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Theorem nvsz 25197
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 2462 . . . 4  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 25172 . . 3  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 eqid 2462 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 25160 . . . 4  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvsz.4 . . . . 5  |-  S  =  ( .sOLD `  U )
65smfval 25162 . . . 4  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 eqid 2462 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
87, 3bafval 25161 . . . 4  |-  ( BaseSet `  U )  =  ran  ( +v `  U )
9 eqid 2462 . . . 4  |-  (GId `  ( +v `  U ) )  =  (GId `  ( +v `  U ) )
104, 6, 8, 9vcz 25127 . . 3  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  A  e.  CC )  ->  ( A S (GId `  ( +v `  U ) ) )  =  (GId `  ( +v `  U ) ) )
112, 10sylan 471 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S (GId `  ( +v `  U ) ) )  =  (GId `  ( +v `  U ) ) )
12 nvsz.6 . . . . 5  |-  Z  =  ( 0vec `  U
)
133, 120vfval 25163 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  ( +v `  U ) ) )
1413adantr 465 . . 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 2513 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   ` cfv 5581  (class class class)co 6277   1stc1st 6774   CCcc 9481  GIdcgi 24853   CVecOLDcvc 25102   NrmCVeccnv 25141   +vcpv 25142   BaseSetcba 25143   .sOLDcns 25144   0veccn0v 25145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-1st 6776  df-2nd 6777  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-ltxr 9624  df-grpo 24857  df-gid 24858  df-ginv 24859  df-ablo 24948  df-vc 25103  df-nv 25149  df-va 25152  df-ba 25153  df-sm 25154  df-0v 25155  df-nmcv 25157
This theorem is referenced by:  nvmul0or  25211  nvnd  25258  dip0r  25294  0lno  25369
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