MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vcm Structured version   Unicode version

Theorem vcm 22052
Description: Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 25-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
vcm.1  |-  G  =  ( 1st `  W
)
vcm.2  |-  S  =  ( 2nd `  W
)
vcm.3  |-  X  =  ran  G
vcm.4  |-  M  =  ( inv `  G
)
Assertion
Ref Expression
vcm  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( M `  A ) )

Proof of Theorem vcm
StepHypRef Expression
1 vcm.1 . . . . 5  |-  G  =  ( 1st `  W
)
21vcgrp 22039 . . . 4  |-  ( W  e.  CVec OLD  ->  G  e. 
GrpOp )
32adantr 453 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  G  e.  GrpOp )
4 neg1cn 10069 . . . 4  |-  -u 1  e.  CC
5 vcm.2 . . . . 5  |-  S  =  ( 2nd `  W
)
6 vcm.3 . . . . 5  |-  X  =  ran  G
71, 5, 6vccl 22031 . . . 4  |-  ( ( W  e.  CVec OLD  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X
)
84, 7mp3an2 1268 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X
)
9 eqid 2438 . . . 4  |-  (GId `  G )  =  (GId
`  G )
106, 9grporid 21810 . . 3  |-  ( ( G  e.  GrpOp  /\  ( -u 1 S A )  e.  X )  -> 
( ( -u 1 S A ) G (GId
`  G ) )  =  ( -u 1 S A ) )
113, 8, 10syl2anc 644 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G (GId `  G
) )  =  (
-u 1 S A ) )
12 simpr 449 . . . . . 6  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  A  e.  X
)
13 vcm.4 . . . . . . . 8  |-  M  =  ( inv `  G
)
146, 13grpoinvcl 21816 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( M `  A )  e.  X )
152, 14sylan 459 . . . . . 6  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( M `  A )  e.  X
)
166grpoass 21793 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  (
( -u 1 S A )  e.  X  /\  A  e.  X  /\  ( M `  A )  e.  X ) )  ->  ( ( (
-u 1 S A ) G A ) G ( M `  A ) )  =  ( ( -u 1 S A ) G ( A G ( M `
 A ) ) ) )
173, 8, 12, 15, 16syl13anc 1187 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( (
-u 1 S A ) G A ) G ( M `  A ) )  =  ( ( -u 1 S A ) G ( A G ( M `
 A ) ) ) )
181, 5, 6vcid 22032 . . . . . . . 8  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
1918oveq2d 6099 . . . . . . 7  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G ( 1 S A ) )  =  ( ( -u 1 S A ) G A ) )
20 ax-1cn 9050 . . . . . . . . . 10  |-  1  e.  CC
2120negidi 9371 . . . . . . . . . 10  |-  ( 1  +  -u 1 )  =  0
2220, 4, 21addcomli 9260 . . . . . . . . 9  |-  ( -u
1  +  1 )  =  0
2322oveq1i 6093 . . . . . . . 8  |-  ( (
-u 1  +  1 ) S A )  =  ( 0 S A )
241, 5, 6vcdir 22034 . . . . . . . . . 10  |-  ( ( W  e.  CVec OLD  /\  ( -u 1  e.  CC  /\  1  e.  CC  /\  A  e.  X ) )  -> 
( ( -u 1  +  1 ) S A )  =  ( ( -u 1 S A ) G ( 1 S A ) ) )
254, 24mp3anr1 1277 . . . . . . . . 9  |-  ( ( W  e.  CVec OLD  /\  ( 1  e.  CC  /\  A  e.  X ) )  ->  ( ( -u 1  +  1 ) S A )  =  ( ( -u 1 S A ) G ( 1 S A ) ) )
2620, 25mpanr1 666 . . . . . . . 8  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1  +  1 ) S A )  =  ( ( -u 1 S A ) G ( 1 S A ) ) )
271, 5, 6, 9vc0 22050 . . . . . . . 8  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  =  (GId
`  G ) )
2823, 26, 273eqtr3a 2494 . . . . . . 7  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G ( 1 S A ) )  =  (GId `  G )
)
2919, 28eqtr3d 2472 . . . . . 6  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G A )  =  (GId `  G )
)
3029oveq1d 6098 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( (
-u 1 S A ) G A ) G ( M `  A ) )  =  ( (GId `  G
) G ( M `
 A ) ) )
3117, 30eqtr3d 2472 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G ( A G ( M `  A
) ) )  =  ( (GId `  G
) G ( M `
 A ) ) )
326, 9, 13grporinv 21819 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( M `  A ) )  =  (GId `  G )
)
332, 32sylan 459 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G ( M `  A
) )  =  (GId
`  G ) )
3433oveq2d 6099 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G ( A G ( M `  A
) ) )  =  ( ( -u 1 S A ) G (GId
`  G ) ) )
3531, 34eqtr3d 2472 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( (GId `  G ) G ( M `  A ) )  =  ( (
-u 1 S A ) G (GId `  G ) ) )
366, 9grpolid 21809 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( M `  A )  e.  X )  ->  (
(GId `  G ) G ( M `  A ) )  =  ( M `  A
) )
373, 15, 36syl2anc 644 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( (GId `  G ) G ( M `  A ) )  =  ( M `
 A ) )
3835, 37eqtr3d 2472 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G (GId `  G
) )  =  ( M `  A ) )
3911, 38eqtr3d 2472 1  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( M `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   ran crn 4881   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350   CCcc 8990   0cc0 8992   1c1 8993    + caddc 8995   -ucneg 9294   GrpOpcgr 21776  GIdcgi 21777   invcgn 21778   CVec OLDcvc 22026
This theorem is referenced by:  vcrinv  22053  vclinv  22054  nvinv  22122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-ltxr 9127  df-sub 9295  df-neg 9296  df-grpo 21781  df-gid 21782  df-ginv 21783  df-ablo 21872  df-vc 22027
  Copyright terms: Public domain W3C validator