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Theorem nv0lid 25729
Description: The zero vector is a left 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
nv0lid  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( Z G A )  =  A )

Proof of Theorem nv0lid
StepHypRef Expression
1 nv0id.2 . . . . 5  |-  G  =  ( +v `  U
)
2 nv0id.6 . . . . 5  |-  Z  =  ( 0vec `  U
)
31, 20vfval 25697 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  G ) )
43oveq1d 6285 . . 3  |-  ( U  e.  NrmCVec  ->  ( Z G A )  =  ( (GId `  G ) G A ) )
54adantr 463 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( Z G A )  =  ( (GId `  G
) G A ) )
61nvgrp 25708 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
7 nv0id.1 . . . . 5  |-  X  =  ( BaseSet `  U )
87, 1bafval 25695 . . . 4  |-  X  =  ran  G
9 eqid 2454 . . . 4  |-  (GId `  G )  =  (GId
`  G )
108, 9grpolid 25419 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
116, 10sylan 469 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
125, 11eqtrd 2495 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( Z G A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   GrpOpcgr 25386  GIdcgi 25387   NrmCVeccnv 25675   +vcpv 25676   BaseSetcba 25677   0veccn0v 25679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-1st 6773  df-2nd 6774  df-grpo 25391  df-gid 25392  df-ablo 25482  df-vc 25637  df-nv 25683  df-va 25686  df-ba 25687  df-sm 25688  df-0v 25689  df-nmcv 25691
This theorem is referenced by:  nvzs  25738  nvpncan2  25749  nvnncan  25756  nvmeq0  25757  imsmetlem  25794  ipdirilem  25942
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