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Theorem vcz 21118
Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
vc0.1  |-  G  =  ( 1st `  W
)
vc0.2  |-  S  =  ( 2nd `  W
)
vc0.3  |-  X  =  ran  G
vc0.4  |-  Z  =  (GId `  G )
Assertion
Ref Expression
vcz  |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( A S Z )  =  Z )

Proof of Theorem vcz
StepHypRef Expression
1 vc0.1 . . . . . 6  |-  G  =  ( 1st `  W
)
2 vc0.3 . . . . . 6  |-  X  =  ran  G
3 vc0.4 . . . . . 6  |-  Z  =  (GId `  G )
41, 2, 3vczcl 21114 . . . . 5  |-  ( W  e.  CVec OLD  ->  Z  e.  X )
54anim2i 555 . . . 4  |-  ( ( A  e.  CC  /\  W  e.  CVec OLD )  ->  ( A  e.  CC  /\  Z  e.  X ) )
65ancoms 441 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( A  e.  CC  /\  Z  e.  X ) )
7 0cn 8826 . . . 4  |-  0  e.  CC
8 vc0.2 . . . . 5  |-  S  =  ( 2nd `  W
)
91, 8, 2vcass 21102 . . . 4  |-  ( ( W  e.  CVec OLD  /\  ( A  e.  CC  /\  0  e.  CC  /\  Z  e.  X )
)  ->  ( ( A  x.  0 ) S Z )  =  ( A S ( 0 S Z ) ) )
107, 9mp3anr2 1280 . . 3  |-  ( ( W  e.  CVec OLD  /\  ( A  e.  CC  /\  Z  e.  X ) )  ->  ( ( A  x.  0 ) S Z )  =  ( A S ( 0 S Z ) ) )
116, 10syldan 458 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( ( A  x.  0 ) S Z )  =  ( A S ( 0 S Z ) ) )
12 mul01 8986 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
1312oveq1d 5834 . . 3  |-  ( A  e.  CC  ->  (
( A  x.  0 ) S Z )  =  ( 0 S Z ) )
141, 8, 2, 3vc0 21117 . . . 4  |-  ( ( W  e.  CVec OLD  /\  Z  e.  X )  ->  ( 0 S Z )  =  Z )
154, 14mpdan 652 . . 3  |-  ( W  e.  CVec OLD  ->  ( 0 S Z )  =  Z )
1613, 15sylan9eqr 2338 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( ( A  x.  0 ) S Z )  =  Z )
1715oveq2d 5835 . . 3  |-  ( W  e.  CVec OLD  ->  ( A S ( 0 S Z ) )  =  ( A S Z ) )
1817adantr 453 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( A S ( 0 S Z ) )  =  ( A S Z ) )
1911, 16, 183eqtr3rd 2325 1  |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1628    e. wcel 1688   ran crn 4689   ` cfv 5221  (class class class)co 5819   1stc1st 6081   2ndc2nd 6082   CCcc 8730   0cc0 8732    x. cmul 8737  GIdcgi 20846   CVec OLDcvc 21093
This theorem is referenced by:  vcoprne  21127  nvsz  21188
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-ltxr 8867  df-grpo 20850  df-gid 20851  df-ginv 20852  df-ablo 20941  df-vc 21094
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