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Theorem sspmval 24284
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 24277 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
3 neg1cn 10537 . . . . . . . . 9  |-  -u 1  e.  CC
4 sspm.y . . . . . . . . . 10  |-  Y  =  ( BaseSet `  W )
5 eqid 2454 . . . . . . . . . 10  |-  ( .sOLD `  W )  =  ( .sOLD `  W )
64, 5nvscl 24159 . . . . . . . . 9  |-  ( ( W  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  Y )  ->  ( -u 1 ( .sOLD `  W ) B )  e.  Y )
73, 6mp3an2 1303 . . . . . . . 8  |-  ( ( W  e.  NrmCVec  /\  B  e.  Y )  ->  ( -u 1 ( .sOLD `  W ) B )  e.  Y )
87ex 434 . . . . . . 7  |-  ( W  e.  NrmCVec  ->  ( B  e.  Y  ->  ( -u 1
( .sOLD `  W ) B )  e.  Y ) )
92, 8syl 16 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( B  e.  Y  ->  (
-u 1 ( .sOLD `  W ) B )  e.  Y
) )
109anim2d 565 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( A  e.  Y  /\  B  e.  Y
)  ->  ( A  e.  Y  /\  ( -u 1 ( .sOLD `  W ) B )  e.  Y ) ) )
1110imp 429 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A  e.  Y  /\  ( -u 1 ( .sOLD `  W ) B )  e.  Y
) )
12 eqid 2454 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
13 eqid 2454 . . . . 5  |-  ( +v
`  W )  =  ( +v `  W
)
144, 12, 13, 1sspgval 24280 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  ( -u 1 ( .sOLD `  W
) B )  e.  Y ) )  -> 
( A ( +v
`  W ) (
-u 1 ( .sOLD `  W ) B ) )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  W ) B ) ) )
1511, 14syldan 470 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( +v `  W ) ( -u
1 ( .sOLD `  W ) B ) )  =  ( A ( +v `  U
) ( -u 1
( .sOLD `  W ) B ) ) )
16 eqid 2454 . . . . . . 7  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
174, 16, 5, 1sspsval 24282 . . . . . 6  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( -u 1  e.  CC  /\  B  e.  Y ) )  -> 
( -u 1 ( .sOLD `  W ) B )  =  (
-u 1 ( .sOLD `  U ) B ) )
183, 17mpanr1 683 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  B  e.  Y
)  ->  ( -u 1
( .sOLD `  W ) B )  =  ( -u 1
( .sOLD `  U ) B ) )
1918adantrl 715 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( -u 1 ( .sOLD `  W ) B )  =  ( -u 1
( .sOLD `  U ) B ) )
2019oveq2d 6217 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( +v `  U ) ( -u
1 ( .sOLD `  W ) B ) )  =  ( A ( +v `  U
) ( -u 1
( .sOLD `  U ) B ) ) )
2115, 20eqtrd 2495 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A ( +v `  W ) ( -u
1 ( .sOLD `  W ) B ) )  =  ( A ( +v `  U
) ( -u 1
( .sOLD `  U ) B ) ) )
22 sspm.l . . . . 5  |-  L  =  ( -v `  W
)
234, 13, 5, 22nvmval 24175 . . . 4  |-  ( ( W  e.  NrmCVec  /\  A  e.  Y  /\  B  e.  Y )  ->  ( A L B )  =  ( A ( +v
`  W ) (
-u 1 ( .sOLD `  W ) B ) ) )
24233expb 1189 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( A  e.  Y  /\  B  e.  Y )
)  ->  ( A L B )  =  ( A ( +v `  W ) ( -u
1 ( .sOLD `  W ) B ) ) )
252, 24sylan 471 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A L B )  =  ( A ( +v
`  W ) (
-u 1 ( .sOLD `  W ) B ) ) )
26 eqid 2454 . . . . . . 7  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2726, 4, 1sspba 24278 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
2827sseld 3464 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( A  e.  Y  ->  A  e.  ( BaseSet `  U
) ) )
2927sseld 3464 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( B  e.  Y  ->  B  e.  ( BaseSet `  U
) ) )
3028, 29anim12d 563 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( A  e.  Y  /\  B  e.  Y
)  ->  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet
`  U ) ) ) )
3130imp 429 . . 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 24175 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet
`  U ) )  ->  ( A M B )  =  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) B ) ) )
34333expb 1189 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  ( BaseSet `  U )  /\  B  e.  ( BaseSet `  U )
) )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) B ) ) )
3534adantlr 714 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  (
BaseSet `  U )  /\  B  e.  ( BaseSet `  U ) ) )  ->  ( A M B )  =  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) B ) ) )
3631, 35syldan 470 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) B ) ) )
3721, 25, 363eqtr4d 2505 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A L B )  =  ( A M B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ` cfv 5527  (class class class)co 6201   CCcc 9392   1c1 9395   -ucneg 9708   NrmCVeccnv 24115   +vcpv 24116   BaseSetcba 24117   .sOLDcns 24118   -vcnsb 24120   SubSpcss 24272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-ltxr 9535  df-sub 9709  df-neg 9710  df-grpo 23831  df-gid 23832  df-ginv 23833  df-gdiv 23834  df-ablo 23922  df-vc 24077  df-nv 24123  df-va 24126  df-ba 24127  df-sm 24128  df-0v 24129  df-vs 24130  df-nmcv 24131  df-ssp 24273
This theorem is referenced by:  sspm  24285  sspz  24286  sspimsval  24291  sspph  24408
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