Theorem mulgneg 13807

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 9463	. . 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 1029	. . . . 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 13799	. . . . 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 1027	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> G e. Grp )
10		eqid 2231	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
11	10, 6	grpinvid 13723	. . . . . 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 6043	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( N .x. X ) = ( 0 .x. X ) )
15		simpl3 1029	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> X e. B )
16	4, 10, 5	mulg0 13792	. . . . . . . 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 2264	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( N .x. X ) = ( 0g ` G ) )
19	18	fveq2d 5652	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( I ` ( N .x. X ) ) = ( I ` ( 0g ` G ) ) )
20	13	negeqd 8433	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> -u N = -u 0 )
21		neg0 8484	. . . . . . . 8 |- -u 0 = 0
22	20, 21	eqtrdi 2280	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> -u N = 0 )
23	22	oveq1d 6043	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( -u N .x. X ) = ( 0 .x. X ) )
24	23, 17	eqtrd 2264	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( -u N .x. X ) = ( 0g ` G ) )
25	12, 19, 24	3eqtr4rd 2275	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
26	8, 25	jaodan 805	. . 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 1027	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> G e. Grp )
29		simprr 533	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN )
30	29	nnzd 9662	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
31		simpl3 1029	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> X e. B )
32	4, 5	mulgcl 13806	. . . . 5 |- ( ( G e. Grp /\ -u N e. ZZ /\ X e. B ) -> ( -u N .x. X ) e. B )
33	28, 30, 31, 32	syl3anc 1274	. . . 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 13730	. . . 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 13799	. . . . . 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 531	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR )
39	38	recnd 8267	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
40	39	negnegd 8540	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u -u N = N )
41	40	oveq1d 6043	. . . . 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 2266	. . . 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 5652	. . 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 2266	. 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 1025	. . 3 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> N e. ZZ )
46		elznn0nn 9554	. . 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 806	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 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   RRcr 8091   0cc0 8092   -ucneg 8410   NNcn 9202   NN0cn0 9461   ZZcz 9540   Basecbs 13162   0gc0g 13419   Grpcgrp 13663   invgcminusg 13664  ._gcmg 13786

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-2 9261  df-n0 9462  df-z 9541  df-uz 9817  df-seqfrec 10773  df-ndx 13165  df-slot 13166  df-base 13168  df-plusg 13253  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-mulg 13787

This theorem is referenced by:  mulgnegneg  13808  mulgm1  13809  mulgaddcomlem  13812  mulginvcom  13814  mulgz  13817  mulgdirlem  13820  mulgdir  13821  mulgneg2  13823  mulgass  13826  mulgsubdir  13829  ghmmulg  13923  mulgass2  14152