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Theorem nvsz 22076
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  =  ( .s OLD `  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 2408 . . . 4  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 22051 . . 3  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 eqid 2408 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 22039 . . . 4  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvsz.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
65smfval 22041 . . . 4  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 eqid 2408 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
87, 3bafval 22040 . . . 4  |-  ( BaseSet `  U )  =  ran  ( +v `  U )
9 eqid 2408 . . . 4  |-  (GId `  ( +v `  U ) )  =  (GId `  ( +v `  U ) )
104, 6, 8, 9vcz 22006 . . 3  |-  ( ( ( 1st `  U
)  e.  CVec OLD  /\  A  e.  CC )  ->  ( A S (GId `  ( +v `  U ) ) )  =  (GId `  ( +v `  U ) ) )
112, 10sylan 458 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S (GId `  ( +v `  U ) ) )  =  (GId `  ( +v `  U ) ) )
12 nvsz.6 . . . . 5  |-  Z  =  ( 0vec `  U
)
133, 120vfval 22042 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  ( +v `  U ) ) )
1413adantr 452 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  Z  =  (GId `  ( +v `  U ) ) )
1514oveq2d 6060 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  ( A S (GId
`  ( +v `  U ) ) ) )
1611, 15, 143eqtr4d 2450 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5417  (class class class)co 6044   1stc1st 6310   CCcc 8948  GIdcgi 21732   CVec OLDcvc 21981   NrmCVeccnv 22020   +vcpv 22021   BaseSetcba 22022   .s
OLDcns 22023   0veccn0v 22024
This theorem is referenced by:  nvmul0or  22090  nvnd  22137  dip0r  22173  0lno  22248
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-1st 6312  df-2nd 6313  df-riota 6512  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-ltxr 9085  df-grpo 21736  df-gid 21737  df-ginv 21738  df-ablo 21827  df-vc 21982  df-nv 22028  df-va 22031  df-ba 22032  df-sm 22033  df-0v 22034  df-nmcv 22036
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