Theorem psgnghm 19926

Description: The sign is a homomorphism from the finitary permutation group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)

Hypotheses

Ref	Expression
psgnghm.s	|- S = ( SymGrp ` D )
psgnghm.n	|- N = (pmSgn ` D )
psgnghm.f	|- F = ( S ↾s dom N )
psgnghm.u	|- U = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )

Assertion

Ref	Expression
psgnghm	|- ( D e. V -> N e. ( F GrpHom U ) )

Proof of Theorem psgnghm

Dummy variables x w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgnghm.s	. . . . . 6 |- S = ( SymGrp ` D )
2		eqid 2622	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
3		eqid 2622	. . . . . 6 |- { x e. ( Base ` S ) | dom ( x \ _I ) e. Fin } = { x e. ( Base ` S ) | dom ( x \ _I ) e. Fin }
4		psgnghm.n	. . . . . 6 |- N = (pmSgn ` D )
5	1, 2, 3, 4	psgnfn 17921	. . . . 5 |- N Fn { x e. ( Base ` S ) | dom ( x \ _I ) e. Fin }
6		fndm 5990	. . . . 5 |- ( N Fn { x e. ( Base ` S ) | dom ( x \ _I ) e. Fin } -> dom N = { x e. ( Base ` S ) | dom ( x \ _I ) e. Fin } )
7	5, 6	ax-mp 5	. . . 4 |- dom N = { x e. ( Base ` S ) | dom ( x \ _I ) e. Fin }
8		ssrab2 3687	. . . 4 |- { x e. ( Base ` S ) | dom ( x \ _I ) e. Fin } C_ ( Base ` S )
9	7, 8	eqsstri 3635	. . 3 |- dom N C_ ( Base ` S )
10		psgnghm.f	. . . 4 |- F = ( S ↾s dom N )
11	10, 2	ressbas2 15931	. . 3 |- ( dom N C_ ( Base ` S ) -> dom N = ( Base ` F ) )
12	9, 11	ax-mp 5	. 2 |- dom N = ( Base ` F )
13		psgnghm.u	. . 3 |- U = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
14	13	cnmsgnbas 19924	. 2 |- { 1 , -u 1 } = ( Base ` U )
15		fvex 6201	. . . 4 |- ( Base ` F ) e. _V
16	12, 15	eqeltri 2697	. . 3 |- dom N e. _V
17		eqid 2622	. . . 4 |- ( +g ` S ) = ( +g ` S )
18	10, 17	ressplusg 15993	. . 3 |- ( dom N e. _V -> ( +g ` S ) = ( +g ` F ) )
19	16, 18	ax-mp 5	. 2 |- ( +g ` S ) = ( +g ` F )
20		prex 4909	. . 3 |- { 1 , -u 1 } e. _V
21		eqid 2622	. . . . 5 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
22		cnfldmul 19752	. . . . 5 |- x. = ( .r ` ℂfld )
23	21, 22	mgpplusg 18493	. . . 4 |- x. = ( +g ` (mulGrp ` ℂfld ) )
24	13, 23	ressplusg 15993	. . 3 |- ( { 1 , -u 1 } e. _V -> x. = ( +g ` U ) )
25	20, 24	ax-mp 5	. 2 |- x. = ( +g ` U )
26	1, 4	psgndmsubg 17922	. . 3 |- ( D e. V -> dom N e. (SubGrp ` S ) )
27	10	subggrp 17597	. . 3 |- ( dom N e. (SubGrp ` S ) -> F e. Grp )
28	26, 27	syl 17	. 2 |- ( D e. V -> F e. Grp )
29	13	cnmsgngrp 19925	. . 3 |- U e. Grp
30	29	a1i 11	. 2 |- ( D e. V -> U e. Grp )
31		fnfun 5988	. . . . . 6 |- ( N Fn { x e. ( Base ` S ) | dom ( x \ _I ) e. Fin } -> Fun N )
32	5, 31	ax-mp 5	. . . . 5 |- Fun N
33		funfn 5918	. . . . 5 |- ( Fun N <-> N Fn dom N )
34	32, 33	mpbi 220	. . . 4 |- N Fn dom N
35	34	a1i 11	. . 3 |- ( D e. V -> N Fn dom N )
36		eqid 2622	. . . . . 6 |- ran (pmTrsp ` D ) = ran (pmTrsp ` D )
37	1, 36, 4	psgnvali 17928	. . . . 5 |- ( x e. dom N -> E. z e. Word ran (pmTrsp ` D ) ( x = ( S gsumg z ) /\ ( N ` x ) = ( -u 1 ^ ( # ` z ) ) ) )
38		lencl 13324	. . . . . . . . . . 11 |- ( z e. Word ran (pmTrsp ` D ) -> ( # ` z ) e. NN0 )
39	38	nn0zd 11480	. . . . . . . . . 10 |- ( z e. Word ran (pmTrsp ` D ) -> ( # ` z ) e. ZZ )
40		m1expcl2 12882	. . . . . . . . . . 11 |- ( ( # ` z ) e. ZZ -> ( -u 1 ^ ( # ` z ) ) e. { -u 1 , 1 } )
41		prcom 4267	. . . . . . . . . . 11 |- { -u 1 , 1 } = { 1 , -u 1 }
42	40, 41	syl6eleq 2711	. . . . . . . . . 10 |- ( ( # ` z ) e. ZZ -> ( -u 1 ^ ( # ` z ) ) e. { 1 , -u 1 } )
43	39, 42	syl 17	. . . . . . . . 9 |- ( z e. Word ran (pmTrsp ` D ) -> ( -u 1 ^ ( # ` z ) ) e. { 1 , -u 1 } )
44	43	adantl 482	. . . . . . . 8 |- ( ( D e. V /\ z e. Word ran (pmTrsp ` D ) ) -> ( -u 1 ^ ( # ` z ) ) e. { 1 , -u 1 } )
45		eleq1a 2696	. . . . . . . 8 |- ( ( -u 1 ^ ( # ` z ) ) e. { 1 , -u 1 } -> ( ( N ` x ) = ( -u 1 ^ ( # ` z ) ) -> ( N ` x ) e. { 1 , -u 1 } ) )
46	44, 45	syl 17	. . . . . . 7 |- ( ( D e. V /\ z e. Word ran (pmTrsp ` D ) ) -> ( ( N ` x ) = ( -u 1 ^ ( # ` z ) ) -> ( N ` x ) e. { 1 , -u 1 } ) )
47	46	adantld 483	. . . . . 6 |- ( ( D e. V /\ z e. Word ran (pmTrsp ` D ) ) -> ( ( x = ( S gsumg z ) /\ ( N ` x ) = ( -u 1 ^ ( # ` z ) ) ) -> ( N ` x ) e. { 1 , -u 1 } ) )
48	47	rexlimdva 3031	. . . . 5 |- ( D e. V -> ( E. z e. Word ran (pmTrsp ` D ) ( x = ( S gsumg z ) /\ ( N ` x ) = ( -u 1 ^ ( # ` z ) ) ) -> ( N ` x ) e. { 1 , -u 1 } ) )
49	37, 48	syl5 34	. . . 4 |- ( D e. V -> ( x e. dom N -> ( N ` x ) e. { 1 , -u 1 } ) )
50	49	ralrimiv 2965	. . 3 |- ( D e. V -> A. x e. dom N ( N ` x ) e. { 1 , -u 1 } )
51		ffnfv 6388	. . 3 |- ( N : dom N --> { 1 , -u 1 } <-> ( N Fn dom N /\ A. x e. dom N ( N ` x ) e. { 1 , -u 1 } ) )
52	35, 50, 51	sylanbrc 698	. 2 |- ( D e. V -> N : dom N --> { 1 , -u 1 } )
53	1, 36, 4	psgnvali 17928	. . . . . 6 |- ( y e. dom N -> E. w e. Word ran (pmTrsp ` D ) ( y = ( S gsumg w ) /\ ( N ` y ) = ( -u 1 ^ ( # ` w ) ) ) )
54	37, 53	anim12i 590	. . . . 5 |- ( ( x e. dom N /\ y e. dom N ) -> ( E. z e. Word ran (pmTrsp ` D ) ( x = ( S gsumg z ) /\ ( N ` x ) = ( -u 1 ^ ( # ` z ) ) ) /\ E. w e. Word ran (pmTrsp ` D ) ( y = ( S gsumg w ) /\ ( N ` y ) = ( -u 1 ^ ( # ` w ) ) ) ) )
55		reeanv 3107	. . . . 5 |- ( E. z e. Word ran (pmTrsp ` D ) E. w e. Word ran (pmTrsp ` D ) ( ( x = ( S gsumg z ) /\ ( N ` x ) = ( -u 1 ^ ( # ` z ) ) ) /\ ( y = ( S gsumg w ) /\ ( N ` y ) = ( -u 1 ^ ( # ` w ) ) ) ) <-> ( E. z e. Word ran (pmTrsp ` D ) ( x = ( S gsumg z ) /\ ( N ` x ) = ( -u 1 ^ ( # ` z ) ) ) /\ E. w e. Word ran (pmTrsp ` D ) ( y = ( S gsumg w ) /\ ( N ` y ) = ( -u 1 ^ ( # ` w ) ) ) ) )
56	54, 55	sylibr 224	. . . 4 |- ( ( x e. dom N /\ y e. dom N ) -> E. z e. Word ran (pmTrsp ` D ) E. w e. Word ran (pmTrsp ` D ) ( ( x = ( S gsumg z ) /\ ( N ` x ) = ( -u 1 ^ ( # ` z ) ) ) /\ ( y = ( S gsumg w ) /\ ( N ` y ) = ( -u 1 ^ ( # ` w ) ) ) ) )
57		ccatcl 13359	. . . . . . . 8 |- ( ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) -> ( z ++ w ) e. Word ran (pmTrsp ` D ) )
58	1, 36, 4	psgnvalii 17929	. . . . . . . 8 |- ( ( D e. V /\ ( z ++ w ) e. Word ran (pmTrsp ` D ) ) -> ( N ` ( S gsumg ( z ++ w ) ) ) = ( -u 1 ^ ( # ` ( z ++ w ) ) ) )
59	57, 58	sylan2 491	. . . . . . 7 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> ( N ` ( S gsumg ( z ++ w ) ) ) = ( -u 1 ^ ( # ` ( z ++ w ) ) ) )
60	1	symggrp 17820	. . . . . . . . . . 11 |- ( D e. V -> S e. Grp )
61		grpmnd 17429	. . . . . . . . . . 11 |- ( S e. Grp -> S e. Mnd )
62	60, 61	syl 17	. . . . . . . . . 10 |- ( D e. V -> S e. Mnd )
63	36, 1, 2	symgtrf 17889	. . . . . . . . . . . 12 |- ran (pmTrsp ` D ) C_ ( Base ` S )
64		sswrd 13313	. . . . . . . . . . . 12 |- ( ran (pmTrsp ` D ) C_ ( Base ` S ) -> Word ran (pmTrsp ` D ) C_ Word ( Base ` S ) )
65	63, 64	ax-mp 5	. . . . . . . . . . 11 |- Word ran (pmTrsp ` D ) C_ Word ( Base ` S )
66	65	sseli 3599	. . . . . . . . . 10 |- ( z e. Word ran (pmTrsp ` D ) -> z e. Word ( Base ` S ) )
67	65	sseli 3599	. . . . . . . . . 10 |- ( w e. Word ran (pmTrsp ` D ) -> w e. Word ( Base ` S ) )
68	2, 17	gsumccat 17378	. . . . . . . . . 10 |- ( ( S e. Mnd /\ z e. Word ( Base ` S ) /\ w e. Word ( Base ` S ) ) -> ( S gsumg ( z ++ w ) ) = ( ( S gsumg z ) ( +g ` S ) ( S gsumg w ) ) )
69	62, 66, 67, 68	syl3an 1368	. . . . . . . . 9 |- ( ( D e. V /\ z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) -> ( S gsumg ( z ++ w ) ) = ( ( S gsumg z ) ( +g ` S ) ( S gsumg w ) ) )
70	69	3expb 1266	. . . . . . . 8 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> ( S gsumg ( z ++ w ) ) = ( ( S gsumg z ) ( +g ` S ) ( S gsumg w ) ) )
71	70	fveq2d 6195	. . . . . . 7 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> ( N ` ( S gsumg ( z ++ w ) ) ) = ( N ` ( ( S gsumg z ) ( +g ` S ) ( S gsumg w ) ) ) )
72		ccatlen 13360	. . . . . . . . . 10 |- ( ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) -> ( # ` ( z ++ w ) ) = ( ( # ` z ) + ( # ` w ) ) )
73	72	adantl 482	. . . . . . . . 9 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> ( # ` ( z ++ w ) ) = ( ( # ` z ) + ( # ` w ) ) )
74	73	oveq2d 6666	. . . . . . . 8 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> ( -u 1 ^ ( # ` ( z ++ w ) ) ) = ( -u 1 ^ ( ( # ` z ) + ( # ` w ) ) ) )
75		neg1cn 11124	. . . . . . . . . 10 |- -u 1 e. CC
76	75	a1i 11	. . . . . . . . 9 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> -u 1 e. CC )
77		lencl 13324	. . . . . . . . . 10 |- ( w e. Word ran (pmTrsp ` D ) -> ( # ` w ) e. NN0 )
78	77	ad2antll 765	. . . . . . . . 9 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> ( # ` w ) e. NN0 )
79	38	ad2antrl 764	. . . . . . . . 9 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> ( # ` z ) e. NN0 )
80	76, 78, 79	expaddd 13010	. . . . . . . 8 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> ( -u 1 ^ ( ( # ` z ) + ( # ` w ) ) ) = ( ( -u 1 ^ ( # ` z ) ) x. ( -u 1 ^ ( # ` w ) ) ) )
81	74, 80	eqtrd 2656	. . . . . . 7 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> ( -u 1 ^ ( # ` ( z ++ w ) ) ) = ( ( -u 1 ^ ( # ` z ) ) x. ( -u 1 ^ ( # ` w ) ) ) )
82	59, 71, 81	3eqtr3d 2664	. . . . . 6 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> ( N ` ( ( S gsumg z ) ( +g ` S ) ( S gsumg w ) ) ) = ( ( -u 1 ^ ( # ` z ) ) x. ( -u 1 ^ ( # ` w ) ) ) )
83		oveq12 6659	. . . . . . . . 9 |- ( ( x = ( S gsumg z ) /\ y = ( S gsumg w ) ) -> ( x ( +g ` S ) y ) = ( ( S gsumg z ) ( +g ` S ) ( S gsumg w ) ) )
84	83	fveq2d 6195	. . . . . . . 8 |- ( ( x = ( S gsumg z ) /\ y = ( S gsumg w ) ) -> ( N ` ( x ( +g ` S ) y ) ) = ( N ` ( ( S gsumg z ) ( +g ` S ) ( S gsumg w ) ) ) )
85		oveq12 6659	. . . . . . . 8 |- ( ( ( N ` x ) = ( -u 1 ^ ( # ` z ) ) /\ ( N ` y ) = ( -u 1 ^ ( # ` w ) ) ) -> ( ( N ` x ) x. ( N ` y ) ) = ( ( -u 1 ^ ( # ` z ) ) x. ( -u 1 ^ ( # ` w ) ) ) )
86	84, 85	eqeqan12d 2638	. . . . . . 7 |- ( ( ( x = ( S gsumg z ) /\ y = ( S gsumg w ) ) /\ ( ( N ` x ) = ( -u 1 ^ ( # ` z ) ) /\ ( N ` y ) = ( -u 1 ^ ( # ` w ) ) ) ) -> ( ( N ` ( x ( +g ` S ) y ) ) = ( ( N ` x ) x. ( N ` y ) ) <-> ( N ` ( ( S gsumg z ) ( +g ` S ) ( S gsumg w ) ) ) = ( ( -u 1 ^ ( # ` z ) ) x. ( -u 1 ^ ( # ` w ) ) ) ) )
87	86	an4s 869	. . . . . 6 |- ( ( ( x = ( S gsumg z ) /\ ( N ` x ) = ( -u 1 ^ ( # ` z ) ) ) /\ ( y = ( S gsumg w ) /\ ( N ` y ) = ( -u 1 ^ ( # ` w ) ) ) ) -> ( ( N ` ( x ( +g ` S ) y ) ) = ( ( N ` x ) x. ( N ` y ) ) <-> ( N ` ( ( S gsumg z ) ( +g ` S ) ( S gsumg w ) ) ) = ( ( -u 1 ^ ( # ` z ) ) x. ( -u 1 ^ ( # ` w ) ) ) ) )
88	82, 87	syl5ibrcom 237	. . . . 5 |- ( ( D e. V /\ ( z e. Word ran (pmTrsp ` D ) /\ w e. Word ran (pmTrsp ` D ) ) ) -> ( ( ( x = ( S gsumg z ) /\ ( N ` x ) = ( -u 1 ^ ( # ` z ) ) ) /\ ( y = ( S gsumg w ) /\ ( N ` y ) = ( -u 1 ^ ( # ` w ) ) ) ) -> ( N ` ( x ( +g ` S ) y ) ) = ( ( N ` x ) x. ( N ` y ) ) ) )
89	88	rexlimdvva 3038	. . . 4 |- ( D e. V -> ( E. z e. Word ran (pmTrsp ` D ) E. w e. Word ran (pmTrsp ` D ) ( ( x = ( S gsumg z ) /\ ( N ` x ) = ( -u 1 ^ ( # ` z ) ) ) /\ ( y = ( S gsumg w ) /\ ( N ` y ) = ( -u 1 ^ ( # ` w ) ) ) ) -> ( N ` ( x ( +g ` S ) y ) ) = ( ( N ` x ) x. ( N ` y ) ) ) )
90	56, 89	syl5 34	. . 3 |- ( D e. V -> ( ( x e. dom N /\ y e. dom N ) -> ( N ` ( x ( +g ` S ) y ) ) = ( ( N ` x ) x. ( N ` y ) ) ) )
91	90	imp 445	. 2 |- ( ( D e. V /\ ( x e. dom N /\ y e. dom N ) ) -> ( N ` ( x ( +g ` S ) y ) ) = ( ( N ` x ) x. ( N ` y ) ) )
92	12, 14, 19, 25, 28, 30, 52, 91	isghmd 17669	1 |- ( D e. V -> N e. ( F GrpHom U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {cpr 4179   _I cid 5023   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   -ucneg 10267   NN0cn0 11292   ZZcz 11377   ^cexp 12860   #chash 13117  Word cword 13291   ++ cconcat 13293   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   gsum_g cgsu 16101   Mndcmnd 17294   Grpcgrp 17422  SubGrpcsubg 17588   GrpHom cghm 17657   SymGrpcsymg 17797  pmTrspcpmtr 17861  pmSgncpsgn 17909  mulGrpcmgp 18489  ℂ_fldccnfld 19746

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  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-div 10685  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-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  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-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-ghm 17658  df-gim 17701  df-oppg 17776  df-symg 17798  df-pmtr 17862  df-psgn 17911  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-cnfld 19747

This theorem is referenced by:  psgnghm2  19927  evpmss  19932