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Theorem pwssub 15666
Description: Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.)
Hypotheses
Ref Expression
pwsgrp.y  |-  Y  =  ( R  ^s  I )
pwsinvg.b  |-  B  =  ( Base `  Y
)
pwssub.m  |-  M  =  ( -g `  R
)
pwssub.n  |-  .-  =  ( -g `  Y )
Assertion
Ref Expression
pwssub  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( F  .-  G
)  =  ( F  oF M G ) )

Proof of Theorem pwssub
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 754 . . . 4  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  I  e.  V )
2 pwsgrp.y . . . . . 6  |-  Y  =  ( R  ^s  I )
3 eqid 2441 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
4 pwsinvg.b . . . . . 6  |-  B  =  ( Base `  Y
)
5 simpll 753 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  R  e.  Grp )
6 simprl 755 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  F  e.  B )
72, 3, 4, 5, 1, 6pwselbas 14425 . . . . 5  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  F : I --> ( Base `  R ) )
87ffvelrnda 5841 . . . 4  |-  ( ( ( ( R  e. 
Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B )
)  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  R
) )
9 fvex 5699 . . . . 5  |-  ( ( invg `  R
) `  ( G `  x ) )  e. 
_V
109a1i 11 . . . 4  |-  ( ( ( ( R  e. 
Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B )
)  /\  x  e.  I )  ->  (
( invg `  R ) `  ( G `  x )
)  e.  _V )
117feqmptd 5742 . . . 4  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
12 simprr 756 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  G  e.  B )
13 eqid 2441 . . . . . . 7  |-  ( invg `  R )  =  ( invg `  R )
14 eqid 2441 . . . . . . 7  |-  ( invg `  Y )  =  ( invg `  Y )
152, 4, 13, 14pwsinvg 15665 . . . . . 6  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  G  e.  B )  ->  ( ( invg `  Y ) `  G
)  =  ( ( invg `  R
)  o.  G ) )
165, 1, 12, 15syl3anc 1218 . . . . 5  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( invg `  Y ) `  G
)  =  ( ( invg `  R
)  o.  G ) )
172, 3, 4, 5, 1, 12pwselbas 14425 . . . . . . 7  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  G : I --> ( Base `  R ) )
1817ffvelrnda 5841 . . . . . 6  |-  ( ( ( ( R  e. 
Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B )
)  /\  x  e.  I )  ->  ( G `  x )  e.  ( Base `  R
) )
1917feqmptd 5742 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  G  =  ( x  e.  I  |->  ( G `
 x ) ) )
203, 13grpinvf 15580 . . . . . . . 8  |-  ( R  e.  Grp  ->  ( invg `  R ) : ( Base `  R
) --> ( Base `  R
) )
2120ad2antrr 725 . . . . . . 7  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( invg `  R ) : (
Base `  R ) --> ( Base `  R )
)
2221feqmptd 5742 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( invg `  R )  =  ( y  e.  ( Base `  R )  |->  ( ( invg `  R
) `  y )
) )
23 fveq2 5689 . . . . . 6  |-  ( y  =  ( G `  x )  ->  (
( invg `  R ) `  y
)  =  ( ( invg `  R
) `  ( G `  x ) ) )
2418, 19, 22, 23fmptco 5874 . . . . 5  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( invg `  R )  o.  G
)  =  ( x  e.  I  |->  ( ( invg `  R
) `  ( G `  x ) ) ) )
2516, 24eqtrd 2473 . . . 4  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( invg `  Y ) `  G
)  =  ( x  e.  I  |->  ( ( invg `  R
) `  ( G `  x ) ) ) )
261, 8, 10, 11, 25offval2 6334 . . 3  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( F  oF ( +g  `  R
) ( ( invg `  Y ) `
 G ) )  =  ( x  e.  I  |->  ( ( F `
 x ) ( +g  `  R ) ( ( invg `  R ) `  ( G `  x )
) ) ) )
272pwsgrp 15664 . . . . . 6  |-  ( ( R  e.  Grp  /\  I  e.  V )  ->  Y  e.  Grp )
2827adantr 465 . . . . 5  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  Y  e.  Grp )
294, 14grpinvcl 15581 . . . . 5  |-  ( ( Y  e.  Grp  /\  G  e.  B )  ->  ( ( invg `  Y ) `  G
)  e.  B )
3028, 12, 29syl2anc 661 . . . 4  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( invg `  Y ) `  G
)  e.  B )
31 eqid 2441 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
32 eqid 2441 . . . 4  |-  ( +g  `  Y )  =  ( +g  `  Y )
332, 4, 5, 1, 6, 30, 31, 32pwsplusgval 14426 . . 3  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( F ( +g  `  Y ) ( ( invg `  Y
) `  G )
)  =  ( F  oF ( +g  `  R ) ( ( invg `  Y
) `  G )
) )
34 pwssub.m . . . . . 6  |-  M  =  ( -g `  R
)
353, 31, 13, 34grpsubval 15579 . . . . 5  |-  ( ( ( F `  x
)  e.  ( Base `  R )  /\  ( G `  x )  e.  ( Base `  R
) )  ->  (
( F `  x
) M ( G `
 x ) )  =  ( ( F `
 x ) ( +g  `  R ) ( ( invg `  R ) `  ( G `  x )
) ) )
368, 18, 35syl2anc 661 . . . 4  |-  ( ( ( ( R  e. 
Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B )
)  /\  x  e.  I )  ->  (
( F `  x
) M ( G `
 x ) )  =  ( ( F `
 x ) ( +g  `  R ) ( ( invg `  R ) `  ( G `  x )
) ) )
3736mpteq2dva 4376 . . 3  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( x  e.  I  |->  ( ( F `  x ) M ( G `  x ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) ( +g  `  R
) ( ( invg `  R ) `
 ( G `  x ) ) ) ) )
3826, 33, 373eqtr4d 2483 . 2  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( F ( +g  `  Y ) ( ( invg `  Y
) `  G )
)  =  ( x  e.  I  |->  ( ( F `  x ) M ( G `  x ) ) ) )
39 pwssub.n . . . 4  |-  .-  =  ( -g `  Y )
404, 32, 14, 39grpsubval 15579 . . 3  |-  ( ( F  e.  B  /\  G  e.  B )  ->  ( F  .-  G
)  =  ( F ( +g  `  Y
) ( ( invg `  Y ) `
 G ) ) )
4140adantl 466 . 2  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( F  .-  G
)  =  ( F ( +g  `  Y
) ( ( invg `  Y ) `
 G ) ) )
421, 8, 18, 11, 19offval2 6334 . 2  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( F  oF M G )  =  ( x  e.  I  |->  ( ( F `  x ) M ( G `  x ) ) ) )
4338, 41, 423eqtr4d 2483 1  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( F  .-  G
)  =  ( F  oF M G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2970    e. cmpt 4348    o. ccom 4842   -->wf 5412   ` cfv 5416  (class class class)co 6089    oFcof 6316   Basecbs 14172   +g cplusg 14236    ^s cpws 14383   Grpcgrp 15408   invgcminusg 15409   -gcsg 15411
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-fz 11436  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-plusg 14249  df-mulr 14250  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-hom 14260  df-cco 14261  df-0g 14378  df-prds 14384  df-pws 14386  df-mnd 15413  df-grp 15543  df-minusg 15544  df-sbg 15545
This theorem is referenced by:  evl1subd  17774  frlmsubgval  18190
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