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Theorem mulgnegnn 12882
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 8916 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  CC )
21negnegd 8249 . . . . 5  |-  ( N  e.  NN  ->  -u -u N  =  N )
32adantr 276 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  B )  -> 
-u -u N  =  N )
43fveq2d 5515 . . 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 5515 . 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 9245 . . . 4  |-  ( N  e.  NN  ->  -u N  e.  ZZ )
7 mulg1.b . . . . 5  |-  B  =  ( Base `  G
)
8 eqid 2177 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
9 eqid 2177 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
10 mulgnegnn.i . . . . 5  |-  I  =  ( invg `  G )
11 mulg1.m . . . . 5  |-  .x.  =  (.g
`  G )
12 eqid 2177 . . . . 5  |-  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
137, 8, 9, 10, 11, 12mulgval 12875 . . . 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 8936 . . . . . . 7  |-  ( N  e.  NN  ->  N  =/=  0 )
16 negeq0 8201 . . . . . . . . 9  |-  ( N  e.  CC  ->  ( N  =  0  <->  -u N  =  0 ) )
1716necon3abid 2386 . . . . . . . 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 3544 . . . . 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 8915 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  RR )
2221renegcld 8327 . . . . . . 7  |-  ( N  e.  NN  ->  -u N  e.  RR )
23 nngt0 8933 . . . . . . . 8  |-  ( N  e.  NN  ->  0  <  N )
2421lt0neg2d 8463 . . . . . . . 8  |-  ( N  e.  NN  ->  (
0  <  N  <->  -u N  <  0 ) )
2523, 24mpbid 147 . . . . . . 7  |-  ( N  e.  NN  ->  -u N  <  0 )
26 0re 7948 . . . . . . . 8  |-  0  e.  RR
27 ltnsym 8033 . . . . . . . 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 3544 . . . . 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 2210 . . . 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 2210 . 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 12878 . . 3  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( N  .x.  X
)  =  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) )
3534fveq2d 5515 . 2  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( I `  ( N  .x.  X ) )  =  ( I `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 N ) ) )
365, 33, 353eqtr4d 2220 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 1353    e. wcel 2148    =/= wne 2347   ifcif 3534   {csn 3591   class class class wbr 4000    X. cxp 4621   ` cfv 5212  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803    < clt 7982   -ucneg 8119   NNcn 8908   ZZcz 9242    seqcseq 10431   Basecbs 12445   +g cplusg 12518   0gc0g 12653   invgcminusg 12768  .gcmg 12872
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518  df-seqfrec 10432  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-0g 12655  df-minusg 12771  df-mulg 12873
This theorem is referenced by:  mulgsubcl  12886  mulgneg  12890  mulgneg2  12905
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