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Theorem vc0 21141
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 21139 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G Z )  =  A )
5 ax-1cn 8811 . . . . . 6  |-  1  e.  CC
65addid1i 9015 . . . . 5  |-  ( 1  +  0 )  =  1
76oveq1i 5884 . . . 4  |-  ( ( 1  +  0 ) S A )  =  ( 1 S A )
8 0cn 8847 . . . . 5  |-  0  e.  CC
9 vc0.2 . . . . . . 7  |-  S  =  ( 2nd `  W
)
101, 9, 2vcdir 21125 . . . . . 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 1274 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  ( 0  e.  CC  /\  A  e.  X ) )  ->  ( (
1  +  0 ) S A )  =  ( ( 1 S A ) G ( 0 S A ) ) )
128, 11mpanr1 664 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 1  +  0 ) S A )  =  ( ( 1 S A ) G ( 0 S A ) ) )
131, 9, 2vcid 21123 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
147, 12, 133eqtr3a 2352 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 0 S A ) )  =  A )
1513oveq1d 5889 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 0 S A ) )  =  ( A G ( 0 S A ) ) )
164, 14, 153eqtr2rd 2335 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( A G ( 0 S A ) )  =  ( A G Z ) )
171, 9, 2vccl 21122 . . . . 5  |-  ( ( W  e.  CVec OLD  /\  0  e.  CC  /\  A  e.  X )  ->  ( 0 S A )  e.  X )
188, 17mp3an2 1265 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  e.  X
)
191, 2, 3vczcl 21138 . . . . 5  |-  ( W  e.  CVec OLD  ->  Z  e.  X )
2019adantr 451 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  Z  e.  X
)
21 simpr 447 . . . 4  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  A  e.  X
)
2218, 20, 213jca 1132 . . 3  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( 0 S A )  e.  X  /\  Z  e.  X  /\  A  e.  X ) )
231, 2vclcan 21137 . . 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 456 . 2  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( ( A G ( 0 S A ) )  =  ( A G Z )  <->  ( 0 S A )  =  Z ) )
2516, 24mpbid 201 1  |-  ( ( W  e.  CVec OLD  /\  A  e.  X )  ->  ( 0 S A )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756  GIdcgi 20870   CVec OLDcvc 21117
This theorem is referenced by:  vcz  21142  vcm  21143  nv0  21211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-vc 21118
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