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Theorem nv0 24170
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  =  ( .sOLD `  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 2454 . . . 4  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 24146 . . 3  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 eqid 2454 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 24134 . . . 4  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nv0.4 . . . . 5  |-  S  =  ( .sOLD `  U )
65smfval 24136 . . . 4  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nv0.1 . . . . 5  |-  X  =  ( BaseSet `  U )
87, 3bafval 24135 . . . 4  |-  X  =  ran  ( +v `  U )
9 eqid 2454 . . . 4  |-  (GId `  ( +v `  U ) )  =  (GId `  ( +v `  U ) )
104, 6, 8, 9vc0 24100 . . 3  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  A  e.  X )  ->  ( 0 S A )  =  (GId
`  ( +v `  U ) ) )
112, 10sylan 471 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  (GId `  ( +v `  U ) ) )
12 nv0.6 . . . 4  |-  Z  =  ( 0vec `  U
)
133, 120vfval 24137 . . 3  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  ( +v `  U ) ) )
1413adantr 465 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  Z  =  (GId `  ( +v `  U ) ) )
1511, 14eqtr4d 2498 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ` cfv 5527  (class class class)co 6201   1stc1st 6686   0cc0 9394  GIdcgi 23827   CVecOLDcvc 24076   NrmCVeccnv 24115   +vcpv 24116   BaseSetcba 24117   .sOLDcns 24118   0veccn0v 24119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-1st 6688  df-2nd 6689  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-ltxr 9535  df-grpo 23831  df-gid 23832  df-ginv 23833  df-ablo 23922  df-vc 24077  df-nv 24123  df-va 24126  df-ba 24127  df-sm 24128  df-0v 24129  df-nmcv 24131
This theorem is referenced by:  nvmul0or  24185  nvz0  24209  nvge0  24215  ipasslem1  24384  hlmul0  24463
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