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Theorem pwssub 15980
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 2467 . . . . . 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 14737 . . . . 5  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  F : I --> ( Base `  R ) )
87ffvelrnda 6019 . . . 4  |-  ( ( ( ( R  e. 
Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B )
)  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  R
) )
9 fvex 5874 . . . . 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 5918 . . . 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 2467 . . . . . . 7  |-  ( invg `  R )  =  ( invg `  R )
14 eqid 2467 . . . . . . 7  |-  ( invg `  Y )  =  ( invg `  Y )
152, 4, 13, 14pwsinvg 15979 . . . . . 6  |-  ( ( R  e.  Grp  /\  I  e.  V  /\  G  e.  B )  ->  ( ( invg `  Y ) `  G
)  =  ( ( invg `  R
)  o.  G ) )
165, 1, 12, 15syl3anc 1228 . . . . 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 14737 . . . . . . 7  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  G : I --> ( Base `  R ) )
1817ffvelrnda 6019 . . . . . 6  |-  ( ( ( ( R  e. 
Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B )
)  /\  x  e.  I )  ->  ( G `  x )  e.  ( Base `  R
) )
1917feqmptd 5918 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  G  =  ( x  e.  I  |->  ( G `
 x ) ) )
203, 13grpinvf 15892 . . . . . . . 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 5918 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( invg `  R )  =  ( y  e.  ( Base `  R )  |->  ( ( invg `  R
) `  y )
) )
23 fveq2 5864 . . . . . 6  |-  ( y  =  ( G `  x )  ->  (
( invg `  R ) `  y
)  =  ( ( invg `  R
) `  ( G `  x ) ) )
2418, 19, 22, 23fmptco 6052 . . . . 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 2508 . . . 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 6538 . . 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 15978 . . . . . 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 15893 . . . . 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 2467 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
32 eqid 2467 . . . 4  |-  ( +g  `  Y )  =  ( +g  `  Y )
332, 4, 5, 1, 6, 30, 31, 32pwsplusgval 14738 . . 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 15891 . . . . 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 4533 . . 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 2518 . 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 15891 . . 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 6538 . 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 2518 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 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505    o. ccom 5003   -->wf 5582   ` cfv 5586  (class class class)co 6282    oFcof 6520   Basecbs 14483   +g cplusg 14548    ^s cpws 14695   Grpcgrp 15720   invgcminusg 15721   -gcsg 15723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-plusg 14561  df-mulr 14562  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-hom 14572  df-cco 14573  df-0g 14690  df-prds 14696  df-pws 14698  df-mnd 15725  df-grp 15855  df-minusg 15856  df-sbg 15857
This theorem is referenced by:  evl1subd  18146  frlmsubgval  18562
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