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Theorem vcm 21119
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 21106 . . . 4  |-  ( W  e.  CVec OLD  ->  G  e. 
GrpOp )
32adantr 453 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  G  e.  GrpOp )
4 neg1cn 9808 . . . 4  |-  -u 1  e.  CC
5 vcm.2 . . . . 5  |-  S  =  ( 2nd `  W
)
6 vcm.3 . . . . 5  |-  X  =  ran  G
71, 5, 6vccl 21098 . . . 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 2284 . . . 4  |-  (GId `  G )  =  (GId
`  G )
106, 9grporid 20879 . . 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 20885 . . . . . . 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 20862 . . . . . 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 21099 . . . . . . . 8  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
1918oveq2d 5835 . . . . . . 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 8790 . . . . . . . . . 10  |-  1  e.  CC
2120negidi 9110 . . . . . . . . . 10  |-  ( 1  +  -u 1 )  =  0
2220, 4, 21addcomli 8999 . . . . . . . . 9  |-  ( -u
1  +  1 )  =  0
2322oveq1i 5829 . . . . . . . 8  |-  ( (
-u 1  +  1 ) S A )  =  ( 0 S A )
241, 5, 6vcdir 21101 . . . . . . . . . 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 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 21117 . . . . . . . 8  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  =  (GId
`  G ) )
2823, 26, 273eqtr3a 2340 . . . . . . 7  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G ( 1 S A ) )  =  (GId `  G )
)
2919, 28eqtr3d 2318 . . . . . 6  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G A )  =  (GId `  G )
)
3029oveq1d 5834 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( (
-u 1 S A ) G A ) G ( M `  A ) )  =  ( (GId `  G
) G ( M `
 A ) ) )
3117, 30eqtr3d 2318 . . . 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 20888 . . . . . 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 5835 . . . 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 2318 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( (GId `  G ) G ( M `  A ) )  =  ( (
-u 1 S A ) G (GId `  G ) ) )
366, 9grpolid 20878 . . . 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 2318 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G (GId `  G
) )  =  ( M `  A ) )
3911, 38eqtr3d 2318 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 1624    e. wcel 1685   ran crn 4689   ` cfv 5221  (class class class)co 5819   1stc1st 6081   2ndc2nd 6082   CCcc 8730   0cc0 8732   1c1 8733    + caddc 8735   -ucneg 9033   GrpOpcgr 20845  GIdcgi 20846   invcgn 20847   CVec OLDcvc 21093
This theorem is referenced by:  vcrinv  21120  vclinv  21121  nvinv  21189
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-ltxr 8867  df-sub 9034  df-neg 9035  df-grpo 20850  df-gid 20851  df-ginv 20852  df-ablo 20941  df-vc 21094
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