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Theorem sspimsval 21371
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 21357 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
3 sspims.y . . . . . . 7  |-  Y  =  ( BaseSet `  W )
4 eqid 2316 . . . . . . 7  |-  ( -v
`  W )  =  ( -v `  W
)
53, 4nvmcl 21260 . . . . . 6  |-  ( ( W  e.  NrmCVec  /\  A  e.  Y  /\  B  e.  Y )  ->  ( A ( -v `  W ) B )  e.  Y )
653expb 1152 . . . . 5  |-  ( ( W  e.  NrmCVec  /\  ( A  e.  Y  /\  B  e.  Y )
)  ->  ( A
( -v `  W
) B )  e.  Y )
72, 6sylan 457 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( -v `  W ) B )  e.  Y )
8 eqid 2316 . . . . . 6  |-  ( normCV `  U )  =  (
normCV
`  U )
9 eqid 2316 . . . . . 6  |-  ( normCV `  W )  =  (
normCV
`  W )
103, 8, 9, 1sspnval 21368 . . . . 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 1151 . . . 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 456 . . 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 2316 . . . . 5  |-  ( -v
`  U )  =  ( -v `  U
)
143, 13, 4, 1sspmval 21364 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( -v `  W ) B )  =  ( A ( -v `  U ) B ) )
1514fveq2d 5567 . . 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 2348 . 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 21310 . . . 4  |-  ( ( W  e.  NrmCVec  /\  A  e.  Y  /\  B  e.  Y )  ->  ( A C B )  =  ( ( normCV `  W
) `  ( A
( -v `  W
) B ) ) )
19183expb 1152 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( A  e.  Y  /\  B  e.  Y )
)  ->  ( A C B )  =  ( ( normCV `  W ) `  ( A ( -v `  W ) B ) ) )
202, 19sylan 457 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A C B )  =  ( ( normCV `  W
) `  ( A
( -v `  W
) B ) ) )
21 eqid 2316 . . . . . . 7  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2221, 3, 1sspba 21358 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
2322sseld 3213 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( A  e.  Y  ->  A  e.  ( BaseSet `  U
) ) )
2422sseld 3213 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( B  e.  Y  ->  B  e.  ( BaseSet `  U
) ) )
2523, 24anim12d 546 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( A  e.  Y  /\  B  e.  Y
)  ->  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet
`  U ) ) ) )
2625imp 418 . . 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 21310 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet
`  U ) )  ->  ( A D B )  =  ( ( normCV `  U ) `  ( A ( -v `  U ) B ) ) )
29283expb 1152 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet `  U )
) )  ->  ( A D B )  =  ( ( normCV `  U
) `  ( A
( -v `  U
) B ) ) )
3029adantlr 695 . . 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 456 . 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 2358 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 358    = wceq 1633    e. wcel 1701   ` cfv 5292  (class class class)co 5900   NrmCVeccnv 21195   BaseSetcba 21197   -vcnsb 21200   normCVcnmcv 21201   IndMetcims 21202   SubSpcss 21352
This theorem is referenced by:  sspims  21372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-ltxr 8917  df-sub 9084  df-neg 9085  df-grpo 20911  df-gid 20912  df-ginv 20913  df-gdiv 20914  df-ablo 21002  df-vc 21157  df-nv 21203  df-va 21206  df-ba 21207  df-sm 21208  df-0v 21209  df-vs 21210  df-nmcv 21211  df-ims 21212  df-ssp 21353
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