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Theorem nv0 26103
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 2429 . . . 4  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 26079 . . 3  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 eqid 2429 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 26067 . . . 4  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nv0.4 . . . . 5  |-  S  =  ( .sOLD `  U )
65smfval 26069 . . . 4  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nv0.1 . . . . 5  |-  X  =  ( BaseSet `  U )
87, 3bafval 26068 . . . 4  |-  X  =  ran  ( +v `  U )
9 eqid 2429 . . . 4  |-  (GId `  ( +v `  U ) )  =  (GId `  ( +v `  U ) )
104, 6, 8, 9vc0 26033 . . 3  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  A  e.  X )  ->  ( 0 S A )  =  (GId
`  ( +v `  U ) ) )
112, 10sylan 473 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  (GId `  ( +v `  U ) ) )
12 nv0.6 . . . 4  |-  Z  =  ( 0vec `  U
)
133, 120vfval 26070 . . 3  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  ( +v `  U ) ) )
1413adantr 466 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  Z  =  (GId `  ( +v `  U ) ) )
1511, 14eqtr4d 2473 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   1stc1st 6805   0cc0 9538  GIdcgi 25760   CVecOLDcvc 26009   NrmCVeccnv 26048   +vcpv 26049   BaseSetcba 26050   .sOLDcns 26051   0veccn0v 26052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-ltxr 9679  df-grpo 25764  df-gid 25765  df-ginv 25766  df-ablo 25855  df-vc 26010  df-nv 26056  df-va 26059  df-ba 26060  df-sm 26061  df-0v 26062  df-nmcv 26064
This theorem is referenced by:  nvmul0or  26118  nvz0  26142  nvge0  26148  ipasslem1  26317  hlmul0  26396
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