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Theorem mulgnegnn 13709
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 9141 . . . . . 6 |- ( N e. NN -> N e. CC )
21negnegd 8471 . . . . 5 |- ( N e. NN -> -u -u N = N )
32adantr 276 . . . 4 |- ( ( N e. NN /\ X e. B ) -> -u -u N = N )
43fveq2d 5639 . . 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 5639 . 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 9472 . . . 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 13699 . . . 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 9161 . . . . . . 7 |- ( N e. NN -> N =/= 0 )
16 negeq0 8423 . . . . . . . . 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 3613 . . . . 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 9140 . . . . . . . 8 |- ( N e. NN -> N e. RR )
2221renegcld 8549 . . . . . . 7 |- ( N e. NN -> -u N e. RR )
23 nngt0 9158 . . . . . . . 8 |- ( N e. NN -> 0 < N )
2421lt0neg2d 8686 . . . . . . . 8 |- ( N e. NN -> ( 0 < N <-> -u N < 0 ) )
2523, 24mpbid 147 . . . . . . 7 |- ( N e. NN -> -u N < 0 )
26 0re 8169 . . . . . . . 8 |- 0 e. RR
27 ltnsym 8255 . . . . . . . 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 3613 . . . . 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 13703 . . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
3534fveq2d 5639 . 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 3603   {csn 3667   class class class wbr 4086   X. cxp 4721   ` cfv 5324  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   1c1 8023   < clt 8204   -ucneg 8341   NNcn 9133   ZZcz 9469   seqcseq 10699   Basecbs 13072   +g cplusg 13150   0gc0g 13329   invgcminusg 13574  .gcmg 13696
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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138
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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-seqfrec 10700  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-0g 13331  df-minusg 13577  df-mulg 13697
This theorem is referenced by:  mulgsubcl  13713  mulgneg  13717  mulgneg2  13733  cnfldmulg  14580
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