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Theorem vc0 21898
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 21896 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G Z )  =  A )
5 ax-1cn 8983 . . . . . 6  |-  1  e.  CC
65addid1i 9187 . . . . 5  |-  ( 1  +  0 )  =  1
76oveq1i 6032 . . . 4  |-  ( ( 1  +  0 ) S A )  =  ( 1 S A )
8 0cn 9019 . . . . 5  |-  0  e.  CC
9 vc0.2 . . . . . . 7  |-  S  =  ( 2nd `  W
)
101, 9, 2vcdir 21882 . . . . . 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 21880 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
147, 12, 133eqtr3a 2445 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 0 S A ) )  =  A )
1513oveq1d 6037 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 0 S A ) )  =  ( A G ( 0 S A ) ) )
164, 14, 153eqtr2rd 2428 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G ( 0 S A ) )  =  ( A G Z ) )
171, 9, 2vccl 21879 . . . . 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 21895 . . . . 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 21894 . . 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 1717   ran crn 4821   ` cfv 5396  (class class class)co 6022   1stc1st 6288   2ndc2nd 6289   CCcc 8923   0cc0 8925   1c1 8926    + caddc 8928  GIdcgi 21625   CVec OLDcvc 21874
This theorem is referenced by:  vcz  21899  vcm  21900  nv0  21968
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-1st 6290  df-2nd 6291  df-riota 6487  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-ltxr 9060  df-grpo 21629  df-gid 21630  df-ginv 21631  df-ablo 21720  df-vc 21875
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