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Theorem vcm 21073
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 21060 . . . 4  |-  ( W  e.  CVec OLD  ->  G  e. 
GrpOp )
32adantr 453 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  G  e.  GrpOp )
4 neg1cn 9767 . . . 4  |-  -u 1  e.  CC
5 vcm.2 . . . . 5  |-  S  =  ( 2nd `  W
)
6 vcm.3 . . . . 5  |-  X  =  ran  G
71, 5, 6vccl 21052 . . . 4  |-  ( ( W  e.  CVec OLD  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X
)
84, 7mp3an2 1270 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X
)
9 eqid 2256 . . . 4  |-  (GId `  G )  =  (GId
`  G )
106, 9grporid 20833 . . 3  |-  ( ( G  e.  GrpOp  /\  ( -u 1 S A )  e.  X )  -> 
( ( -u 1 S A ) G (GId
`  G ) )  =  ( -u 1 S A ) )
113, 8, 10syl2anc 645 . 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 20839 . . . . . . 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 20816 . . . . . 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 1189 . . . . 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 21053 . . . . . . . 8  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
1918oveq2d 5794 . . . . . . 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 8749 . . . . . . . . . 10  |-  1  e.  CC
2120negidi 9069 . . . . . . . . . 10  |-  ( 1  +  -u 1 )  =  0
2220, 4, 21addcomli 8958 . . . . . . . . 9  |-  ( -u
1  +  1 )  =  0
2322oveq1i 5788 . . . . . . . 8  |-  ( (
-u 1  +  1 ) S A )  =  ( 0 S A )
241, 5, 6vcdir 21055 . . . . . . . . . 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 1279 . . . . . . . . 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 667 . . . . . . . 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 21071 . . . . . . . 8  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  =  (GId
`  G ) )
2823, 26, 273eqtr3a 2312 . . . . . . 7  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G ( 1 S A ) )  =  (GId `  G )
)
2919, 28eqtr3d 2290 . . . . . 6  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G A )  =  (GId `  G )
)
3029oveq1d 5793 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( (
-u 1 S A ) G A ) G ( M `  A ) )  =  ( (GId `  G
) G ( M `
 A ) ) )
3117, 30eqtr3d 2290 . . . 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 20842 . . . . . 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 5794 . . . 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 2290 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( (GId `  G ) G ( M `  A ) )  =  ( (
-u 1 S A ) G (GId `  G ) ) )
366, 9grpolid 20832 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( M `  A )  e.  X )  ->  (
(GId `  G ) G ( M `  A ) )  =  ( M `  A
) )
373, 15, 36syl2anc 645 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( (GId `  G ) G ( M `  A ) )  =  ( M `
 A ) )
3835, 37eqtr3d 2290 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G (GId `  G
) )  =  ( M `  A ) )
3911, 38eqtr3d 2290 1  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( M `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   ran crn 4648   ` cfv 4659  (class class class)co 5778   1stc1st 6040   2ndc2nd 6041   CCcc 8689   0cc0 8691   1c1 8692    + caddc 8694   -ucneg 8992   GrpOpcgr 20799  GIdcgi 20800   invcgn 20801   CVec OLDcvc 21047
This theorem is referenced by:  vcrinv  21074  vclinv  21075  nvinv  21143
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-po 4272  df-so 4273  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-pnf 8823  df-mnf 8824  df-ltxr 8826  df-sub 8993  df-neg 8994  df-grpo 20804  df-gid 20805  df-ginv 20806  df-ablo 20895  df-vc 21048
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