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Theorem mulgnegnn 13935
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 9312 . . . . . 6 |- ( N e. NN -> N e. CC )
21negnegd 8628 . . . . 5 |- ( N e. NN -> -u -u N = N )
32adantr 276 . . . 4 |- ( ( N e. NN /\ X e. B ) -> -u -u N = N )
43fveq2d 5699 . . 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 5699 . 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 9647 . . . 4 |- ( N e. NN -> -u N e. ZZ )
7 mulg1.b . . . . 5 |- B = ( Base ` G )
8 eqid 2238 . . . . 5 |- ( +g ` G ) = ( +g ` G )
9 eqid 2238 . . . . 5 |- ( 0g ` G ) = ( 0g ` G )
10 mulgnegnn.i . . . . 5 |- I = ( invg ` G )
11 mulg1.m . . . . 5 |- .x. = (.g ` G )
12 eqid 2238 . . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
137, 8, 9, 10, 11, 12mulgval 13925 . . . 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 9332 . . . . . . 7 |- ( N e. NN -> N =/= 0 )
16 negeq0 8580 . . . . . . . . 9 |- ( N e. CC -> ( N = 0 <-> -u N = 0 ) )
1716necon3abid 2459 . . . . . . . 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 3650 . . . . 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 9311 . . . . . . . 8 |- ( N e. NN -> N e. RR )
2221renegcld 8707 . . . . . . 7 |- ( N e. NN -> -u N e. RR )
23 nngt0 9329 . . . . . . . 8 |- ( N e. NN -> 0 < N )
2421lt0neg2d 8844 . . . . . . . 8 |- ( N e. NN -> ( 0 < N <-> -u N < 0 ) )
2523, 24mpbid 147 . . . . . . 7 |- ( N e. NN -> -u N < 0 )
26 0re 8326 . . . . . . . 8 |- 0 e. RR
27 ltnsym 8411 . . . . . . . 8 |- ( ( -u N e. RR /\ 0 e. RR ) -> ( -u N < 0 -> -. 0 < -u N ) )
2826, 27mpan2 429 . . . . . . 7 |- ( -u N e. RR -> ( -u N < 0 -> -. 0 < -u N ) )
2922, 25, 28sylc 62 . . . . . 6 |- ( N e. NN -> -. 0 < -u N )
3029iffalsed 3650 . . . . 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 2271 . . . 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 2271 . 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 13929 . . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
3534fveq2d 5699 . 2 |- ( ( N e. NN /\ X e. B ) -> ( I ` ( N .x. X ) ) = ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) ) )
365, 33, 353eqtr4d 2281 1 |- ( ( N e. NN /\ X e. B ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   ifcif 3638   {csn 3709   class class class wbr 4130   X. cxp 4772   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   < clt 8360   -ucneg 8498   NNcn 9304   ZZcz 9644   seqcseq 10884   Basecbs 13352   +g cplusg 13431   0gc0g 13610   invgcminusg 13806  .gcmg 13922
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-minusg 13809  df-mulg 13923
This theorem is used by:  mulgsubcl  13939  mulgneg  13943  mulgneg2  13959  cnfldmulg  14913
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