MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspmval Unicode version

Theorem sspmval 21325
Description: Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspm.y  |-  Y  =  ( BaseSet `  W )
sspm.m  |-  M  =  ( -v `  U
)
sspm.l  |-  L  =  ( -v `  W
)
sspm.h  |-  H  =  ( SubSp `  U )
Assertion
Ref Expression
sspmval  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A L B )  =  ( A M B ) )

Proof of Theorem sspmval
StepHypRef Expression
1 sspm.h . . . . . . . 8  |-  H  =  ( SubSp `  U )
21sspnv 21318 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
3 neg1cn 9829 . . . . . . . . 9  |-  -u 1  e.  CC
4 sspm.y . . . . . . . . . 10  |-  Y  =  ( BaseSet `  W )
5 eqid 2296 . . . . . . . . . 10  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
64, 5nvscl 21200 . . . . . . . . 9  |-  ( ( W  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  Y )  ->  ( -u 1 ( .s OLD `  W ) B )  e.  Y )
73, 6mp3an2 1265 . . . . . . . 8  |-  ( ( W  e.  NrmCVec  /\  B  e.  Y )  ->  ( -u 1 ( .s OLD `  W ) B )  e.  Y )
87ex 423 . . . . . . 7  |-  ( W  e.  NrmCVec  ->  ( B  e.  Y  ->  ( -u 1
( .s OLD `  W
) B )  e.  Y ) )
92, 8syl 15 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( B  e.  Y  ->  (
-u 1 ( .s
OLD `  W ) B )  e.  Y
) )
109anim2d 548 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( A  e.  Y  /\  B  e.  Y
)  ->  ( A  e.  Y  /\  ( -u 1 ( .s OLD `  W ) B )  e.  Y ) ) )
1110imp 418 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A  e.  Y  /\  ( -u 1 ( .s
OLD `  W ) B )  e.  Y
) )
12 eqid 2296 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
13 eqid 2296 . . . . 5  |-  ( +v
`  W )  =  ( +v `  W
)
144, 12, 13, 1sspgval 21321 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  ( -u 1 ( .s OLD `  W
) B )  e.  Y ) )  -> 
( A ( +v
`  W ) (
-u 1 ( .s
OLD `  W ) B ) )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  W ) B ) ) )
1511, 14syldan 456 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( +v `  W ) ( -u
1 ( .s OLD `  W ) B ) )  =  ( A ( +v `  U
) ( -u 1
( .s OLD `  W
) B ) ) )
16 eqid 2296 . . . . . . 7  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
174, 16, 5, 1sspsval 21323 . . . . . 6  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( -u 1  e.  CC  /\  B  e.  Y ) )  -> 
( -u 1 ( .s
OLD `  W ) B )  =  (
-u 1 ( .s
OLD `  U ) B ) )
183, 17mpanr1 664 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  B  e.  Y
)  ->  ( -u 1
( .s OLD `  W
) B )  =  ( -u 1 ( .s OLD `  U
) B ) )
1918adantrl 696 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( -u 1 ( .s OLD `  W ) B )  =  ( -u 1
( .s OLD `  U
) B ) )
2019oveq2d 5890 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  W ) B ) )  =  ( A ( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
2115, 20eqtrd 2328 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( +v `  W ) ( -u
1 ( .s OLD `  W ) B ) )  =  ( A ( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
22 sspm.l . . . . 5  |-  L  =  ( -v `  W
)
234, 13, 5, 22nvmval 21216 . . . 4  |-  ( ( W  e.  NrmCVec  /\  A  e.  Y  /\  B  e.  Y )  ->  ( A L B )  =  ( A ( +v
`  W ) (
-u 1 ( .s
OLD `  W ) B ) ) )
24233expb 1152 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( A  e.  Y  /\  B  e.  Y )
)  ->  ( A L B )  =  ( A ( +v `  W ) ( -u
1 ( .s OLD `  W ) B ) ) )
252, 24sylan 457 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A L B )  =  ( A ( +v
`  W ) (
-u 1 ( .s
OLD `  W ) B ) ) )
26 eqid 2296 . . . . . . 7  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2726, 4, 1sspba 21319 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
2827sseld 3192 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( A  e.  Y  ->  A  e.  ( BaseSet `  U
) ) )
2927sseld 3192 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( B  e.  Y  ->  B  e.  ( BaseSet `  U
) ) )
3028, 29anim12d 546 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( A  e.  Y  /\  B  e.  Y
)  ->  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet
`  U ) ) ) )
3130imp 418 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet `  U )
) )
32 sspm.m . . . . . 6  |-  M  =  ( -v `  U
)
3326, 12, 16, 32nvmval 21216 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet
`  U ) )  ->  ( A M B )  =  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) )
34333expb 1152 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet `  U )
) )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
3534adantlr 695 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  (
BaseSet `  U )  /\  B  e.  ( BaseSet `  U ) ) )  ->  ( A M B )  =  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) )
3631, 35syldan 456 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
3721, 25, 363eqtr4d 2338 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A L B )  =  ( A M B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754   -ucneg 9054   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159   -vcnsb 21161   SubSpcss 21313
This theorem is referenced by:  sspm  21326  sspz  21327  sspimsval  21332  sspph  21449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ssp 21314
  Copyright terms: Public domain W3C validator