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Theorem vc0 21118
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 21116 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G Z )  =  A )
5 ax-1cn 8791 . . . . . 6  |-  1  e.  CC
65addid1i 8995 . . . . 5  |-  ( 1  +  0 )  =  1
76oveq1i 5830 . . . 4  |-  ( ( 1  +  0 ) S A )  =  ( 1 S A )
8 0cn 8827 . . . . 5  |-  0  e.  CC
9 vc0.2 . . . . . . 7  |-  S  =  ( 2nd `  W
)
101, 9, 2vcdir 21102 . . . . . 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 666 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 1  +  0 ) S A )  =  ( ( 1 S A ) G ( 0 S A ) ) )
131, 9, 2vcid 21100 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
147, 12, 133eqtr3a 2341 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 0 S A ) )  =  A )
1513oveq1d 5835 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 0 S A ) )  =  ( A G ( 0 S A ) ) )
164, 14, 153eqtr2rd 2324 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G ( 0 S A ) )  =  ( A G Z ) )
171, 9, 2vccl 21099 . . . . 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 21115 . . . . 5  |-  ( W  e.  CVec OLD  ->  Z  e.  X )
2019adantr 453 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  Z  e.  X
)
21 simpr 449 . . . 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 21114 . . 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 458 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( A G ( 0 S A ) )  =  ( A G Z )  <->  ( 0 S A )  =  Z ) )
2516, 24mpbid 203 1  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   ran crn 4690   ` cfv 5222  (class class class)co 5820   1stc1st 6082   2ndc2nd 6083   CCcc 8731   0cc0 8733   1c1 8734    + caddc 8736  GIdcgi 20847   CVec OLDcvc 21094
This theorem is referenced by:  vcz  21119  vcm  21120  nv0  21188
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-ltxr 8868  df-grpo 20851  df-gid 20852  df-ginv 20853  df-ablo 20942  df-vc 21095
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