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Theorem sspims 22082
Description: The induced metric on a subspace is a restriction of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspims.y  |-  Y  =  ( BaseSet `  W )
sspims.d  |-  D  =  ( IndMet `  U )
sspims.c  |-  C  =  ( IndMet `  W )
sspims.h  |-  H  =  ( SubSp `  U )
Assertion
Ref Expression
sspims  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  C  =  ( D  |`  ( Y  X.  Y
) ) )

Proof of Theorem sspims
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sspims.y . 2  |-  Y  =  ( BaseSet `  W )
2 sspims.h . 2  |-  H  =  ( SubSp `  U )
3 sspims.d . . 3  |-  D  =  ( IndMet `  U )
4 sspims.c . . 3  |-  C  =  ( IndMet `  W )
51, 3, 4, 2sspimsval 22081 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
x C y )  =  ( x D y ) )
61, 4imsdf 22023 . 2  |-  ( W  e.  NrmCVec  ->  C : ( Y  X.  Y ) --> RR )
7 eqid 2381 . . 3  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
87, 3imsdf 22023 . 2  |-  ( U  e.  NrmCVec  ->  D : ( ( BaseSet `  U )  X.  ( BaseSet `  U )
) --> RR )
91, 2, 5, 6, 8sspmlem 22073 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  C  =  ( D  |`  ( Y  X.  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    X. cxp 4810    |` cres 4814   ` cfv 5388   RRcr 8916   NrmCVeccnv 21905   BaseSetcba 21907   IndMetcims 21912   SubSpcss 22062
This theorem is referenced by:  bnsscmcl  22212  minvecolem4a  22221
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 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-po 4438  df-so 4439  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-riota 6479  df-er 6835  df-en 7040  df-dom 7041  df-sdom 7042  df-pnf 9049  df-mnf 9050  df-ltxr 9052  df-sub 9219  df-neg 9220  df-grpo 21621  df-gid 21622  df-ginv 21623  df-gdiv 21624  df-ablo 21712  df-vc 21867  df-nv 21913  df-va 21916  df-ba 21917  df-sm 21918  df-0v 21919  df-vs 21920  df-nmcv 21921  df-ims 21922  df-ssp 22063
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