MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nv0 Unicode version

Theorem nv0 21966
Description: Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nv0.1  |-  X  =  ( BaseSet `  U )
nv0.4  |-  S  =  ( .s OLD `  U
)
nv0.6  |-  Z  =  ( 0vec `  U
)
Assertion
Ref Expression
nv0  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  Z )

Proof of Theorem nv0
StepHypRef Expression
1 eqid 2387 . . . 4  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 21942 . . 3  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 eqid 2387 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 21930 . . . 4  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nv0.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
65smfval 21932 . . . 4  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nv0.1 . . . . 5  |-  X  =  ( BaseSet `  U )
87, 3bafval 21931 . . . 4  |-  X  =  ran  ( +v `  U )
9 eqid 2387 . . . 4  |-  (GId `  ( +v `  U ) )  =  (GId `  ( +v `  U ) )
104, 6, 8, 9vc0 21896 . . 3  |-  ( ( ( 1st `  U
)  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  =  (GId
`  ( +v `  U ) ) )
112, 10sylan 458 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  (GId `  ( +v `  U ) ) )
12 nv0.6 . . . 4  |-  Z  =  ( 0vec `  U
)
133, 120vfval 21933 . . 3  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  ( +v `  U ) ) )
1413adantr 452 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  Z  =  (GId `  ( +v `  U ) ) )
1511, 14eqtr4d 2422 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   1stc1st 6286   0cc0 8923  GIdcgi 21623   CVec OLDcvc 21872   NrmCVeccnv 21911   +vcpv 21912   BaseSetcba 21913   .s
OLDcns 21914   0veccn0v 21915
This theorem is referenced by:  nvmul0or  21981  nvz0  22005  nvge0  22011  ipasslem1  22180  hlmul0  22259
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-1st 6288  df-2nd 6289  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-grpo 21627  df-gid 21628  df-ginv 21629  df-ablo 21718  df-vc 21873  df-nv 21919  df-va 21922  df-ba 21923  df-sm 21924  df-0v 21925  df-nmcv 21927
  Copyright terms: Public domain W3C validator