Theorem mulgneg 13717

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 9394	. . 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 1026	. . . . 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 13709	. . . . 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 1024	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> G e. Grp )
10		eqid 2229	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
11	10, 6	grpinvid 13633	. . . . . 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 6028	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( N .x. X ) = ( 0 .x. X ) )
15		simpl3 1026	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> X e. B )
16	4, 10, 5	mulg0 13702	. . . . . . . 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 2262	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( N .x. X ) = ( 0g ` G ) )
19	18	fveq2d 5639	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( I ` ( N .x. X ) ) = ( I ` ( 0g ` G ) ) )
20	13	negeqd 8364	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> -u N = -u 0 )
21		neg0 8415	. . . . . . . 8 |- -u 0 = 0
22	20, 21	eqtrdi 2278	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> -u N = 0 )
23	22	oveq1d 6028	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( -u N .x. X ) = ( 0 .x. X ) )
24	23, 17	eqtrd 2262	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( -u N .x. X ) = ( 0g ` G ) )
25	12, 19, 24	3eqtr4rd 2273	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
26	8, 25	jaodan 802	. . 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 1024	. . . 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 9591	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
31		simpl3 1026	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> X e. B )
32	4, 5	mulgcl 13716	. . . . 5 |- ( ( G e. Grp /\ -u N e. ZZ /\ X e. B ) -> ( -u N .x. X ) e. B )
33	28, 30, 31, 32	syl3anc 1271	. . . 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 13640	. . . 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 13709	. . . . . 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 8198	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
40	39	negnegd 8471	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u -u N = N )
41	40	oveq1d 6028	. . . . 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 2264	. . . 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 5639	. . 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 2264	. 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 1022	. . 3 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> N e. ZZ )
46		elznn0nn 9483	. . 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 803	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 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5324  (class class class)co 6013   RRcr 8021   0cc0 8022   -ucneg 8341   NNcn 9133   NN0cn0 9392   ZZcz 9469   Basecbs 13072   0gc0g 13329   Grpcgrp 13573   invgcminusg 13574  ._gcmg 13696

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-seqfrec 10700  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576  df-minusg 13577  df-mulg 13697

This theorem is referenced by:  mulgnegneg  13718  mulgm1  13719  mulgaddcomlem  13722  mulginvcom  13724  mulgz  13727  mulgdirlem  13730  mulgdir  13731  mulgneg2  13733  mulgass  13736  mulgsubdir  13739  ghmmulg  13833  mulgass2  14061