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Theorem mulgnegnn 13669
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
StepHypRef Expression
1 nncn 9118 . . . . . 6 |- ( N e. NN -> N e. CC )
21negnegd 8448 . . . . 5 |- ( N e. NN -> -u -u N = N )
32adantr 276 . . . 4 |- ( ( N e. NN /\ X e. B ) -> -u -u N = N )
43fveq2d 5631 . . 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 ) )
54fveq2d 5631 . 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 9449 . . . 4 |- ( N e. NN -> -u N e. ZZ )
7 mulg1.b . . . . 5 |- B = ( Base ` G )
8 eqid 2229 . . . . 5 |- ( +g ` G ) = ( +g ` G )
9 eqid 2229 . . . . 5 |- ( 0g ` G ) = ( 0g ` G )
10 mulgnegnn.i . . . . 5 |- I = ( invg ` G )
11 mulg1.m . . . . 5 |- .x. = (.g ` G )
12 eqid 2229 . . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
137, 8, 9, 10, 11, 12mulgval 13659 . . . 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 ) ) ) ) )
146, 13sylan 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 9138 . . . . . . 7 |- ( N e. NN -> N =/= 0 )
16 negeq0 8400 . . . . . . . . 9 |- ( N e. CC -> ( N = 0 <-> -u N = 0 ) )
1716necon3abid 2439 . . . . . . . 8 |- ( N e. CC -> ( N =/= 0 <-> -. -u N = 0 ) )
181, 17syl 14 . . . . . . 7 |- ( N e. NN -> ( N =/= 0 <-> -. -u N = 0 ) )
1915, 18mpbid 147 . . . . . 6 |- ( N e. NN -> -. -u N = 0 )
2019iffalsed 3612 . . . . 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 9117 . . . . . . . 8 |- ( N e. NN -> N e. RR )
2221renegcld 8526 . . . . . . 7 |- ( N e. NN -> -u N e. RR )
23 nngt0 9135 . . . . . . . 8 |- ( N e. NN -> 0 < N )
2421lt0neg2d 8663 . . . . . . . 8 |- ( N e. NN -> ( 0 < N <-> -u N < 0 ) )
2523, 24mpbid 147 . . . . . . 7 |- ( N e. NN -> -u N < 0 )
26 0re 8146 . . . . . . . 8 |- 0 e. RR
27 ltnsym 8232 . . . . . . . 8 |- ( ( -u N e. RR /\ 0 e. RR ) -> ( -u N < 0 -> -. 0 < -u N ) )
2826, 27mpan2 425 . . . . . . 7 |- ( -u N e. RR -> ( -u N < 0 -> -. 0 < -u N ) )
2922, 25, 28sylc 62 . . . . . 6 |- ( N e. NN -> -. 0 < -u N )
3029iffalsed 3612 . . . . 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 ) ) )
3120, 30eqtrd 2262 . . . 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 ) ) )
3231adantr 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 ) ) )
3314, 32eqtrd 2262 . 2 |- ( ( N e. NN /\ X e. B ) -> ( -u N .x. X ) = ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u -u N ) ) )
347, 8, 11, 12mulgnn 13663 . . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
3534fveq2d 5631 . 2 |- ( ( N e. NN /\ X e. B ) -> ( I ` ( N .x. X ) ) = ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) ) )
365, 33, 353eqtr4d 2272 1 |- ( ( N e. NN /\ X e. B ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   =/= wne 2400   ifcif 3602   {csn 3666   class class class wbr 4083   X. cxp 4717   ` cfv 5318  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   1c1 8000   < clt 8181   -ucneg 8318   NNcn 9110   ZZcz 9446   seqcseq 10669   Basecbs 13032   +g cplusg 13110   0gc0g 13289   invgcminusg 13534  .gcmg 13656
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447  df-uz 9723  df-seqfrec 10670  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-minusg 13537  df-mulg 13657
This theorem is referenced by:  mulgsubcl  13673  mulgneg  13677  mulgneg2  13693  cnfldmulg  14540
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