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Theorem vcz 22037
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 22033 . . . . 5  |-  ( W  e.  CVec OLD  ->  Z  e.  X )
54anim2i 553 . . . 4  |-  ( ( A  e.  CC  /\  W  e.  CVec OLD )  ->  ( A  e.  CC  /\  Z  e.  X ) )
65ancoms 440 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( A  e.  CC  /\  Z  e.  X ) )
7 0cn 9073 . . . 4  |-  0  e.  CC
8 vc0.2 . . . . 5  |-  S  =  ( 2nd `  W
)
91, 8, 2vcass 22021 . . . 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 1277 . . 3  |-  ( ( W  e.  CVec OLD  /\  ( A  e.  CC  /\  Z  e.  X ) )  ->  ( ( A  x.  0 ) S Z )  =  ( A S ( 0 S Z ) ) )
116, 10syldan 457 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( ( A  x.  0 ) S Z )  =  ( A S ( 0 S Z ) ) )
12 mul01 9234 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
1312oveq1d 6087 . . 3  |-  ( A  e.  CC  ->  (
( A  x.  0 ) S Z )  =  ( 0 S Z ) )
141, 8, 2, 3vc0 22036 . . . 4  |-  ( ( W  e.  CVec OLD  /\  Z  e.  X )  ->  ( 0 S Z )  =  Z )
154, 14mpdan 650 . . 3  |-  ( W  e.  CVec OLD  ->  ( 0 S Z )  =  Z )
1613, 15sylan9eqr 2489 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( ( A  x.  0 ) S Z )  =  Z )
1715oveq2d 6088 . . 3  |-  ( W  e.  CVec OLD  ->  ( A S ( 0 S Z ) )  =  ( A S Z ) )
1817adantr 452 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( A S ( 0 S Z ) )  =  ( A S Z ) )
1911, 16, 183eqtr3rd 2476 1  |-  ( ( W  e.  CVec OLD  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ran crn 4870   ` cfv 5445  (class class class)co 6072   1stc1st 6338   2ndc2nd 6339   CCcc 8977   0cc0 8979    x. cmul 8984  GIdcgi 21763   CVec OLDcvc 22012
This theorem is referenced by:  vcoprne  22046  nvsz  22107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-1st 6340  df-2nd 6341  df-riota 6540  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-ltxr 9114  df-grpo 21767  df-gid 21768  df-ginv 21769  df-ablo 21858  df-vc 22013
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