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Theorem pwssub 16309
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 755 . . . 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 2457 . . . . . 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 756 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  F  e.  B )
72, 3, 4, 5, 1, 6pwselbas 14905 . . . . 5  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  F : I --> ( Base `  R ) )
87ffvelrnda 6032 . . . 4  |-  ( ( ( ( R  e. 
Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B )
)  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  R
) )
9 fvex 5882 . . . . 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 5926 . . . 4  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
12 simprr 757 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  G  e.  B )
13 eqid 2457 . . . . . . 7  |-  ( invg `  R )  =  ( invg `  R )
14 eqid 2457 . . . . . . 7  |-  ( invg `  Y )  =  ( invg `  Y )
152, 4, 13, 14pwsinvg 16308 . . . . . 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 14905 . . . . . . 7  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  G : I --> ( Base `  R ) )
1817ffvelrnda 6032 . . . . . 6  |-  ( ( ( ( R  e. 
Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B )
)  /\  x  e.  I )  ->  ( G `  x )  e.  ( Base `  R
) )
1917feqmptd 5926 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  ->  G  =  ( x  e.  I  |->  ( G `
 x ) ) )
203, 13grpinvf 16220 . . . . . . . 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 5926 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( invg `  R )  =  ( y  e.  ( Base `  R )  |->  ( ( invg `  R
) `  y )
) )
23 fveq2 5872 . . . . . 6  |-  ( y  =  ( G `  x )  ->  (
( invg `  R ) `  y
)  =  ( ( invg `  R
) `  ( G `  x ) ) )
2418, 19, 22, 23fmptco 6065 . . . . 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 2498 . . . 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 6555 . . 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 16307 . . . . . 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 16221 . . . . 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 2457 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
32 eqid 2457 . . . 4  |-  ( +g  `  Y )  =  ( +g  `  Y )
332, 4, 5, 1, 6, 30, 31, 32pwsplusgval 14906 . . 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 16219 . . . . 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 4543 . . 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 2508 . 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 16219 . . 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 6555 . 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 2508 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 1395    e. wcel 1819   _Vcvv 3109    |-> cmpt 4515    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537   Basecbs 14643   +g cplusg 14711    ^s cpws 14863   Grpcgrp 16179   invgcminusg 16180   -gcsg 16181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-hom 14735  df-cco 14736  df-0g 14858  df-prds 14864  df-pws 14866  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183  df-minusg 16184  df-sbg 16185
This theorem is referenced by:  evl1subd  18504  frlmsubgval  18924
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