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Theorem vcm 22003
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 21990 . . . 4  |-  ( W  e.  CVec OLD  ->  G  e. 
GrpOp )
32adantr 452 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  G  e.  GrpOp )
4 neg1cn 10023 . . . 4  |-  -u 1  e.  CC
5 vcm.2 . . . . 5  |-  S  =  ( 2nd `  W
)
6 vcm.3 . . . . 5  |-  X  =  ran  G
71, 5, 6vccl 21982 . . . 4  |-  ( ( W  e.  CVec OLD  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X
)
84, 7mp3an2 1267 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X
)
9 eqid 2404 . . . 4  |-  (GId `  G )  =  (GId
`  G )
106, 9grporid 21761 . . 3  |-  ( ( G  e.  GrpOp  /\  ( -u 1 S A )  e.  X )  -> 
( ( -u 1 S A ) G (GId
`  G ) )  =  ( -u 1 S A ) )
113, 8, 10syl2anc 643 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G (GId `  G
) )  =  (
-u 1 S A ) )
12 simpr 448 . . . . . 6  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  A  e.  X
)
13 vcm.4 . . . . . . . 8  |-  M  =  ( inv `  G
)
146, 13grpoinvcl 21767 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( M `  A )  e.  X )
152, 14sylan 458 . . . . . 6  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( M `  A )  e.  X
)
166grpoass 21744 . . . . . 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 1186 . . . . 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 21983 . . . . . . . 8  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
1918oveq2d 6056 . . . . . . 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 9004 . . . . . . . . . 10  |-  1  e.  CC
2120negidi 9325 . . . . . . . . . 10  |-  ( 1  +  -u 1 )  =  0
2220, 4, 21addcomli 9214 . . . . . . . . 9  |-  ( -u
1  +  1 )  =  0
2322oveq1i 6050 . . . . . . . 8  |-  ( (
-u 1  +  1 ) S A )  =  ( 0 S A )
241, 5, 6vcdir 21985 . . . . . . . . . 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 1276 . . . . . . . . 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 665 . . . . . . . 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 22001 . . . . . . . 8  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  =  (GId
`  G ) )
2823, 26, 273eqtr3a 2460 . . . . . . 7  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G ( 1 S A ) )  =  (GId `  G )
)
2919, 28eqtr3d 2438 . . . . . 6  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G A )  =  (GId `  G )
)
3029oveq1d 6055 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( (
-u 1 S A ) G A ) G ( M `  A ) )  =  ( (GId `  G
) G ( M `
 A ) ) )
3117, 30eqtr3d 2438 . . . 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 21770 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( M `  A ) )  =  (GId `  G )
)
332, 32sylan 458 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G ( M `  A
) )  =  (GId
`  G ) )
3433oveq2d 6056 . . . 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 2438 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( (GId `  G ) G ( M `  A ) )  =  ( (
-u 1 S A ) G (GId `  G ) ) )
366, 9grpolid 21760 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( M `  A )  e.  X )  ->  (
(GId `  G ) G ( M `  A ) )  =  ( M `  A
) )
373, 15, 36syl2anc 643 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( (GId `  G ) G ( M `  A ) )  =  ( M `
 A ) )
3835, 37eqtr3d 2438 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G (GId `  G
) )  =  ( M `  A ) )
3911, 38eqtr3d 2438 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 359    = wceq 1649    e. wcel 1721   ran crn 4838   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949   -ucneg 9248   GrpOpcgr 21727  GIdcgi 21728   invcgn 21729   CVec OLDcvc 21977
This theorem is referenced by:  vcrinv  22004  vclinv  22005  nvinv  22073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250  df-grpo 21732  df-gid 21733  df-ginv 21734  df-ablo 21823  df-vc 21978
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