Theorem mulgnegnn 12849

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 8895	. . . . . 6 |- ( N e. NN -> N e. CC )
2	1	negnegd 8230	. . . . 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 5508	. . 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 5508	. 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 9224	. . . 4 |- ( N e. NN -> -u N e. ZZ )
7		mulg1.b	. . . . 5 |- B = ( Base ` G )
8		eqid 2173	. . . . 5 |- ( +g ` G ) = ( +g ` G )
9		eqid 2173	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
10		mulgnegnn.i	. . . . 5 |- I = ( invg ` G )
11		mulg1.m	. . . . 5 |- .x. = (.g ` G )
12		eqid 2173	. . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
13	7, 8, 9, 10, 11, 12	mulgval 12842	. . . 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 8915	. . . . . . 7 |- ( N e. NN -> N =/= 0 )
16		negeq0 8182	. . . . . . . . 9 |- ( N e. CC -> ( N = 0 <-> -u N = 0 ) )
17	16	necon3abid 2382	. . . . . . . 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 148	. . . . . 6 |- ( N e. NN -> -. -u N = 0 )
20	19	iffalsed 3539	. . . . 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 8894	. . . . . . . 8 |- ( N e. NN -> N e. RR )
22	21	renegcld 8308	. . . . . . 7 |- ( N e. NN -> -u N e. RR )
23		nngt0 8912	. . . . . . . 8 |- ( N e. NN -> 0 < N )
24	21	lt0neg2d 8444	. . . . . . . 8 |- ( N e. NN -> ( 0 < N <-> -u N < 0 ) )
25	23, 24	mpbid 148	. . . . . . 7 |- ( N e. NN -> -u N < 0 )
26		0re 7929	. . . . . . . 8 |- 0 e. RR
27		ltnsym 8014	. . . . . . . 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 3539	. . . . 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 2206	. . . 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 2206	. 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 12845	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
35	34	fveq2d 5508	. 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 2216	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 1351   e. wcel 2144   =/= wne 2343   ifcif 3529   {csn 3586   class class class wbr 3995   X. cxp 4615   ` cfv 5205  (class class class)co 5862   CCcc 7781   RRcr 7782   0cc0 7783   1c1 7784   < clt 7963   -ucneg 8100   NNcn 8887   ZZcz 9221   seqcseq 10410   Basecbs 12425   +g cplusg 12489   0gc0g 12623   invgcminusg 12736  ._gcmg 12839

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 612  ax-in2 613  ax-io 707  ax-5 1443  ax-7 1444  ax-gen 1445  ax-ie1 1489  ax-ie2 1490  ax-8 1500  ax-10 1501  ax-11 1502  ax-i12 1503  ax-bndl 1505  ax-4 1506  ax-17 1522  ax-i9 1526  ax-ial 1530  ax-i5r 1531  ax-13 2146  ax-14 2147  ax-ext 2155  ax-coll 4110  ax-sep 4113  ax-nul 4121  ax-pow 4166  ax-pr 4200  ax-un 4424  ax-setind 4527  ax-iinf 4578  ax-cnex 7874  ax-resscn 7875  ax-1cn 7876  ax-1re 7877  ax-icn 7878  ax-addcl 7879  ax-addrcl 7880  ax-mulcl 7881  ax-addcom 7883  ax-addass 7885  ax-distr 7887  ax-i2m1 7888  ax-0lt1 7889  ax-0id 7891  ax-rnegex 7892  ax-cnre 7894  ax-pre-ltirr 7895  ax-pre-ltwlin 7896  ax-pre-lttrn 7897  ax-pre-ltadd 7899

This theorem depends on definitions:  df-bi 117  df-dc 833  df-3or 977  df-3an 978  df-tru 1354  df-fal 1357  df-nf 1457  df-sb 1759  df-eu 2025  df-mo 2026  df-clab 2160  df-cleq 2166  df-clel 2169  df-nfc 2304  df-ne 2344  df-nel 2439  df-ral 2456  df-rex 2457  df-reu 2458  df-rab 2460  df-v 2735  df-sbc 2959  df-csb 3053  df-dif 3126  df-un 3128  df-in 3130  df-ss 3137  df-nul 3418  df-if 3530  df-pw 3571  df-sn 3592  df-pr 3593  df-op 3595  df-uni 3803  df-int 3838  df-iun 3881  df-br 3996  df-opab 4057  df-mpt 4058  df-tr 4094  df-id 4284  df-iord 4357  df-on 4359  df-ilim 4360  df-suc 4362  df-iom 4581  df-xp 4623  df-rel 4624  df-cnv 4625  df-co 4626  df-dm 4627  df-rn 4628  df-res 4629  df-ima 4630  df-iota 5167  df-fun 5207  df-fn 5208  df-f 5209  df-f1 5210  df-fo 5211  df-f1o 5212  df-fv 5213  df-riota 5818  df-ov 5865  df-oprab 5866  df-mpo 5867  df-1st 6128  df-2nd 6129  df-recs 6293  df-frec 6379  df-pnf 7965  df-mnf 7966  df-xr 7967  df-ltxr 7968  df-le 7969  df-sub 8101  df-neg 8102  df-inn 8888  df-2 8946  df-n0 9145  df-z 9222  df-uz 9497  df-seqfrec 10411  df-ndx 12428  df-slot 12429  df-base 12431  df-plusg 12502  df-0g 12625  df-minusg 12739  df-mulg 12840

This theorem is referenced by:  mulgsubcl  12853  mulgneg  12857  mulgneg2  12872