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Theorem mulgval 12875
Description: Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
Hypotheses
Ref Expression
mulgval.b  |-  B  =  ( Base `  G
)
mulgval.p  |-  .+  =  ( +g  `  G )
mulgval.o  |-  .0.  =  ( 0g `  G )
mulgval.i  |-  I  =  ( invg `  G )
mulgval.t  |-  .x.  =  (.g
`  G )
mulgval.s  |-  S  =  seq 1 (  .+  ,  ( NN  X.  { X } ) )
Assertion
Ref Expression
mulgval  |-  ( ( N  e.  ZZ  /\  X  e.  B )  ->  ( N  .x.  X
)  =  if ( N  =  0 ,  .0.  ,  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) ) ) )

Proof of Theorem mulgval
Dummy variables  x  n  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulgval.b . . . 4  |-  B  =  ( Base `  G
)
21basmex 12503 . . 3  |-  ( X  e.  B  ->  G  e.  _V )
32adantl 277 . 2  |-  ( ( N  e.  ZZ  /\  X  e.  B )  ->  G  e.  _V )
4 mulgval.p . . . . 5  |-  .+  =  ( +g  `  G )
5 mulgval.o . . . . 5  |-  .0.  =  ( 0g `  G )
6 mulgval.i . . . . 5  |-  I  =  ( invg `  G )
7 mulgval.t . . . . 5  |-  .x.  =  (.g
`  G )
81, 4, 5, 6, 7mulgfvalg 12874 . . . 4  |-  ( G  e.  _V  ->  .x.  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq 1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq 1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) ) )
98adantl 277 . . 3  |-  ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  ->  .x.  =  (
n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq 1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq 1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) ) )
10 simpl 109 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  ->  n  =  N )
1110eqeq1d 2186 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  ( n  =  0  <-> 
N  =  0 ) )
1210breq2d 4012 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  ->  ( 0  <  n  <->  0  <  N ) )
13 simpr 110 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  x  =  X )  ->  x  =  X )
1413sneqd 3604 . . . . . . . . . 10  |-  ( ( n  =  N  /\  x  =  X )  ->  { x }  =  { X } )
1514xpeq2d 4647 . . . . . . . . 9  |-  ( ( n  =  N  /\  x  =  X )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { X } ) )
1615seqeq3d 10439 . . . . . . . 8  |-  ( ( n  =  N  /\  x  =  X )  ->  seq 1 (  .+  ,  ( NN  X.  { x } ) )  =  seq 1
(  .+  ,  ( NN  X.  { X }
) ) )
17 mulgval.s . . . . . . . 8  |-  S  =  seq 1 (  .+  ,  ( NN  X.  { X } ) )
1816, 17eqtr4di 2228 . . . . . . 7  |-  ( ( n  =  N  /\  x  =  X )  ->  seq 1 (  .+  ,  ( NN  X.  { x } ) )  =  S )
1918, 10fveq12d 5518 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq 1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
)  =  ( S `
 N ) )
2010negeqd 8142 . . . . . . . 8  |-  ( ( n  =  N  /\  x  =  X )  -> 
-u n  =  -u N )
2118, 20fveq12d 5518 . . . . . . 7  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq 1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  -u n
)  =  ( S `
 -u N ) )
2221fveq2d 5515 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  ->  ( I `  (  seq 1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) )  =  ( I `  ( S `  -u N
) ) )
2312, 19, 22ifbieq12d 3560 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  if ( 0  < 
n ,  (  seq 1 (  .+  , 
( NN  X.  {
x } ) ) `
 n ) ,  ( I `  (  seq 1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) ) )  =  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) ) )
2411, 23ifbieq2d 3558 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq 1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq 1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) )  =  if ( N  =  0 ,  .0.  ,  if ( 0  < 
N ,  ( S `
 N ) ,  ( I `  ( S `  -u N ) ) ) ) )
2524adantl 277 . . 3  |-  ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  (
n  =  N  /\  x  =  X )
)  ->  if (
n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq 1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq 1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) )  =  if ( N  =  0 ,  .0.  ,  if ( 0  < 
N ,  ( S `
 N ) ,  ( I `  ( S `  -u N ) ) ) ) )
26 simpll 527 . . 3  |-  ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  ->  N  e.  ZZ )
27 simplr 528 . . 3  |-  ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  ->  X  e.  B
)
28 fn0g 12686 . . . . . . 7  |-  0g  Fn  _V
29 funfvex 5528 . . . . . . . 8  |-  ( ( Fun  0g  /\  G  e.  dom  0g )  -> 
( 0g `  G
)  e.  _V )
3029funfni 5312 . . . . . . 7  |-  ( ( 0g  Fn  _V  /\  G  e.  _V )  ->  ( 0g `  G
)  e.  _V )
3128, 30mpan 424 . . . . . 6  |-  ( G  e.  _V  ->  ( 0g `  G )  e. 
_V )
325, 31eqeltrid 2264 . . . . 5  |-  ( G  e.  _V  ->  .0.  e.  _V )
3332ad2antlr 489 . . . 4  |-  ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  N  =  0 )  ->  .0.  e.  _V )
34 nnuz 9552 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
35 1zzd 9269 . . . . . . . . 9  |-  ( ( X  e.  B  /\  G  e.  _V )  ->  1  e.  ZZ )
36 fvconst2g 5726 . . . . . . . . . . . 12  |-  ( ( X  e.  B  /\  u  e.  NN )  ->  ( ( NN  X.  { X } ) `  u )  =  X )
37 simpl 109 . . . . . . . . . . . 12  |-  ( ( X  e.  B  /\  u  e.  NN )  ->  X  e.  B )
3836, 37eqeltrd 2254 . . . . . . . . . . 11  |-  ( ( X  e.  B  /\  u  e.  NN )  ->  ( ( NN  X.  { X } ) `  u )  e.  B
)
3938elexd 2750 . . . . . . . . . 10  |-  ( ( X  e.  B  /\  u  e.  NN )  ->  ( ( NN  X.  { X } ) `  u )  e.  _V )
4039adantlr 477 . . . . . . . . 9  |-  ( ( ( X  e.  B  /\  G  e.  _V )  /\  u  e.  NN )  ->  ( ( NN 
X.  { X }
) `  u )  e.  _V )
41 simprl 529 . . . . . . . . . 10  |-  ( ( ( X  e.  B  /\  G  e.  _V )  /\  ( u  e. 
_V  /\  v  e.  _V ) )  ->  u  e.  _V )
42 plusgslid 12551 . . . . . . . . . . . . 13  |-  ( +g  = Slot  ( +g  `  ndx )  /\  ( +g  `  ndx )  e.  NN )
4342slotex 12472 . . . . . . . . . . . 12  |-  ( G  e.  _V  ->  ( +g  `  G )  e. 
_V )
444, 43eqeltrid 2264 . . . . . . . . . . 11  |-  ( G  e.  _V  ->  .+  e.  _V )
4544ad2antlr 489 . . . . . . . . . 10  |-  ( ( ( X  e.  B  /\  G  e.  _V )  /\  ( u  e. 
_V  /\  v  e.  _V ) )  ->  .+  e.  _V )
46 simprr 531 . . . . . . . . . 10  |-  ( ( ( X  e.  B  /\  G  e.  _V )  /\  ( u  e. 
_V  /\  v  e.  _V ) )  ->  v  e.  _V )
47 ovexg 5903 . . . . . . . . . 10  |-  ( ( u  e.  _V  /\  .+  e.  _V  /\  v  e.  _V )  ->  (
u  .+  v )  e.  _V )
4841, 45, 46, 47syl3anc 1238 . . . . . . . . 9  |-  ( ( ( X  e.  B  /\  G  e.  _V )  /\  ( u  e. 
_V  /\  v  e.  _V ) )  ->  (
u  .+  v )  e.  _V )
4934, 35, 40, 48seqf 10447 . . . . . . . 8  |-  ( ( X  e.  B  /\  G  e.  _V )  ->  seq 1 (  .+  ,  ( NN  X.  { X } ) ) : NN --> _V )
5017feq1i 5354 . . . . . . . 8  |-  ( S : NN --> _V  <->  seq 1
(  .+  ,  ( NN  X.  { X }
) ) : NN --> _V )
5149, 50sylibr 134 . . . . . . 7  |-  ( ( X  e.  B  /\  G  e.  _V )  ->  S : NN --> _V )
5251ad5ant23 522 . . . . . 6  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  0  <  N
)  ->  S : NN
--> _V )
53 simp-4l 541 . . . . . . 7  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  0  <  N
)  ->  N  e.  ZZ )
54 simpr 110 . . . . . . 7  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  0  <  N
)  ->  0  <  N )
55 elnnz 9252 . . . . . . 7  |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  < 
N ) )
5653, 54, 55sylanbrc 417 . . . . . 6  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  0  <  N
)  ->  N  e.  NN )
5752, 56ffvelcdmd 5648 . . . . 5  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  0  <  N
)  ->  ( S `  N )  e.  _V )
581, 6grpinvfng 12807 . . . . . . . 8  |-  ( G  e.  _V  ->  I  Fn  B )
59 basfn 12502 . . . . . . . . . 10  |-  Base  Fn  _V
60 funfvex 5528 . . . . . . . . . . 11  |-  ( ( Fun  Base  /\  G  e. 
dom  Base )  ->  ( Base `  G )  e. 
_V )
6160funfni 5312 . . . . . . . . . 10  |-  ( (
Base  Fn  _V  /\  G  e.  _V )  ->  ( Base `  G )  e. 
_V )
6259, 61mpan 424 . . . . . . . . 9  |-  ( G  e.  _V  ->  ( Base `  G )  e. 
_V )
631, 62eqeltrid 2264 . . . . . . . 8  |-  ( G  e.  _V  ->  B  e.  _V )
64 fnex 5734 . . . . . . . 8  |-  ( ( I  Fn  B  /\  B  e.  _V )  ->  I  e.  _V )
6558, 63, 64syl2anc 411 . . . . . . 7  |-  ( G  e.  _V  ->  I  e.  _V )
6665ad3antlr 493 . . . . . 6  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  I  e.  _V )
6751ad5ant23 522 . . . . . . 7  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  S : NN --> _V )
68 znegcl 9273 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
6968ad4antr 494 . . . . . . . 8  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  -u N  e.  ZZ )
70 simplr 528 . . . . . . . . . 10  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  -.  N  =  0 )
71 simpr 110 . . . . . . . . . 10  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  -.  0  <  N )
72 ztri3or0 9284 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N  <  0  \/  N  =  0  \/  0  <  N ) )
7372ad4antr 494 . . . . . . . . . 10  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  ( N  <  0  \/  N  =  0  \/  0  <  N ) )
7470, 71, 73ecase23d 1350 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  N  <  0 )
75 zre 9246 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  RR )
7675ad4antr 494 . . . . . . . . . 10  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  N  e.  RR )
7776lt0neg1d 8462 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  ( N  <  0  <->  0  <  -u N ) )
7874, 77mpbid 147 . . . . . . . 8  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  0  <  -u N )
79 elnnz 9252 . . . . . . . 8  |-  ( -u N  e.  NN  <->  ( -u N  e.  ZZ  /\  0  <  -u N ) )
8069, 78, 79sylanbrc 417 . . . . . . 7  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  -u N  e.  NN )
8167, 80ffvelcdmd 5648 . . . . . 6  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  ( S `  -u N )  e.  _V )
82 fvexg 5530 . . . . . 6  |-  ( ( I  e.  _V  /\  ( S `  -u N
)  e.  _V )  ->  ( I `  ( S `  -u N ) )  e.  _V )
8366, 81, 82syl2anc 411 . . . . 5  |-  ( ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  /\  -.  0  < 
N )  ->  (
I `  ( S `  -u N ) )  e.  _V )
84 0zd 9254 . . . . . 6  |-  ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  ->  0  e.  ZZ )
85 simplll 533 . . . . . 6  |-  ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  ->  N  e.  ZZ )
86 zdclt 9319 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  -> DECID  0  <  N )
8784, 85, 86syl2anc 411 . . . . 5  |-  ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  -> DECID  0  <  N )
8857, 83, 87ifcldadc 3563 . . . 4  |-  ( ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  /\  -.  N  =  0 )  ->  if ( 0  <  N ,  ( S `  N ) ,  ( I `  ( S `  -u N
) ) )  e. 
_V )
89 0zd 9254 . . . . 5  |-  ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  ->  0  e.  ZZ )
90 zdceq 9317 . . . . 5  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
9126, 89, 90syl2anc 411 . . . 4  |-  ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  -> DECID 
N  =  0 )
9233, 88, 91ifcldadc 3563 . . 3  |-  ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  ->  if ( N  =  0 ,  .0.  ,  if ( 0  < 
N ,  ( S `
 N ) ,  ( I `  ( S `  -u N ) ) ) )  e. 
_V )
939, 25, 26, 27, 92ovmpod 5996 . 2  |-  ( ( ( N  e.  ZZ  /\  X  e.  B )  /\  G  e.  _V )  ->  ( N  .x.  X )  =  if ( N  =  0 ,  .0.  ,  if ( 0  <  N ,  ( S `  N ) ,  ( I `  ( S `
 -u N ) ) ) ) )
943, 93mpdan 421 1  |-  ( ( N  e.  ZZ  /\  X  e.  B )  ->  ( N  .x.  X
)  =  if ( N  =  0 ,  .0.  ,  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104  DECID wdc 834    \/ w3o 977    = wceq 1353    e. wcel 2148   _Vcvv 2737   ifcif 3534   {csn 3591   class class class wbr 4000    X. cxp 4621    Fn wfn 5207   -->wf 5208   ` cfv 5212  (class class class)co 5869    e. cmpo 5871   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:  mulg0  12877  mulgnn  12878  mulgnegnn  12882
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