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Theorem obsipid 16622
Description: A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
obsipid.h  |-  .,  =  ( .i `  W )
obsipid.f  |-  F  =  (Scalar `  W )
obsipid.u  |-  .1.  =  ( 1r `  F )
Assertion
Ref Expression
obsipid  |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B )  ->  ( A  .,  A )  =  .1.  )

Proof of Theorem obsipid
StepHypRef Expression
1 eqid 2283 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 obsipid.h . . . 4  |-  .,  =  ( .i `  W )
3 obsipid.f . . . 4  |-  F  =  (Scalar `  W )
4 obsipid.u . . . 4  |-  .1.  =  ( 1r `  F )
5 eqid 2283 . . . 4  |-  ( 0g
`  F )  =  ( 0g `  F
)
61, 2, 3, 4, 5obsip 16621 . . 3  |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B  /\  A  e.  B )  ->  ( A  .,  A )  =  if ( A  =  A ,  .1.  , 
( 0g `  F
) ) )
763anidm23 1241 . 2  |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B )  ->  ( A  .,  A )  =  if ( A  =  A ,  .1.  , 
( 0g `  F
) ) )
8 eqid 2283 . . 3  |-  A  =  A
9 iftrue 3571 . . 3  |-  ( A  =  A  ->  if ( A  =  A ,  .1.  ,  ( 0g
`  F ) )  =  .1.  )
108, 9ax-mp 8 . 2  |-  if ( A  =  A ,  .1.  ,  ( 0g `  F ) )  =  .1.
117, 10syl6eq 2331 1  |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B )  ->  ( A  .,  A )  =  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ifcif 3565   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .icip 13213   0gc0g 13400   1rcur 15339  OBasiscobs 16602
This theorem is referenced by:  obsne0  16625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-obs 16605
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