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Theorem vcm 20957
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 20944 . . . 4  |-  ( W  e.  CVec OLD  ->  G  e. 
GrpOp )
32adantr 453 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  G  e.  GrpOp )
4 neg1cn 9693 . . . 4  |-  -u 1  e.  CC
5 vcm.2 . . . . 5  |-  S  =  ( 2nd `  W
)
6 vcm.3 . . . . 5  |-  X  =  ran  G
71, 5, 6vccl 20936 . . . 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 2253 . . . 4  |-  (GId `  G )  =  (GId
`  G )
106, 9grporid 20717 . . 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 20723 . . . . . . 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 20700 . . . . . 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 20937 . . . . . . . 8  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
1918oveq2d 5726 . . . . . . 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 8675 . . . . . . . . . 10  |-  1  e.  CC
2120negidi 8995 . . . . . . . . . 10  |-  ( 1  +  -u 1 )  =  0
2220, 4, 21addcomli 8884 . . . . . . . . 9  |-  ( -u
1  +  1 )  =  0
2322oveq1i 5720 . . . . . . . 8  |-  ( (
-u 1  +  1 ) S A )  =  ( 0 S A )
241, 5, 6vcdir 20939 . . . . . . . . . 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 20955 . . . . . . . 8  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  =  (GId
`  G ) )
2823, 26, 273eqtr3a 2309 . . . . . . 7  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G ( 1 S A ) )  =  (GId `  G )
)
2919, 28eqtr3d 2287 . . . . . 6  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G A )  =  (GId `  G )
)
3029oveq1d 5725 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( (
-u 1 S A ) G A ) G ( M `  A ) )  =  ( (GId `  G
) G ( M `
 A ) ) )
3117, 30eqtr3d 2287 . . . 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 20726 . . . . . 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 5726 . . . 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 2287 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( (GId `  G ) G ( M `  A ) )  =  ( (
-u 1 S A ) G (GId `  G ) ) )
366, 9grpolid 20716 . . . 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 2287 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( -u
1 S A ) G (GId `  G
) )  =  ( M `  A ) )
3911, 38eqtr3d 2287 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 4581   ` cfv 4592  (class class class)co 5710   1stc1st 5972   2ndc2nd 5973   CCcc 8615   0cc0 8617   1c1 8618    + caddc 8620   -ucneg 8918   GrpOpcgr 20683  GIdcgi 20684   invcgn 20685   CVec OLDcvc 20931
This theorem is referenced by:  vcrinv  20958  vclinv  20959  nvinv  21027
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-ltxr 8752  df-sub 8919  df-neg 8920  df-grpo 20688  df-gid 20689  df-ginv 20690  df-ablo 20779  df-vc 20932
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