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

Theorem nvsz 25933
Description: Anything times the zero vector is the zero vector. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvsz.4  |-  S  =  ( .sOLD `  U )
nvsz.6  |-  Z  =  ( 0vec `  U
)
Assertion
Ref Expression
nvsz  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  Z )

Proof of Theorem nvsz
StepHypRef Expression
1 eqid 2402 . . . 4  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 25908 . . 3  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 eqid 2402 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 25896 . . . 4  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvsz.4 . . . . 5  |-  S  =  ( .sOLD `  U )
65smfval 25898 . . . 4  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 eqid 2402 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
87, 3bafval 25897 . . . 4  |-  ( BaseSet `  U )  =  ran  ( +v `  U )
9 eqid 2402 . . . 4  |-  (GId `  ( +v `  U ) )  =  (GId `  ( +v `  U ) )
104, 6, 8, 9vcz 25863 . . 3  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  A  e.  CC )  ->  ( A S (GId `  ( +v `  U ) ) )  =  (GId `  ( +v `  U ) ) )
112, 10sylan 469 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S (GId `  ( +v `  U ) ) )  =  (GId `  ( +v `  U ) ) )
12 nvsz.6 . . . . 5  |-  Z  =  ( 0vec `  U
)
133, 120vfval 25899 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  ( +v `  U ) ) )
1413adantr 463 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  Z  =  (GId `  ( +v `  U ) ) )
1514oveq2d 6293 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  ( A S (GId
`  ( +v `  U ) ) ) )
1611, 15, 143eqtr4d 2453 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277   1stc1st 6781   CCcc 9519  GIdcgi 25589   CVecOLDcvc 25838   NrmCVeccnv 25877   +vcpv 25878   BaseSetcba 25879   .sOLDcns 25880   0veccn0v 25881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-1st 6783  df-2nd 6784  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-ltxr 9662  df-grpo 25593  df-gid 25594  df-ginv 25595  df-ablo 25684  df-vc 25839  df-nv 25885  df-va 25888  df-ba 25889  df-sm 25890  df-0v 25891  df-nmcv 25893
This theorem is referenced by:  nvmul0or  25947  nvnd  25994  dip0r  26030  0lno  26105
  Copyright terms: Public domain W3C validator