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Theorem nv0rid 22121
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 22090 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  G ) )
43oveq2d 6100 . . 3  |-  ( U  e.  NrmCVec  ->  ( A G Z )  =  ( A G (GId `  G ) ) )
54adantr 453 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G Z )  =  ( A G (GId
`  G ) ) )
61nvgrp 22101 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
7 nv0id.1 . . . . 5  |-  X  =  ( BaseSet `  U )
87, 1bafval 22088 . . . 4  |-  X  =  ran  G
9 eqid 2438 . . . 4  |-  (GId `  G )  =  (GId
`  G )
108, 9grporid 21813 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
116, 10sylan 459 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
125, 11eqtrd 2470 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G Z )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   GrpOpcgr 21779  GIdcgi 21780   NrmCVeccnv 22068   +vcpv 22069   BaseSetcba 22070   0veccn0v 22072
This theorem is referenced by:  nvsubadd  22141  nvabs  22167  nvnd  22185  imsmetlem  22187  lnomul  22266  0lno  22296  ipdirilem  22335  hladdid  22410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-1st 6352  df-2nd 6353  df-riota 6552  df-grpo 21784  df-gid 21785  df-ablo 21875  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-nmcv 22084
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