Theorem mulgnegnn 13205

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 8992	. . . . . 6 |- ( N e. NN -> N e. CC )
2	1	negnegd 8323	. . . . 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 5559	. . 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 5559	. 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 9323	. . . 4 |- ( N e. NN -> -u N e. ZZ )
7		mulg1.b	. . . . 5 |- B = ( Base ` G )
8		eqid 2193	. . . . 5 |- ( +g ` G ) = ( +g ` G )
9		eqid 2193	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
10		mulgnegnn.i	. . . . 5 |- I = ( invg ` G )
11		mulg1.m	. . . . 5 |- .x. = (.g ` G )
12		eqid 2193	. . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
13	7, 8, 9, 10, 11, 12	mulgval 13195	. . . 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 9012	. . . . . . 7 |- ( N e. NN -> N =/= 0 )
16		negeq0 8275	. . . . . . . . 9 |- ( N e. CC -> ( N = 0 <-> -u N = 0 ) )
17	16	necon3abid 2403	. . . . . . . 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 3568	. . . . 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 8991	. . . . . . . 8 |- ( N e. NN -> N e. RR )
22	21	renegcld 8401	. . . . . . 7 |- ( N e. NN -> -u N e. RR )
23		nngt0 9009	. . . . . . . 8 |- ( N e. NN -> 0 < N )
24	21	lt0neg2d 8537	. . . . . . . 8 |- ( N e. NN -> ( 0 < N <-> -u N < 0 ) )
25	23, 24	mpbid 147	. . . . . . 7 |- ( N e. NN -> -u N < 0 )
26		0re 8021	. . . . . . . 8 |- 0 e. RR
27		ltnsym 8107	. . . . . . . 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 3568	. . . . 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 2226	. . . 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 2226	. 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 13199	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
35	34	fveq2d 5559	. 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 2236	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 2164   =/= wne 2364   ifcif 3558   {csn 3619   class class class wbr 4030   X. cxp 4658   ` cfv 5255  (class class class)co 5919   CCcc 7872   RRcr 7873   0cc0 7874   1c1 7875   < clt 8056   -ucneg 8193   NNcn 8984   ZZcz 9320   seqcseq 10521   Basecbs 12621   +g cplusg 12698   0gc0g 12870   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-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-minusg 13079  df-mulg 13193

This theorem is referenced by:  mulgsubcl  13209  mulgneg  13213  mulgneg2  13229  cnfldmulg  14075