Theorem mulgnegnn 13338

Description: Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)

Hypotheses

Ref	Expression
mulg1.b	|- B = ( Base ` G )
mulg1.m	|- .x. = (.g ` G )
mulgnegnn.i	|- I = ( invg ` G )

Assertion

Ref	Expression
mulgnegnn	|- ( ( N e. NN /\ X e. B ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )

Proof of Theorem mulgnegnn

Step	Hyp	Ref	Expression
1		nncn 9015	. . . . . 6 |- ( N e. NN -> N e. CC )
2	1	negnegd 8345	. . . . 5 |- ( N e. NN -> -u -u N = N )
3	2	adantr 276	. . . 4 |- ( ( N e. NN /\ X e. B ) -> -u -u N = N )
4	3	fveq2d 5565	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
5	4	fveq2d 5565	. 2 |- ( ( N e. NN /\ X e. B ) -> ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) = ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) ) )
6		nnnegz 9346	. . . 4 |- ( N e. NN -> -u N e. ZZ )
7		mulg1.b	. . . . 5 |- B = ( Base ` G )
8		eqid 2196	. . . . 5 |- ( +g ` G ) = ( +g ` G )
9		eqid 2196	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
10		mulgnegnn.i	. . . . 5 |- I = ( invg ` G )
11		mulg1.m	. . . . 5 |- .x. = (.g ` G )
12		eqid 2196	. . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
13	7, 8, 9, 10, 11, 12	mulgval 13328	. . . 4 |- ( ( -u N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = if ( -u N = 0 , ( 0g ` G ) , if ( 0 < -u N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) , ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) ) ) )
14	6, 13	sylan 283	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( -u N .x. X ) = if ( -u N = 0 , ( 0g ` G ) , if ( 0 < -u N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) , ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) ) ) )
15		nnne0 9035	. . . . . . 7 |- ( N e. NN -> N =/= 0 )
16		negeq0 8297	. . . . . . . . 9 |- ( N e. CC -> ( N = 0 <-> -u N = 0 ) )
17	16	necon3abid 2406	. . . . . . . 8 |- ( N e. CC -> ( N =/= 0 <-> -. -u N = 0 ) )
18	1, 17	syl 14	. . . . . . 7 |- ( N e. NN -> ( N =/= 0 <-> -. -u N = 0 ) )
19	15, 18	mpbid 147	. . . . . 6 |- ( N e. NN -> -. -u N = 0 )
20	19	iffalsed 3572	. . . . 5 |- ( N e. NN -> if ( -u N = 0 , ( 0g ` G ) , if ( 0 < -u N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) , ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) ) ) = if ( 0 < -u N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) , ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) ) )
21		nnre 9014	. . . . . . . 8 |- ( N e. NN -> N e. RR )
22	21	renegcld 8423	. . . . . . 7 |- ( N e. NN -> -u N e. RR )
23		nngt0 9032	. . . . . . . 8 |- ( N e. NN -> 0 < N )
24	21	lt0neg2d 8560	. . . . . . . 8 |- ( N e. NN -> ( 0 < N <-> -u N < 0 ) )
25	23, 24	mpbid 147	. . . . . . 7 |- ( N e. NN -> -u N < 0 )
26		0re 8043	. . . . . . . 8 |- 0 e. RR
27		ltnsym 8129	. . . . . . . 8 |- ( ( -u N e. RR /\ 0 e. RR ) -> ( -u N < 0 -> -. 0 < -u N ) )
28	26, 27	mpan2 425	. . . . . . 7 |- ( -u N e. RR -> ( -u N < 0 -> -. 0 < -u N ) )
29	22, 25, 28	sylc 62	. . . . . 6 |- ( N e. NN -> -. 0 < -u N )
30	29	iffalsed 3572	. . . . 5 |- ( N e. NN -> if ( 0 < -u N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) , ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) ) = ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) )
31	20, 30	eqtrd 2229	. . . 4 |- ( N e. NN -> if ( -u N = 0 , ( 0g ` G ) , if ( 0 < -u N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) , ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) ) ) = ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) )
32	31	adantr 276	. . 3 |- ( ( N e. NN /\ X e. B ) -> if ( -u N = 0 , ( 0g ` G ) , if ( 0 < -u N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) , ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) ) ) = ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) )
33	14, 32	eqtrd 2229	. 2 |- ( ( N e. NN /\ X e. B ) -> ( -u N .x. X ) = ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) )
34	7, 8, 11, 12	mulgnn 13332	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
35	34	fveq2d 5565	. 2 |- ( ( N e. NN /\ X e. B ) -> ( I ` ( N .x. X ) ) = ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) ) )
36	5, 33, 35	3eqtr4d 2239	1 |- ( ( N e. NN /\ X e. B ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   =/= wne 2367   ifcif 3562   {csn 3623   class class class wbr 4034   X. cxp 4662   ` cfv 5259  (class class class)co 5925   CCcc 7894   RRcr 7895   0cc0 7896   1c1 7897   < clt 8078   -ucneg 8215   NNcn 9007   ZZcz 9343   seqcseq 10556   Basecbs 12703   +g cplusg 12780   0gc0g 12958   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-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-minusg 13206  df-mulg 13326

This theorem is referenced by:  mulgsubcl  13342  mulgneg  13346  mulgneg2  13362  cnfldmulg  14208