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Theorem sspimsval 25851
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 25837 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
3 sspims.y . . . . . . 7  |-  Y  =  ( BaseSet `  W )
4 eqid 2454 . . . . . . 7  |-  ( -v
`  W )  =  ( -v `  W
)
53, 4nvmcl 25740 . . . . . 6  |-  ( ( W  e.  NrmCVec  /\  A  e.  Y  /\  B  e.  Y )  ->  ( A ( -v `  W ) B )  e.  Y )
653expb 1195 . . . . 5  |-  ( ( W  e.  NrmCVec  /\  ( A  e.  Y  /\  B  e.  Y )
)  ->  ( A
( -v `  W
) B )  e.  Y )
72, 6sylan 469 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( -v `  W ) B )  e.  Y )
8 eqid 2454 . . . . . 6  |-  ( normCV `  U )  =  (
normCV
`  U )
9 eqid 2454 . . . . . 6  |-  ( normCV `  W )  =  (
normCV
`  W )
103, 8, 9, 1sspnval 25848 . . . . 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 1194 . . . 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 468 . . 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 2454 . . . . 5  |-  ( -v
`  U )  =  ( -v `  U
)
143, 13, 4, 1sspmval 25844 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( -v `  W ) B )  =  ( A ( -v `  U ) B ) )
1514fveq2d 5852 . . 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 2495 . 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 25790 . . . 4  |-  ( ( W  e.  NrmCVec  /\  A  e.  Y  /\  B  e.  Y )  ->  ( A C B )  =  ( ( normCV `  W
) `  ( A
( -v `  W
) B ) ) )
19183expb 1195 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( A  e.  Y  /\  B  e.  Y )
)  ->  ( A C B )  =  ( ( normCV `  W ) `  ( A ( -v `  W ) B ) ) )
202, 19sylan 469 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A C B )  =  ( ( normCV `  W
) `  ( A
( -v `  W
) B ) ) )
21 eqid 2454 . . . . . . 7  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2221, 3, 1sspba 25838 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
2322sseld 3488 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( A  e.  Y  ->  A  e.  ( BaseSet `  U
) ) )
2422sseld 3488 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( B  e.  Y  ->  B  e.  ( BaseSet `  U
) ) )
2523, 24anim12d 561 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( A  e.  Y  /\  B  e.  Y
)  ->  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet
`  U ) ) ) )
2625imp 427 . . 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 25790 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet
`  U ) )  ->  ( A D B )  =  ( ( normCV `  U ) `  ( A ( -v `  U ) B ) ) )
29283expb 1195 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet `  U )
) )  ->  ( A D B )  =  ( ( normCV `  U
) `  ( A
( -v `  U
) B ) ) )
3029adantlr 712 . . 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 468 . 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 2505 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A C B )  =  ( A D B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   NrmCVeccnv 25675   BaseSetcba 25677   -vcnsb 25680   normCVcnmcv 25681   IndMetcims 25682   SubSpcss 25832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-neg 9799  df-grpo 25391  df-gid 25392  df-ginv 25393  df-gdiv 25394  df-ablo 25482  df-vc 25637  df-nv 25683  df-va 25686  df-ba 25687  df-sm 25688  df-0v 25689  df-vs 25690  df-nmcv 25691  df-ims 25692  df-ssp 25833
This theorem is referenced by:  sspims  25852
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