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Theorem pwsinvg 15666
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  =  ( invg `  R )
pwsinvg.n  |-  N  =  ( invg `  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 2442 . . . 4  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
2 simp2 989 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  I  e.  V )
3 fvex 5700 . . . . 5  |-  (Scalar `  R )  e.  _V
43a1i 11 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  (Scalar `  R )  e.  _V )
5 simp1 988 . . . . 5  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  R  e.  Grp )
6 fconst6g 5598 . . . . 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 2442 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  =  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
9 eqid 2442 . . . 4  |-  ( invg `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  ( invg `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )
10 simp3 990 . . . . 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 2442 . . . . . . . . 9  |-  (Scalar `  R )  =  (Scalar `  R )
1412, 13pwsval 14423 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  I  e.  V )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
15143adant3 1008 . . . . . . 7  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
1615fveq2d 5694 . . . . . 6  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( Base `  Y
)  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
1711, 16syl5eq 2486 . . . . 5  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  B  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
1810, 17eleqtrd 2518 . . . 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 15664 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( ( invg `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) `
 X )  =  ( x  e.  I  |->  ( ( invg `  ( ( I  X.  { R } ) `  x ) ) `  ( X `  x ) ) ) )
20 fvconst2g 5930 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  x  e.  I )  ->  ( ( I  X.  { R } ) `  x )  =  R )
215, 20sylan 471 . . . . . . 7  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( (
I  X.  { R } ) `  x
)  =  R )
2221fveq2d 5694 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( invg `  ( (
I  X.  { R } ) `  x
) )  =  ( invg `  R
) )
23 pwsinvg.m . . . . . 6  |-  M  =  ( invg `  R )
2422, 23syl6eqr 2492 . . . . 5  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( invg `  ( (
I  X.  { R } ) `  x
) )  =  M )
2524fveq1d 5692 . . . 4  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( ( invg `  ( ( I  X.  { R } ) `  x
) ) `  ( X `  x )
)  =  ( M `
 ( X `  x ) ) )
2625mpteq2dva 4377 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( x  e.  I  |->  ( ( invg `  ( ( I  X.  { R } ) `  x ) ) `  ( X `  x ) ) )  =  ( x  e.  I  |->  ( M `  ( X `
 x ) ) ) )
2719, 26eqtrd 2474 . 2  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( ( invg `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) `
 X )  =  ( x  e.  I  |->  ( M `  ( X `  x )
) ) )
28 pwsinvg.n . . . 4  |-  N  =  ( invg `  Y )
2915fveq2d 5694 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( invg `  Y )  =  ( invg `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
3028, 29syl5eq 2486 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  N  =  ( invg `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
3130fveq1d 5692 . 2  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( N `  X
)  =  ( ( invg `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) ) `
 X ) )
32 eqid 2442 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
3312, 32, 11, 5, 2, 10pwselbas 14426 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  X : I --> ( Base `  R ) )
3433ffvelrnda 5842 . . 3  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( X `  x )  e.  (
Base `  R )
)
3533feqmptd 5743 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  X  =  ( x  e.  I  |->  ( X `
 x ) ) )
3632, 23grpinvf 15581 . . . . 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 5743 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  M  =  ( y  e.  ( Base `  R
)  |->  ( M `  y ) ) )
39 fveq2 5690 . . 3  |-  ( y  =  ( X `  x )  ->  ( M `  y )  =  ( M `  ( X `  x ) ) )
4034, 35, 38, 39fmptco 5875 . 2  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( M  o.  X
)  =  ( x  e.  I  |->  ( M `
 ( X `  x ) ) ) )
4127, 31, 403eqtr4d 2484 1  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( N `  X
)  =  ( M  o.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2971   {csn 3876    e. cmpt 4349    X. cxp 4837    o. ccom 4843   -->wf 5413   ` cfv 5417  (class class class)co 6090   Basecbs 14173  Scalarcsca 14240   X_scprds 14383    ^s cpws 14384   Grpcgrp 15409   invgcminusg 15410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-fz 11437  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-plusg 14250  df-mulr 14251  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-hom 14261  df-cco 14262  df-0g 14379  df-prds 14385  df-pws 14387  df-mnd 15414  df-grp 15544  df-minusg 15545
This theorem is referenced by:  pwssub  15667
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