Theorem mulgneg 13346

Description: Group multiple (exponentiation) operation at a negative integer. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 11-Dec-2014.)

Hypotheses

Ref	Expression
mulgnncl.b	|- B = ( Base ` G )
mulgnncl.t	|- .x. = (.g ` G )
mulgneg.i	|- I = ( invg ` G )

Assertion

Ref	Expression
mulgneg	|- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )

Proof of Theorem mulgneg

Step	Hyp	Ref	Expression
1		elnn0 9268	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		simpr 110	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N e. NN ) -> N e. NN )
3		simpl3 1004	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N e. NN ) -> X e. B )
4		mulgnncl.b	. . . . . 6 |- B = ( Base ` G )
5		mulgnncl.t	. . . . . 6 |- .x. = (.g ` G )
6		mulgneg.i	. . . . . 6 |- I = ( invg ` G )
7	4, 5, 6	mulgnegnn 13338	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
8	2, 3, 7	syl2anc 411	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N e. NN ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
9		simpl1 1002	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> G e. Grp )
10		eqid 2196	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
11	10, 6	grpinvid 13262	. . . . . 6 |- ( G e. Grp -> ( I ` ( 0g ` G ) ) = ( 0g ` G ) )
12	9, 11	syl 14	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( I ` ( 0g ` G ) ) = ( 0g ` G ) )
13		simpr 110	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> N = 0 )
14	13	oveq1d 5940	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( N .x. X ) = ( 0 .x. X ) )
15		simpl3 1004	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> X e. B )
16	4, 10, 5	mulg0 13331	. . . . . . . 8 |- ( X e. B -> ( 0 .x. X ) = ( 0g ` G ) )
17	15, 16	syl 14	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( 0 .x. X ) = ( 0g ` G ) )
18	14, 17	eqtrd 2229	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( N .x. X ) = ( 0g ` G ) )
19	18	fveq2d 5565	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( I ` ( N .x. X ) ) = ( I ` ( 0g ` G ) ) )
20	13	negeqd 8238	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> -u N = -u 0 )
21		neg0 8289	. . . . . . . 8 |- -u 0 = 0
22	20, 21	eqtrdi 2245	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> -u N = 0 )
23	22	oveq1d 5940	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( -u N .x. X ) = ( 0 .x. X ) )
24	23, 17	eqtrd 2229	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( -u N .x. X ) = ( 0g ` G ) )
25	12, 19, 24	3eqtr4rd 2240	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
26	8, 25	jaodan 798	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. NN \/ N = 0 ) ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
27	1, 26	sylan2b 287	. 2 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N e. NN0 ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
28		simpl1 1002	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> G e. Grp )
29		simprr 531	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN )
30	29	nnzd 9464	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
31		simpl3 1004	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> X e. B )
32	4, 5	mulgcl 13345	. . . . 5 |- ( ( G e. Grp /\ -u N e. ZZ /\ X e. B ) -> ( -u N .x. X ) e. B )
33	28, 30, 31, 32	syl3anc 1249	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u N .x. X ) e. B )
34	4, 6	grpinvinv 13269	. . . 4 |- ( ( G e. Grp /\ ( -u N .x. X ) e. B ) -> ( I ` ( I ` ( -u N .x. X ) ) ) = ( -u N .x. X ) )
35	28, 33, 34	syl2anc 411	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( I ` ( I ` ( -u N .x. X ) ) ) = ( -u N .x. X ) )
36	4, 5, 6	mulgnegnn 13338	. . . . . 6 |- ( ( -u N e. NN /\ X e. B ) -> ( -u -u N .x. X ) = ( I ` ( -u N .x. X ) ) )
37	29, 31, 36	syl2anc 411	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u -u N .x. X ) = ( I ` ( -u N .x. X ) ) )
38		simprl 529	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR )
39	38	recnd 8072	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
40	39	negnegd 8345	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u -u N = N )
41	40	oveq1d 5940	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u -u N .x. X ) = ( N .x. X ) )
42	37, 41	eqtr3d 2231	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( I ` ( -u N .x. X ) ) = ( N .x. X ) )
43	42	fveq2d 5565	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( I ` ( I ` ( -u N .x. X ) ) ) = ( I ` ( N .x. X ) ) )
44	35, 43	eqtr3d 2231	. 2 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
45		simp2 1000	. . 3 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> N e. ZZ )
46		elznn0nn 9357	. . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
47	45, 46	sylib 122	. 2 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
48	27, 44, 47	mpjaodan 799	1 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5925   RRcr 7895   0cc0 7896   -ucneg 8215   NNcn 9007   NN0cn0 9266   ZZcz 9343   Basecbs 12703   0gc0g 12958   Grpcgrp 13202   invgcminusg 13203  ._gcmg 13325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-seqfrec 10557  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-mulg 13326

This theorem is referenced by:  mulgnegneg  13347  mulgm1  13348  mulgaddcomlem  13351  mulginvcom  13353  mulgz  13356  mulgdirlem  13359  mulgdir  13360  mulgneg2  13362  mulgass  13365  mulgsubdir  13368  ghmmulg  13462  mulgass2  13690