Theorem mulgneg 13726

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 9403	. . 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 1028	. . . . 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 13718	. . . . 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 1026	. . . . . 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 13642	. . . . . 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 6032	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( N .x. X ) = ( 0 .x. X ) )
15		simpl3 1028	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> X e. B )
16	4, 10, 5	mulg0 13711	. . . . . . . 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 5643	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( I ` ( N .x. X ) ) = ( I ` ( 0g ` G ) ) )
20	13	negeqd 8373	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> -u N = -u 0 )
21		neg0 8424	. . . . . . . 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 6032	. . . . . 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 804	. . 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 1026	. . . 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 9600	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
31		simpl3 1028	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> X e. B )
32	4, 5	mulgcl 13725	. . . . 5 |- ( ( G e. Grp /\ -u N e. ZZ /\ X e. B ) -> ( -u N .x. X ) e. B )
33	28, 30, 31, 32	syl3anc 1273	. . . 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 13649	. . . 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 13718	. . . . . 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 8207	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
40	39	negnegd 8480	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u -u N = N )
41	40	oveq1d 6032	. . . . 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 5643	. . 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 1024	. . 3 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> N e. ZZ )
46		elznn0nn 9492	. . 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 805	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 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   RRcr 8030   0cc0 8031   -ucneg 8350   NNcn 9142   NN0cn0 9401   ZZcz 9478   Basecbs 13081   0gc0g 13338   Grpcgrp 13582   invgcminusg 13583  ._gcmg 13705

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-seqfrec 10709  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-mulg 13706

This theorem is referenced by:  mulgnegneg  13727  mulgm1  13728  mulgaddcomlem  13731  mulginvcom  13733  mulgz  13736  mulgdirlem  13739  mulgdir  13740  mulgneg2  13742  mulgass  13745  mulgsubdir  13748  ghmmulg  13842  mulgass2  14070