MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vc0 Unicode version

Theorem vc0 22001
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (Contributed by NM, 4-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
vc0  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  =  Z )

Proof of Theorem vc0
StepHypRef Expression
1 vc0.1 . . . 4  |-  G  =  ( 1st `  W
)
2 vc0.3 . . . 4  |-  X  =  ran  G
3 vc0.4 . . . 4  |-  Z  =  (GId `  G )
41, 2, 3vc0rid 21999 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G Z )  =  A )
5 ax-1cn 9004 . . . . . 6  |-  1  e.  CC
65addid1i 9209 . . . . 5  |-  ( 1  +  0 )  =  1
76oveq1i 6050 . . . 4  |-  ( ( 1  +  0 ) S A )  =  ( 1 S A )
8 0cn 9040 . . . . 5  |-  0  e.  CC
9 vc0.2 . . . . . . 7  |-  S  =  ( 2nd `  W
)
101, 9, 2vcdir 21985 . . . . . 6  |-  ( ( W  e.  CVec OLD  /\  ( 1  e.  CC  /\  0  e.  CC  /\  A  e.  X )
)  ->  ( (
1  +  0 ) S A )  =  ( ( 1 S A ) G ( 0 S A ) ) )
115, 10mp3anr1 1276 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  ( 0  e.  CC  /\  A  e.  X ) )  ->  ( (
1  +  0 ) S A )  =  ( ( 1 S A ) G ( 0 S A ) ) )
128, 11mpanr1 665 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 1  +  0 ) S A )  =  ( ( 1 S A ) G ( 0 S A ) ) )
131, 9, 2vcid 21983 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
147, 12, 133eqtr3a 2460 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 0 S A ) )  =  A )
1513oveq1d 6055 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 0 S A ) )  =  ( A G ( 0 S A ) ) )
164, 14, 153eqtr2rd 2443 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G ( 0 S A ) )  =  ( A G Z ) )
171, 9, 2vccl 21982 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  0  e.  CC  /\  A  e.  X )  ->  ( 0 S A )  e.  X )
188, 17mp3an2 1267 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  e.  X
)
191, 2, 3vczcl 21998 . . . . 5  |-  ( W  e.  CVec OLD  ->  Z  e.  X )
2019adantr 452 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  Z  e.  X
)
21 simpr 448 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  A  e.  X
)
2218, 20, 213jca 1134 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 0 S A )  e.  X  /\  Z  e.  X  /\  A  e.  X ) )
231, 2vclcan 21997 . . 3  |-  ( ( W  e.  CVec OLD  /\  ( ( 0 S A )  e.  X  /\  Z  e.  X  /\  A  e.  X
) )  ->  (
( A G ( 0 S A ) )  =  ( A G Z )  <->  ( 0 S A )  =  Z ) )
2422, 23syldan 457 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( A G ( 0 S A ) )  =  ( A G Z )  <->  ( 0 S A )  =  Z ) )
2516, 24mpbid 202 1  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ran crn 4838   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949  GIdcgi 21728   CVec OLDcvc 21977
This theorem is referenced by:  vcz  22002  vcm  22003  nv0  22071
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-grpo 21732  df-gid 21733  df-ginv 21734  df-ablo 21823  df-vc 21978
  Copyright terms: Public domain W3C validator