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Theorem sspimsval 22239
Description: The induced metric on a subspace in terms 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
sspimsval  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A C B )  =  ( A D B ) )

Proof of Theorem sspimsval
StepHypRef Expression
1 sspims.h . . . . . 6  |-  H  =  ( SubSp `  U )
21sspnv 22225 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
3 sspims.y . . . . . . 7  |-  Y  =  ( BaseSet `  W )
4 eqid 2436 . . . . . . 7  |-  ( -v
`  W )  =  ( -v `  W
)
53, 4nvmcl 22128 . . . . . 6  |-  ( ( W  e.  NrmCVec  /\  A  e.  Y  /\  B  e.  Y )  ->  ( A ( -v `  W ) B )  e.  Y )
653expb 1154 . . . . 5  |-  ( ( W  e.  NrmCVec  /\  ( A  e.  Y  /\  B  e.  Y )
)  ->  ( A
( -v `  W
) B )  e.  Y )
72, 6sylan 458 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( -v `  W ) B )  e.  Y )
8 eqid 2436 . . . . . 6  |-  ( normCV `  U )  =  (
normCV
`  U )
9 eqid 2436 . . . . . 6  |-  ( normCV `  W )  =  (
normCV
`  W )
103, 8, 9, 1sspnval 22236 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H  /\  ( A ( -v `  W ) B )  e.  Y )  -> 
( ( normCV `  W
) `  ( A
( -v `  W
) B ) )  =  ( ( normCV `  U ) `  ( A ( -v `  W ) B ) ) )
11103expa 1153 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A ( -v
`  W ) B )  e.  Y )  ->  ( ( normCV `  W ) `  ( A ( -v `  W ) B ) )  =  ( (
normCV
`  U ) `  ( A ( -v `  W ) B ) ) )
127, 11syldan 457 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  (
( normCV `  W ) `  ( A ( -v `  W ) B ) )  =  ( (
normCV
`  U ) `  ( A ( -v `  W ) B ) ) )
13 eqid 2436 . . . . 5  |-  ( -v
`  U )  =  ( -v `  U
)
143, 13, 4, 1sspmval 22232 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( -v `  W ) B )  =  ( A ( -v `  U ) B ) )
1514fveq2d 5732 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  (
( normCV `  U ) `  ( A ( -v `  W ) B ) )  =  ( (
normCV
`  U ) `  ( A ( -v `  U ) B ) ) )
1612, 15eqtrd 2468 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  (
( normCV `  W ) `  ( A ( -v `  W ) B ) )  =  ( (
normCV
`  U ) `  ( A ( -v `  U ) B ) ) )
17 sspims.c . . . . 5  |-  C  =  ( IndMet `  W )
183, 4, 9, 17imsdval 22178 . . . 4  |-  ( ( W  e.  NrmCVec  /\  A  e.  Y  /\  B  e.  Y )  ->  ( A C B )  =  ( ( normCV `  W
) `  ( A
( -v `  W
) B ) ) )
19183expb 1154 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( A  e.  Y  /\  B  e.  Y )
)  ->  ( A C B )  =  ( ( normCV `  W ) `  ( A ( -v `  W ) B ) ) )
202, 19sylan 458 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A C B )  =  ( ( normCV `  W
) `  ( A
( -v `  W
) B ) ) )
21 eqid 2436 . . . . . . 7  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2221, 3, 1sspba 22226 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
2322sseld 3347 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( A  e.  Y  ->  A  e.  ( BaseSet `  U
) ) )
2422sseld 3347 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( B  e.  Y  ->  B  e.  ( BaseSet `  U
) ) )
2523, 24anim12d 547 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( A  e.  Y  /\  B  e.  Y
)  ->  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet
`  U ) ) ) )
2625imp 419 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet `  U )
) )
27 sspims.d . . . . . 6  |-  D  =  ( IndMet `  U )
2821, 13, 8, 27imsdval 22178 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet
`  U ) )  ->  ( A D B )  =  ( ( normCV `  U ) `  ( A ( -v `  U ) B ) ) )
29283expb 1154 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet `  U )
) )  ->  ( A D B )  =  ( ( normCV `  U
) `  ( A
( -v `  U
) B ) ) )
3029adantlr 696 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  (
BaseSet `  U )  /\  B  e.  ( BaseSet `  U ) ) )  ->  ( A D B )  =  ( ( normCV `  U ) `  ( A ( -v `  U ) B ) ) )
3126, 30syldan 457 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A D B )  =  ( ( normCV `  U
) `  ( A
( -v `  U
) B ) ) )
3216, 20, 313eqtr4d 2478 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A C B )  =  ( A D B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   NrmCVeccnv 22063   BaseSetcba 22065   -vcnsb 22068   normCVcnmcv 22069   IndMetcims 22070   SubSpcss 22220
This theorem is referenced by:  sspims  22240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gdiv 21782  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-vs 22078  df-nmcv 22079  df-ims 22080  df-ssp 22221
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