Theorem mulgneg 13213

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 9245	. . 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 13205	. . . . 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 2193	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
11	10, 6	grpinvid 13135	. . . . . 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 5934	. . . . . . 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 13198	. . . . . . . 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 2226	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( N .x. X ) = ( 0g ` G ) )
19	18	fveq2d 5559	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( I ` ( N .x. X ) ) = ( I ` ( 0g ` G ) ) )
20	13	negeqd 8216	. . . . . . . 8 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> -u N = -u 0 )
21		neg0 8267	. . . . . . . 8 |- -u 0 = 0
22	20, 21	eqtrdi 2242	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> -u N = 0 )
23	22	oveq1d 5934	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( -u N .x. X ) = ( 0 .x. X ) )
24	23, 17	eqtrd 2226	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ N = 0 ) -> ( -u N .x. X ) = ( 0g ` G ) )
25	12, 19, 24	3eqtr4rd 2237	. . . 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 9441	. . . . 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 13212	. . . . 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 13142	. . . 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 13205	. . . . . 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 8050	. . . . . . 7 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
40	39	negnegd 8323	. . . . . 6 |- ( ( ( G e. Grp /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u -u N = N )
41	40	oveq1d 5934	. . . . 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 2228	. . . 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 5559	. . 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 2228	. 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 9334	. . 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 2164   ` cfv 5255  (class class class)co 5919   RRcr 7873   0cc0 7874   -ucneg 8193   NNcn 8984   NN0cn0 9243   ZZcz 9320   Basecbs 12621   0gc0g 12870   Grpcgrp 13075   invgcminusg 13076  ._gcmg 13192

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321  df-uz 9596  df-seqfrec 10522  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-mulg 13193

This theorem is referenced by:  mulgnegneg  13214  mulgm1  13215  mulgaddcomlem  13218  mulginvcom  13220  mulgz  13223  mulgdirlem  13226  mulgdir  13227  mulgneg2  13229  mulgass  13232  mulgsubdir  13235  ghmmulg  13329  mulgass2  13557