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Theorem pwsinvg 14885
Description: Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.)
Hypotheses
Ref Expression
pwsgrp.y  |-  Y  =  ( R  ^s  I )
pwsinvg.b  |-  B  =  ( Base `  Y
)
pwsinvg.m  |-  M  =  ( inv g `  R )
pwsinvg.n  |-  N  =  ( inv g `  Y )
Assertion
Ref Expression
pwsinvg  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( N `  X
)  =  ( M  o.  X ) )

Proof of Theorem pwsinvg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . 4  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
2 simp2 958 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  I  e.  V )
3 fvex 5701 . . . . 5  |-  (Scalar `  R )  e.  _V
43a1i 11 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  (Scalar `  R )  e.  _V )
5 simp1 957 . . . . 5  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  R  e.  Grp )
6 fconst6g 5591 . . . . 5  |-  ( R  e.  Grp  ->  (
I  X.  { R } ) : I --> Grp )
75, 6syl 16 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( I  X.  { R } ) : I --> Grp )
8 eqid 2404 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  =  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
9 eqid 2404 . . . 4  |-  ( inv g `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  ( inv g `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )
10 simp3 959 . . . . 5  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  X  e.  B )
11 pwsinvg.b . . . . . 6  |-  B  =  ( Base `  Y
)
12 pwsgrp.y . . . . . . . . 9  |-  Y  =  ( R  ^s  I )
13 eqid 2404 . . . . . . . . 9  |-  (Scalar `  R )  =  (Scalar `  R )
1412, 13pwsval 13663 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  I  e.  V )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
15143adant3 977 . . . . . . 7  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
1615fveq2d 5691 . . . . . 6  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( Base `  Y
)  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
1711, 16syl5eq 2448 . . . . 5  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  B  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
1810, 17eleqtrd 2480 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  X  e.  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
191, 2, 4, 7, 8, 9, 18prdsinvgd 14883 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( ( inv g `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) `
 X )  =  ( x  e.  I  |->  ( ( inv g `  ( ( I  X.  { R } ) `  x ) ) `  ( X `  x ) ) ) )
20 fvconst2g 5904 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  x  e.  I )  ->  ( ( I  X.  { R } ) `  x )  =  R )
215, 20sylan 458 . . . . . . 7  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( (
I  X.  { R } ) `  x
)  =  R )
2221fveq2d 5691 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( inv g `  ( (
I  X.  { R } ) `  x
) )  =  ( inv g `  R
) )
23 pwsinvg.m . . . . . 6  |-  M  =  ( inv g `  R )
2422, 23syl6eqr 2454 . . . . 5  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( inv g `  ( (
I  X.  { R } ) `  x
) )  =  M )
2524fveq1d 5689 . . . 4  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( ( inv g `  ( ( I  X.  { R } ) `  x
) ) `  ( X `  x )
)  =  ( M `
 ( X `  x ) ) )
2625mpteq2dva 4255 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( x  e.  I  |->  ( ( inv g `  ( ( I  X.  { R } ) `  x ) ) `  ( X `  x ) ) )  =  ( x  e.  I  |->  ( M `  ( X `
 x ) ) ) )
2719, 26eqtrd 2436 . 2  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( ( inv g `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) `
 X )  =  ( x  e.  I  |->  ( M `  ( X `  x )
) ) )
28 pwsinvg.n . . . 4  |-  N  =  ( inv g `  Y )
2915fveq2d 5691 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( inv g `  Y )  =  ( inv g `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
3028, 29syl5eq 2448 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  N  =  ( inv g `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
3130fveq1d 5689 . 2  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( N `  X
)  =  ( ( inv g `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) ) `
 X ) )
32 eqid 2404 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
3312, 32, 11, 5, 2, 10pwselbas 13666 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  X : I --> ( Base `  R ) )
3433ffvelrnda 5829 . . 3  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( X `  x )  e.  (
Base `  R )
)
3533feqmptd 5738 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  X  =  ( x  e.  I  |->  ( X `
 x ) ) )
3632, 23grpinvf 14804 . . . . 5  |-  ( R  e.  Grp  ->  M : ( Base `  R
) --> ( Base `  R
) )
375, 36syl 16 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  M : ( Base `  R ) --> ( Base `  R ) )
3837feqmptd 5738 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  M  =  ( y  e.  ( Base `  R
)  |->  ( M `  y ) ) )
39 fveq2 5687 . . 3  |-  ( y  =  ( X `  x )  ->  ( M `  y )  =  ( M `  ( X `  x ) ) )
4034, 35, 38, 39fmptco 5860 . 2  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( M  o.  X
)  =  ( x  e.  I  |->  ( M `
 ( X `  x ) ) ) )
4127, 31, 403eqtr4d 2446 1  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( N `  X
)  =  ( M  o.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   {csn 3774    e. cmpt 4226    X. cxp 4835    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424  Scalarcsca 13487   X_scprds 13624    ^s cpws 13625   Grpcgrp 14640   inv gcminusg 14641
This theorem is referenced by:  pwssub  14886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-hom 13508  df-cco 13509  df-prds 13626  df-pws 13628  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768
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