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Theorem nv0rid 21193
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 21162 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  G ) )
43oveq2d 5874 . . 3  |-  ( U  e.  NrmCVec  ->  ( A G Z )  =  ( A G (GId `  G ) ) )
54adantr 451 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G Z )  =  ( A G (GId
`  G ) ) )
61nvgrp 21173 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
7 nv0id.1 . . . . 5  |-  X  =  ( BaseSet `  U )
87, 1bafval 21160 . . . 4  |-  X  =  ran  G
9 eqid 2283 . . . 4  |-  (GId `  G )  =  (GId
`  G )
108, 9grporid 20887 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
116, 10sylan 457 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
125, 11eqtrd 2315 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G Z )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   0veccn0v 21144
This theorem is referenced by:  nvsubadd  21213  nvabs  21239  nvnd  21257  imsmetlem  21259  lnomul  21338  0lno  21368  ipdirilem  21407  hladdid  21482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156
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