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Theorem vcz 25661
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.  CVecOLD  /\  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 25657 . . . . 5  |-  ( W  e.  CVecOLD  ->  Z  e.  X )
54anim2i 567 . . . 4  |-  ( ( A  e.  CC  /\  W  e.  CVecOLD )  ->  ( A  e.  CC  /\  Z  e.  X ) )
65ancoms 451 . . 3  |-  ( ( W  e.  CVecOLD  /\  A  e.  CC )  ->  ( A  e.  CC  /\  Z  e.  X ) )
7 0cn 9577 . . . 4  |-  0  e.  CC
8 vc0.2 . . . . 5  |-  S  =  ( 2nd `  W
)
91, 8, 2vcass 25645 . . . 4  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  0  e.  CC  /\  Z  e.  X )
)  ->  ( ( A  x.  0 ) S Z )  =  ( A S ( 0 S Z ) ) )
107, 9mp3anr2 1320 . . 3  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  CC  /\  Z  e.  X ) )  ->  ( ( A  x.  0 ) S Z )  =  ( A S ( 0 S Z ) ) )
116, 10syldan 468 . 2  |-  ( ( W  e.  CVecOLD  /\  A  e.  CC )  ->  ( ( A  x.  0 ) S Z )  =  ( A S ( 0 S Z ) ) )
12 mul01 9748 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
1312oveq1d 6285 . . 3  |-  ( A  e.  CC  ->  (
( A  x.  0 ) S Z )  =  ( 0 S Z ) )
141, 8, 2, 3vc0 25660 . . . 4  |-  ( ( W  e.  CVecOLD  /\  Z  e.  X )  ->  ( 0 S Z )  =  Z )
154, 14mpdan 666 . . 3  |-  ( W  e.  CVecOLD  ->  (
0 S Z )  =  Z )
1613, 15sylan9eqr 2517 . 2  |-  ( ( W  e.  CVecOLD  /\  A  e.  CC )  ->  ( ( A  x.  0 ) S Z )  =  Z )
1715oveq2d 6286 . . 3  |-  ( W  e.  CVecOLD  ->  ( A S ( 0 S Z ) )  =  ( A S Z ) )
1817adantr 463 . 2  |-  ( ( W  e.  CVecOLD  /\  A  e.  CC )  ->  ( A S ( 0 S Z ) )  =  ( A S Z ) )
1911, 16, 183eqtr3rd 2504 1  |-  ( ( W  e.  CVecOLD  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   ran crn 4989   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   CCcc 9479   0cc0 9481    x. cmul 9486  GIdcgi 25387   CVecOLDcvc 25636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-grpo 25391  df-gid 25392  df-ginv 25393  df-ablo 25482  df-vc 25637
This theorem is referenced by:  vcoprne  25670  nvsz  25731
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