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Theorem mulgnegnn 13468
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 9044 . . . . . 6 |- ( N e. NN -> N e. CC )
21negnegd 8374 . . . . 5 |- ( N e. NN -> -u -u N = N )
32adantr 276 . . . 4 |- ( ( N e. NN /\ X e. B ) -> -u -u N = N )
43fveq2d 5580 . . 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 5580 . 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 9375 . . . 4 |- ( N e. NN -> -u N e. ZZ )
7 mulg1.b . . . . 5 |- B = ( Base ` G )
8 eqid 2205 . . . . 5 |- ( +g ` G ) = ( +g ` G )
9 eqid 2205 . . . . 5 |- ( 0g ` G ) = ( 0g ` G )
10 mulgnegnn.i . . . . 5 |- I = ( invg ` G )
11 mulg1.m . . . . 5 |- .x. = (.g ` G )
12 eqid 2205 . . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
137, 8, 9, 10, 11, 12mulgval 13458 . . . 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 9064 . . . . . . 7 |- ( N e. NN -> N =/= 0 )
16 negeq0 8326 . . . . . . . . 9 |- ( N e. CC -> ( N = 0 <-> -u N = 0 ) )
1716necon3abid 2415 . . . . . . . 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 3581 . . . . 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 9043 . . . . . . . 8 |- ( N e. NN -> N e. RR )
2221renegcld 8452 . . . . . . 7 |- ( N e. NN -> -u N e. RR )
23 nngt0 9061 . . . . . . . 8 |- ( N e. NN -> 0 < N )
2421lt0neg2d 8589 . . . . . . . 8 |- ( N e. NN -> ( 0 < N <-> -u N < 0 ) )
2523, 24mpbid 147 . . . . . . 7 |- ( N e. NN -> -u N < 0 )
26 0re 8072 . . . . . . . 8 |- 0 e. RR
27 ltnsym 8158 . . . . . . . 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 3581 . . . . 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 2238 . . . 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 2238 . 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 13462 . . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
3534fveq2d 5580 . 2 |- ( ( N e. NN /\ X e. B ) -> ( I ` ( N .x. X ) ) = ( I ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) ) )
365, 33, 353eqtr4d 2248 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 1373   e. wcel 2176   =/= wne 2376   ifcif 3571   {csn 3633   class class class wbr 4044   X. cxp 4673   ` cfv 5271  (class class class)co 5944   CCcc 7923   RRcr 7924   0cc0 7925   1c1 7926   < clt 8107   -ucneg 8244   NNcn 9036   ZZcz 9372   seqcseq 10592   Basecbs 12832   +g cplusg 12909   0gc0g 13088   invgcminusg 13333  .gcmg 13455
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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-ltadd 8041
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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-2 9095  df-n0 9296  df-z 9373  df-uz 9649  df-seqfrec 10593  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-minusg 13336  df-mulg 13456
This theorem is referenced by:  mulgsubcl  13472  mulgneg  13476  mulgneg2  13492  cnfldmulg  14338
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