Theorem mulgnegnn 13543

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 9064	. . . . . 6 |- ( N e. NN -> N e. CC )
2	1	negnegd 8394	. . . . 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 5593	. . 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 5593	. 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 9395	. . . 4 |- ( N e. NN -> -u N e. ZZ )
7		mulg1.b	. . . . 5 |- B = ( Base ` G )
8		eqid 2206	. . . . 5 |- ( +g ` G ) = ( +g ` G )
9		eqid 2206	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
10		mulgnegnn.i	. . . . 5 |- I = ( invg ` G )
11		mulg1.m	. . . . 5 |- .x. = (.g ` G )
12		eqid 2206	. . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
13	7, 8, 9, 10, 11, 12	mulgval 13533	. . . 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 9084	. . . . . . 7 |- ( N e. NN -> N =/= 0 )
16		negeq0 8346	. . . . . . . . 9 |- ( N e. CC -> ( N = 0 <-> -u N = 0 ) )
17	16	necon3abid 2416	. . . . . . . 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 3585	. . . . 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 9063	. . . . . . . 8 |- ( N e. NN -> N e. RR )
22	21	renegcld 8472	. . . . . . 7 |- ( N e. NN -> -u N e. RR )
23		nngt0 9081	. . . . . . . 8 |- ( N e. NN -> 0 < N )
24	21	lt0neg2d 8609	. . . . . . . 8 |- ( N e. NN -> ( 0 < N <-> -u N < 0 ) )
25	23, 24	mpbid 147	. . . . . . 7 |- ( N e. NN -> -u N < 0 )
26		0re 8092	. . . . . . . 8 |- 0 e. RR
27		ltnsym 8178	. . . . . . . 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 3585	. . . . 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 2239	. . . 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 2239	. 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 13537	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
35	34	fveq2d 5593	. 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 2249	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 1373   e. wcel 2177   =/= wne 2377   ifcif 3575   {csn 3638   class class class wbr 4051   X. cxp 4681   ` cfv 5280  (class class class)co 5957   CCcc 7943   RRcr 7944   0cc0 7945   1c1 7946   < clt 8127   -ucneg 8264   NNcn 9056   ZZcz 9392   seqcseq 10614   Basecbs 12907   +g cplusg 12984   0gc0g 13163   invgcminusg 13408  ._gcmg 13530

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-inn 9057  df-2 9115  df-n0 9316  df-z 9393  df-uz 9669  df-seqfrec 10615  df-ndx 12910  df-slot 12911  df-base 12913  df-plusg 12997  df-0g 13165  df-minusg 13411  df-mulg 13531

This theorem is referenced by:  mulgsubcl  13547  mulgneg  13551  mulgneg2  13567  cnfldmulg  14413