [ u ] .~ = c -> ( E ` ( a ( +g ` G ) [ u ] .~ ) ) = ( E ` ( a ( +g ` G ) c ) ) )
44		fveq2 6191	. . . . . . 7 |- ( [ u ] .~ = c -> ( E ` [ u ] .~ ) = ( E ` c ) )
45	44	oveq2d 6666	. . . . . 6 |- ( [ u ] .~ = c -> ( ( E ` a ) ( +g ` H ) ( E ` [ u ] .~ ) ) = ( ( E ` a ) ( +g ` H ) ( E ` c ) ) )
46	43, 45	eqeq12d 2637	. . . . 5 |- ( [ u ] .~ = c -> ( ( E ` ( a ( +g ` G ) [ u ] .~ ) ) = ( ( E ` a ) ( +g ` H ) ( E ` [ u ] .~ ) ) <-> ( E ` ( a ( +g ` G ) c ) ) = ( ( E ` a ) ( +g ` H ) ( E ` c ) ) ) )
47	38	sselda 3603	. . . . . . . 8 |- ( ( ph /\ a e. X ) -> a e. ( W /. .~ ) )
48	47	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ u e. W ) /\ a e. X ) -> a e. ( W /. .~ ) )
49		oveq1 6657	. . . . . . . . . 10 |- ( [ t ] .~ = a -> ( [ t ] .~ ( +g ` G ) [ u ] .~ ) = ( a ( +g ` G ) [ u ] .~ ) )
50	49	fveq2d 6195	. . . . . . . . 9 |- ( [ t ] .~ = a -> ( E ` ( [ t ] .~ ( +g ` G ) [ u ] .~ ) ) = ( E ` ( a ( +g ` G ) [ u ] .~ ) ) )
51		fveq2 6191	. . . . . . . . . 10 |- ( [ t ] .~ = a -> ( E ` [ t ] .~ ) = ( E ` a ) )
52	51	oveq1d 6665	. . . . . . . . 9 |- ( [ t ] .~ = a -> ( ( E ` [ t ] .~ ) ( +g ` H ) ( E ` [ u ] .~ ) ) = ( ( E ` a ) ( +g ` H ) ( E ` [ u ] .~ ) ) )
53	50, 52	eqeq12d 2637	. . . . . . . 8 |- ( [ t ] .~ = a -> ( ( E ` ( [ t ] .~ ( +g ` G ) [ u ] .~ ) ) = ( ( E ` [ t ] .~ ) ( +g ` H ) ( E ` [ u ] .~ ) ) <-> ( E ` ( a ( +g ` G ) [ u ] .~ ) ) = ( ( E ` a ) ( +g ` H ) ( E ` [ u ] .~ ) ) ) )
54		fviss 6256	. . . . . . . . . . . . . . . 16 |- ( _I ` Word ( I X. 2o ) ) C_ Word ( I X. 2o )
55	13, 54	eqsstri 3635	. . . . . . . . . . . . . . 15 |- W C_ Word ( I X. 2o )
56	55	sseli 3599	. . . . . . . . . . . . . 14 |- ( t e. W -> t e. Word ( I X. 2o ) )
57	55	sseli 3599	. . . . . . . . . . . . . 14 |- ( u e. W -> u e. Word ( I X. 2o ) )
58		ccatcl 13359	. . . . . . . . . . . . . 14 |- ( ( t e. Word ( I X. 2o ) /\ u e. Word ( I X. 2o ) ) -> ( t ++ u ) e. Word ( I X. 2o ) )
59	56, 57, 58	syl2an 494	. . . . . . . . . . . . 13 |- ( ( t e. W /\ u e. W ) -> ( t ++ u ) e. Word ( I X. 2o ) )
60	13	efgrcl 18128	. . . . . . . . . . . . . . 15 |- ( t e. W -> ( I e. _V /\ W = Word ( I X. 2o ) ) )
61	60	adantr 481	. . . . . . . . . . . . . 14 |- ( ( t e. W /\ u e. W ) -> ( I e. _V /\ W = Word ( I X. 2o ) ) )
62	61	simprd 479	. . . . . . . . . . . . 13 |- ( ( t e. W /\ u e. W ) -> W = Word ( I X. 2o ) )
63	59, 62	eleqtrrd 2704	. . . . . . . . . . . 12 |- ( ( t e. W /\ u e. W ) -> ( t ++ u ) e. W )
64	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 18187	. . . . . . . . . . . 12 |- ( ( ph /\ ( t ++ u ) e. W ) -> ( E ` [ ( t ++ u ) ] .~ ) = ( H gsumg ( T o. ( t ++ u ) ) ) )
65	63, 64	sylan2 491	. . . . . . . . . . 11 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( E ` [ ( t ++ u ) ] .~ ) = ( H gsumg ( T o. ( t ++ u ) ) ) )
66	56	ad2antrl 764	. . . . . . . . . . . . 13 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> t e. Word ( I X. 2o ) )
67	57	ad2antll 765	. . . . . . . . . . . . 13 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> u e. Word ( I X. 2o ) )
68	2, 10, 11, 9, 5, 12	frgpuptf 18183	. . . . . . . . . . . . . 14 |- ( ph -> T : ( I X. 2o ) --> B )
69	68	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> T : ( I X. 2o ) --> B )
70		ccatco 13581	. . . . . . . . . . . . 13 |- ( ( t e. Word ( I X. 2o ) /\ u e. Word ( I X. 2o ) /\ T : ( I X. 2o ) --> B ) -> ( T o. ( t ++ u ) ) = ( ( T o. t ) ++ ( T o. u ) ) )
71	66, 67, 69, 70	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( T o. ( t ++ u ) ) = ( ( T o. t ) ++ ( T o. u ) ) )
72	71	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( H gsumg ( T o. ( t ++ u ) ) ) = ( H gsumg ( ( T o. t ) ++ ( T o. u ) ) ) )
73		grpmnd 17429	. . . . . . . . . . . . . 14 |- ( H e. Grp -> H e. Mnd )
74	9, 73	syl 17	. . . . . . . . . . . . 13 |- ( ph -> H e. Mnd )
75	74	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> H e. Mnd )
76		wrdco 13577	. . . . . . . . . . . . . 14 |- ( ( t e. Word ( I X. 2o ) /\ T : ( I X. 2o ) --> B ) -> ( T o. t ) e. Word B )
77	56, 68, 76	syl2anr 495	. . . . . . . . . . . . 13 |- ( ( ph /\ t e. W ) -> ( T o. t ) e. Word B )
78	77	adantrr 753	. . . . . . . . . . . 12 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( T o. t ) e. Word B )
79		wrdco 13577	. . . . . . . . . . . . 13 |- ( ( u e. Word ( I X. 2o ) /\ T : ( I X. 2o ) --> B ) -> ( T o. u ) e. Word B )
80	67, 69, 79	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( T o. u ) e. Word B )
81	2, 4	gsumccat 17378	. . . . . . . . . . . 12 |- ( ( H e. Mnd /\ ( T o. t ) e. Word B /\ ( T o. u ) e. Word B ) -> ( H gsumg ( ( T o. t ) ++ ( T o. u ) ) ) = ( ( H gsumg ( T o. t ) ) ( +g ` H ) ( H gsumg ( T o. u ) ) ) )
82	75, 78, 80, 81	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( H gsumg ( ( T o. t ) ++ ( T o. u ) ) ) = ( ( H gsumg ( T o. t ) ) ( +g ` H ) ( H gsumg ( T o. u ) ) ) )
83	65, 72, 82	3eqtrd 2660	. . . . . . . . . 10 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( E ` [ ( t ++ u ) ] .~ ) = ( ( H gsumg ( T o. t ) ) ( +g ` H ) ( H gsumg ( T o. u ) ) ) )
84	13, 6, 14, 3	frgpadd 18176	. . . . . . . . . . . 12 |- ( ( t e. W /\ u e. W ) -> ( [ t ] .~ ( +g ` G ) [ u ] .~ ) = [ ( t ++ u ) ] .~ )
85	84	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( [ t ] .~ ( +g ` G ) [ u ] .~ ) = [ ( t ++ u ) ] .~ )
86	85	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( E ` ( [ t ] .~ ( +g ` G ) [ u ] .~ ) ) = ( E ` [ ( t ++ u ) ] .~ ) )
87	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 18187	. . . . . . . . . . . 12 |- ( ( ph /\ t e. W ) -> ( E ` [ t ] .~ ) = ( H gsumg ( T o. t ) ) )
88	87	adantrr 753	. . . . . . . . . . 11 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( E ` [ t ] .~ ) = ( H gsumg ( T o. t ) ) )
89	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 18187	. . . . . . . . . . . 12 |- ( ( ph /\ u e. W ) -> ( E ` [ u ] .~ ) = ( H gsumg ( T o. u ) ) )
90	89	adantrl 752	. . . . . . . . . . 11 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( E ` [ u ] .~ ) = ( H gsumg ( T o. u ) ) )
91	88, 90	oveq12d 6668	. . . . . . . . . 10 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( ( E ` [ t ] .~ ) ( +g ` H ) ( E ` [ u ] .~ ) ) = ( ( H gsumg ( T o. t ) ) ( +g ` H ) ( H gsumg ( T o. u ) ) ) )
92	83, 86, 91	3eqtr4d 2666	. . . . . . . . 9 |- ( ( ph /\ ( t e. W /\ u e. W ) ) -> ( E ` ( [ t ] .~ ( +g ` G ) [ u ] .~ ) ) = ( ( E ` [ t ] .~ ) ( +g ` H ) ( E ` [ u ] .~ ) ) )
93	92	anass1rs 849	. . . . . . . 8 |- ( ( ( ph /\ u e. W ) /\ t e. W ) -> ( E ` ( [ t ] .~ ( +g ` G ) [ u ] .~ ) ) = ( ( E ` [ t ] .~ ) ( +g ` H ) ( E ` [ u ] .~ ) ) )
94	41, 53, 93	ectocld 7814	. . . . . . 7 |- ( ( ( ph /\ u e. W ) /\ a e. ( W /. .~ ) ) -> ( E ` ( a ( +g ` G ) [ u ] .~ ) ) = ( ( E ` a ) ( +g ` H ) ( E ` [ u ] .~ ) ) )
95	48, 94	syldan 487	. . . . . 6 |- ( ( ( ph /\ u e. W ) /\ a e. X ) -> ( E ` ( a ( +g ` G ) [ u ] .~ ) ) = ( ( E ` a ) ( +g ` H ) ( E ` [ u ] .~ ) ) )
96	95	an32s 846	. . . . 5 |- ( ( ( ph /\ a e. X ) /\ u e. W ) -> ( E ` ( a ( +g ` G ) [ u ] .~ ) ) = ( ( E ` a ) ( +g ` H ) ( E ` [ u ] .~ ) ) )
97	41, 46, 96	ectocld 7814	. . . 4 |- ( ( ( ph /\ a e. X ) /\ c e. ( W /. .~ ) ) -> ( E ` ( a ( +g ` G ) c ) ) = ( ( E ` a ) ( +g ` H ) ( E ` c ) ) )
98	40, 97	syldan 487	. . 3 |- ( ( ( ph /\ a e. X ) /\ c e. X ) -> ( E ` ( a ( +g ` G ) c ) ) = ( ( E ` a ) ( +g ` H ) ( E ` c ) ) )
99	98	anasss 679	. 2 |- ( ( ph /\ ( a e. X /\ c e. X ) ) -> ( E ` ( a ( +g ` G ) c ) ) = ( ( E ` a ) ( +g ` H ) ( E ` c ) ) )
100	1, 2, 3, 4, 8, 9, 16, 99	isghmd 17669	1 |- ( ph -> E e. ( G GrpHom H ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ifcif 4086   <.cop 4183   |-> cmpt 4729   _I cid 5023   X. cxp 5112   ran crn 5115   o. ccom 5118   Oncon0 5723   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2oc2o 7554   [cec 7740   /.cqs 7741  Word cword 13291   ++ cconcat 13293   Basecbs 15857   +g cplusg 15941   gsum_g cgsu 16101   /._s cqus 16165   Mndcmnd 17294  freeMndcfrmd 17384   Grpcgrp 17422   invgcminusg 17423   GrpHom cghm 17657   ~_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-seq 12802  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-sets 15864  df-ress 15865  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-gsum 16103  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-frmd 17386  df-grp 17425  df-minusg 17426  df-ghm 17658  df-efg 18122  df-frgp 18123

This theorem is referenced by:  frgpup3lem  18190  frgpup3  18191