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Theorem mulgsubdi 16322
Description: Group multiple of a difference. (Contributed by Mario Carneiro, 13-Dec-2014.)
Hypotheses
Ref Expression
mulgsubdi.b  |-  B  =  ( Base `  G
)
mulgsubdi.t  |-  .x.  =  (.g
`  G )
mulgsubdi.d  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
mulgsubdi  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( M  .x.  ( X  .-  Y ) )  =  ( ( M 
.x.  X )  .-  ( M  .x.  Y ) ) )

Proof of Theorem mulgsubdi
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  G  e.  Abel )
2 simpr1 994 . . . 4  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  M  e.  ZZ )
3 simpr2 995 . . . 4  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
4 ablgrp 16287 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
54adantr 465 . . . . 5  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  G  e.  Grp )
6 simpr3 996 . . . . 5  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
7 mulgsubdi.b . . . . . 6  |-  B  =  ( Base `  G
)
8 eqid 2443 . . . . . 6  |-  ( invg `  G )  =  ( invg `  G )
97, 8grpinvcl 15588 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
105, 6, 9syl2anc 661 . . . 4  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( ( invg `  G ) `  Y
)  e.  B )
11 mulgsubdi.t . . . . 5  |-  .x.  =  (.g
`  G )
12 eqid 2443 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
137, 11, 12mulgdi 16319 . . . 4  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  (
( invg `  G ) `  Y
)  e.  B ) )  ->  ( M  .x.  ( X ( +g  `  G ) ( ( invg `  G
) `  Y )
) )  =  ( ( M  .x.  X
) ( +g  `  G
) ( M  .x.  ( ( invg `  G ) `  Y
) ) ) )
141, 2, 3, 10, 13syl13anc 1220 . . 3  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( M  .x.  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) ) )  =  ( ( M  .x.  X ) ( +g  `  G
) ( M  .x.  ( ( invg `  G ) `  Y
) ) ) )
157, 11, 8mulgneg2 15659 . . . . . 6  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  Y  e.  B )  ->  ( -u M  .x.  Y )  =  ( M  .x.  ( ( invg `  G ) `  Y
) ) )
167, 11, 8mulgneg 15650 . . . . . 6  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  Y  e.  B )  ->  ( -u M  .x.  Y )  =  ( ( invg `  G ) `
 ( M  .x.  Y ) ) )
1715, 16eqtr3d 2477 . . . . 5  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  Y  e.  B )  ->  ( M  .x.  ( ( invg `  G ) `
 Y ) )  =  ( ( invg `  G ) `
 ( M  .x.  Y ) ) )
185, 2, 6, 17syl3anc 1218 . . . 4  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( M  .x.  (
( invg `  G ) `  Y
) )  =  ( ( invg `  G ) `  ( M  .x.  Y ) ) )
1918oveq2d 6112 . . 3  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( ( M  .x.  X ) ( +g  `  G ) ( M 
.x.  ( ( invg `  G ) `
 Y ) ) )  =  ( ( M  .x.  X ) ( +g  `  G
) ( ( invg `  G ) `
 ( M  .x.  Y ) ) ) )
2014, 19eqtrd 2475 . 2  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( M  .x.  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) ) )  =  ( ( M  .x.  X ) ( +g  `  G
) ( ( invg `  G ) `
 ( M  .x.  Y ) ) ) )
21 mulgsubdi.d . . . . 5  |-  .-  =  ( -g `  G )
227, 12, 8, 21grpsubval 15586 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) ) )
233, 6, 22syl2anc 661 . . 3  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) ) )
2423oveq2d 6112 . 2  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( M  .x.  ( X  .-  Y ) )  =  ( M  .x.  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) ) ) )
257, 11mulgcl 15649 . . . 4  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  X  e.  B )  ->  ( M  .x.  X )  e.  B )
265, 2, 3, 25syl3anc 1218 . . 3  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( M  .x.  X
)  e.  B )
277, 11mulgcl 15649 . . . 4  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  Y  e.  B )  ->  ( M  .x.  Y )  e.  B )
285, 2, 6, 27syl3anc 1218 . . 3  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( M  .x.  Y
)  e.  B )
297, 12, 8, 21grpsubval 15586 . . 3  |-  ( ( ( M  .x.  X
)  e.  B  /\  ( M  .x.  Y )  e.  B )  -> 
( ( M  .x.  X )  .-  ( M  .x.  Y ) )  =  ( ( M 
.x.  X ) ( +g  `  G ) ( ( invg `  G ) `  ( M  .x.  Y ) ) ) )
3026, 28, 29syl2anc 661 . 2  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( ( M  .x.  X )  .-  ( M  .x.  Y ) )  =  ( ( M 
.x.  X ) ( +g  `  G ) ( ( invg `  G ) `  ( M  .x.  Y ) ) ) )
3120, 24, 303eqtr4d 2485 1  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( M  .x.  ( X  .-  Y ) )  =  ( ( M 
.x.  X )  .-  ( M  .x.  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5423  (class class class)co 6096   -ucneg 9601   ZZcz 10651   Basecbs 14179   +g cplusg 14243   Grpcgrp 15415   invgcminusg 15416   -gcsg 15418  .gcmg 15419   Abelcabel 16283
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-cmn 16284  df-abl 16285
This theorem is referenced by: (None)
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