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Theorem nv0rid 25303
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 25272 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  G ) )
43oveq2d 6301 . . 3  |-  ( U  e.  NrmCVec  ->  ( A G Z )  =  ( A G (GId `  G ) ) )
54adantr 465 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G Z )  =  ( A G (GId
`  G ) ) )
61nvgrp 25283 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
7 nv0id.1 . . . . 5  |-  X  =  ( BaseSet `  U )
87, 1bafval 25270 . . . 4  |-  X  =  ran  G
9 eqid 2467 . . . 4  |-  (GId `  G )  =  (GId
`  G )
108, 9grporid 24995 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
116, 10sylan 471 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
125, 11eqtrd 2508 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G Z )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6285   GrpOpcgr 24961  GIdcgi 24962   NrmCVeccnv 25250   +vcpv 25251   BaseSetcba 25252   0veccn0v 25254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-1st 6785  df-2nd 6786  df-grpo 24966  df-gid 24967  df-ablo 25057  df-vc 25212  df-nv 25258  df-va 25261  df-ba 25262  df-sm 25263  df-0v 25264  df-nmcv 25266
This theorem is referenced by:  nvsubadd  25323  nvabs  25349  nvnd  25367  imsmetlem  25369  lnomul  25448  0lno  25478  ipdirilem  25517  hladdid  25592
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