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Theorem mulgfvalg 12985
Description: 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 )
Assertion
Ref Expression
mulgfvalg  |-  ( 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
) ) ) ) ) )
Distinct variable groups:    x,  .0. , n    x, B, n    x,  .+ , n    x, G, n    x, I, n
Allowed substitution hints:    .x. ( x, n)    V( x, n)

Proof of Theorem mulgfvalg
Dummy variables  w  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulgval.t . 2  |-  .x.  =  (.g
`  G )
2 df-mulg 12984 . . 3  |- .g  =  (
w  e.  _V  |->  ( n  e.  ZZ ,  x  e.  ( Base `  w )  |->  if ( n  =  0 ,  ( 0g `  w
) ,  [_  seq 1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  / 
s ]_ if ( 0  <  n ,  ( s `  n ) ,  ( ( invg `  w ) `
 ( s `  -u n ) ) ) ) ) )
3 eqidd 2178 . . . 4  |-  ( w  =  G  ->  ZZ  =  ZZ )
4 fveq2 5516 . . . . 5  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
5 mulgval.b . . . . 5  |-  B  =  ( Base `  G
)
64, 5eqtr4di 2228 . . . 4  |-  ( w  =  G  ->  ( Base `  w )  =  B )
7 fveq2 5516 . . . . . 6  |-  ( w  =  G  ->  ( 0g `  w )  =  ( 0g `  G
) )
8 mulgval.o . . . . . 6  |-  .0.  =  ( 0g `  G )
97, 8eqtr4di 2228 . . . . 5  |-  ( w  =  G  ->  ( 0g `  w )  =  .0.  )
10 seqex 10447 . . . . . . 7  |-  seq 1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) )  e.  _V
1110a1i 9 . . . . . 6  |-  ( w  =  G  ->  seq 1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  e. 
_V )
12 id 19 . . . . . . . . 9  |-  ( s  =  seq 1 ( ( +g  `  w
) ,  ( NN 
X.  { x }
) )  ->  s  =  seq 1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) ) )
13 fveq2 5516 . . . . . . . . . . 11  |-  ( w  =  G  ->  ( +g  `  w )  =  ( +g  `  G
) )
14 mulgval.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  G )
1513, 14eqtr4di 2228 . . . . . . . . . 10  |-  ( w  =  G  ->  ( +g  `  w )  = 
.+  )
1615seqeq2d 10452 . . . . . . . . 9  |-  ( w  =  G  ->  seq 1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  =  seq 1 (  .+  ,  ( NN  X.  { x } ) ) )
1712, 16sylan9eqr 2232 . . . . . . . 8  |-  ( ( w  =  G  /\  s  =  seq 1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
s  =  seq 1
(  .+  ,  ( NN  X.  { x }
) ) )
1817fveq1d 5518 . . . . . . 7  |-  ( ( w  =  G  /\  s  =  seq 1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( s `  n
)  =  (  seq 1 (  .+  , 
( NN  X.  {
x } ) ) `
 n ) )
19 simpl 109 . . . . . . . . . 10  |-  ( ( w  =  G  /\  s  =  seq 1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  ->  w  =  G )
2019fveq2d 5520 . . . . . . . . 9  |-  ( ( w  =  G  /\  s  =  seq 1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( invg `  w )  =  ( invg `  G
) )
21 mulgval.i . . . . . . . . 9  |-  I  =  ( invg `  G )
2220, 21eqtr4di 2228 . . . . . . . 8  |-  ( ( w  =  G  /\  s  =  seq 1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( invg `  w )  =  I )
2317fveq1d 5518 . . . . . . . 8  |-  ( ( w  =  G  /\  s  =  seq 1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( s `  -u n
)  =  (  seq 1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) )
2422, 23fveq12d 5523 . . . . . . 7  |-  ( ( w  =  G  /\  s  =  seq 1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( ( invg `  w ) `  (
s `  -u n ) )  =  ( I `
 (  seq 1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) )
2518, 24ifeq12d 3554 . . . . . 6  |-  ( ( w  =  G  /\  s  =  seq 1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  ->  if ( 0  <  n ,  ( s `  n ) ,  ( ( invg `  w ) `  (
s `  -u n ) ) )  =  if ( 0  <  n ,  (  seq 1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq 1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) )
2611, 25csbied 3104 . . . . 5  |-  ( w  =  G  ->  [_  seq 1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  / 
s ]_ if ( 0  <  n ,  ( s `  n ) ,  ( ( invg `  w ) `
 ( s `  -u n ) ) )  =  if ( 0  <  n ,  (  seq 1 (  .+  ,  ( NN  X.  { x } ) ) `  n ) ,  ( I `  (  seq 1 (  .+  ,  ( NN  X.  { x } ) ) `  -u n
) ) ) )
279, 26ifeq12d 3554 . . . 4  |-  ( w  =  G  ->  if ( n  =  0 ,  ( 0g `  w ) ,  [_  seq 1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  / 
s ]_ if ( 0  <  n ,  ( s `  n ) ,  ( ( invg `  w ) `
 ( s `  -u n ) ) ) )  =  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq 1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq 1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )
283, 6, 27mpoeq123dv 5937 . . 3  |-  ( w  =  G  ->  (
n  e.  ZZ ,  x  e.  ( Base `  w )  |->  if ( n  =  0 ,  ( 0g `  w
) ,  [_  seq 1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  / 
s ]_ if ( 0  <  n ,  ( s `  n ) ,  ( ( invg `  w ) `
 ( s `  -u n ) ) ) ) )  =  ( 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
) ) ) ) ) )
29 elex 2749 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
30 zex 9262 . . . 4  |-  ZZ  e.  _V
31 basfn 12520 . . . . . 6  |-  Base  Fn  _V
32 funfvex 5533 . . . . . . 7  |-  ( ( Fun  Base  /\  G  e. 
dom  Base )  ->  ( Base `  G )  e. 
_V )
3332funfni 5317 . . . . . 6  |-  ( (
Base  Fn  _V  /\  G  e.  _V )  ->  ( Base `  G )  e. 
_V )
3431, 29, 33sylancr 414 . . . . 5  |-  ( G  e.  V  ->  ( Base `  G )  e. 
_V )
355, 34eqeltrid 2264 . . . 4  |-  ( G  e.  V  ->  B  e.  _V )
36 mpoexga 6213 . . . 4  |-  ( ( ZZ  e.  _V  /\  B  e.  _V )  ->  ( 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
) ) ) ) )  e.  _V )
3730, 35, 36sylancr 414 . . 3  |-  ( G  e.  V  ->  (
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
) ) ) ) )  e.  _V )
382, 28, 29, 37fvmptd3 5610 . 2  |-  ( G  e.  V  ->  (.g `  G )  =  ( 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
) ) ) ) ) )
391, 38eqtrid 2222 1  |-  ( 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
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2738   [_csb 3058   ifcif 3535   {csn 3593   class class class wbr 4004    X. cxp 4625    Fn wfn 5212   ` cfv 5217    e. cmpo 5877   0cc0 7811   1c1 7812    < clt 7992   -ucneg 8129   NNcn 8919   ZZcz 9253    seqcseq 10445   Basecbs 12462   +g cplusg 12536   0gc0g 12705   invgcminusg 12878  .gcmg 12983
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-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 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-neg 8131  df-inn 8920  df-z 9254  df-seqfrec 10446  df-ndx 12465  df-slot 12466  df-base 12468  df-mulg 12984
This theorem is referenced by:  mulgval  12986  mulgfng  12987  mulgpropdg  13025
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