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Theorem pwsinvg 15977
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 2462 . . . 4  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
2 simp2 992 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  I  e.  V )
3 fvex 5869 . . . . 5  |-  (Scalar `  R )  e.  _V
43a1i 11 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  (Scalar `  R )  e.  _V )
5 simp1 991 . . . . 5  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  R  e.  Grp )
6 fconst6g 5767 . . . . 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 2462 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  =  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
9 eqid 2462 . . . 4  |-  ( invg `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  ( invg `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )
10 simp3 993 . . . . 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 2462 . . . . . . . . 9  |-  (Scalar `  R )  =  (Scalar `  R )
1412, 13pwsval 14732 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  I  e.  V )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
15143adant3 1011 . . . . . . 7  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
1615fveq2d 5863 . . . . . 6  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( Base `  Y
)  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
1711, 16syl5eq 2515 . . . . 5  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  B  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
1810, 17eleqtrd 2552 . . . 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 15975 . . 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 6107 . . . . . . . 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 5863 . . . . . 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 2521 . . . . 5  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( invg `  ( (
I  X.  { R } ) `  x
) )  =  M )
2524fveq1d 5861 . . . 4  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( ( invg `  ( ( I  X.  { R } ) `  x
) ) `  ( X `  x )
)  =  ( M `
 ( X `  x ) ) )
2625mpteq2dva 4528 . . 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 2503 . 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 5863 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( invg `  Y )  =  ( invg `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
3028, 29syl5eq 2515 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  N  =  ( invg `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
3130fveq1d 5861 . 2  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( N `  X
)  =  ( ( invg `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) ) `
 X ) )
32 eqid 2462 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
3312, 32, 11, 5, 2, 10pwselbas 14735 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  X : I --> ( Base `  R ) )
3433ffvelrnda 6014 . . 3  |-  ( ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  /\  x  e.  I
)  ->  ( X `  x )  e.  (
Base `  R )
)
3533feqmptd 5913 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  X  =  ( x  e.  I  |->  ( X `
 x ) ) )
3632, 23grpinvf 15890 . . . . 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 5913 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  M  =  ( y  e.  ( Base `  R
)  |->  ( M `  y ) ) )
39 fveq2 5859 . . 3  |-  ( y  =  ( X `  x )  ->  ( M `  y )  =  ( M `  ( X `  x ) ) )
4034, 35, 38, 39fmptco 6047 . 2  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B )  ->  ( M  o.  X
)  =  ( x  e.  I  |->  ( M `
 ( X `  x ) ) ) )
4127, 31, 403eqtr4d 2513 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 968    = wceq 1374    e. wcel 1762   _Vcvv 3108   {csn 4022    |-> cmpt 4500    X. cxp 4992    o. ccom 4998   -->wf 5577   ` cfv 5581  (class class class)co 6277   Basecbs 14481  Scalarcsca 14549   X_scprds 14692    ^s cpws 14693   Grpcgrp 15718   invgcminusg 15719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-plusg 14559  df-mulr 14560  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-hom 14570  df-cco 14571  df-0g 14688  df-prds 14694  df-pws 14696  df-mnd 15723  df-grp 15853  df-minusg 15854
This theorem is referenced by:  pwssub  15978
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