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Theorem obsipid 16951
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 2438 . . . 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 2438 . . . 4  |-  ( 0g
`  F )  =  ( 0g `  F
)
61, 2, 3, 4, 5obsip 16950 . . 3  |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B  /\  A  e.  B )  ->  ( A  .,  A )  =  if ( A  =  A ,  .1.  , 
( 0g `  F
) ) )
763anidm23 1244 . 2  |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B )  ->  ( A  .,  A )  =  if ( A  =  A ,  .1.  , 
( 0g `  F
) ) )
8 eqid 2438 . . 3  |-  A  =  A
9 iftrue 3747 . . 3  |-  ( A  =  A  ->  if ( A  =  A ,  .1.  ,  ( 0g
`  F ) )  =  .1.  )
108, 9ax-mp 8 . 2  |-  if ( A  =  A ,  .1.  ,  ( 0g `  F ) )  =  .1.
117, 10syl6eq 2486 1  |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B )  ->  ( A  .,  A )  =  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   ifcif 3741   ` cfv 5456  (class class class)co 6083   Basecbs 13471  Scalarcsca 13534   .icip 13536   0gc0g 13725   1rcur 15664  OBasiscobs 16931
This theorem is referenced by:  obsne0  16954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-obs 16934
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