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Theorem nv0rid 22069
Description: The zero vector is a right identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nv0id.1  |-  X  =  ( BaseSet `  U )
nv0id.2  |-  G  =  ( +v `  U
)
nv0id.6  |-  Z  =  ( 0vec `  U
)
Assertion
Ref Expression
nv0rid  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G Z )  =  A )

Proof of Theorem nv0rid
StepHypRef Expression
1 nv0id.2 . . . . 5  |-  G  =  ( +v `  U
)
2 nv0id.6 . . . . 5  |-  Z  =  ( 0vec `  U
)
31, 20vfval 22038 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  G ) )
43oveq2d 6056 . . 3  |-  ( U  e.  NrmCVec  ->  ( A G Z )  =  ( A G (GId `  G ) ) )
54adantr 452 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G Z )  =  ( A G (GId
`  G ) ) )
61nvgrp 22049 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
7 nv0id.1 . . . . 5  |-  X  =  ( BaseSet `  U )
87, 1bafval 22036 . . . 4  |-  X  =  ran  G
9 eqid 2404 . . . 4  |-  (GId `  G )  =  (GId
`  G )
108, 9grporid 21761 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
116, 10sylan 458 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
125, 11eqtrd 2436 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G Z )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   GrpOpcgr 21727  GIdcgi 21728   NrmCVeccnv 22016   +vcpv 22017   BaseSetcba 22018   0veccn0v 22020
This theorem is referenced by:  nvsubadd  22089  nvabs  22115  nvnd  22133  imsmetlem  22135  lnomul  22214  0lno  22244  ipdirilem  22283  hladdid  22358
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
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-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-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-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-1st 6308  df-2nd 6309  df-riota 6508  df-grpo 21732  df-gid 21733  df-ablo 21823  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-nmcv 22032
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