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Theorem sspmval 21309
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 21302 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
3 neg1cn 9813 . . . . . . . . 9  |-  -u 1  e.  CC
4 sspm.y . . . . . . . . . 10  |-  Y  =  ( BaseSet `  W )
5 eqid 2283 . . . . . . . . . 10  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
64, 5nvscl 21184 . . . . . . . . 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 2283 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
13 eqid 2283 . . . . 5  |-  ( +v
`  W )  =  ( +v `  W
)
144, 12, 13, 1sspgval 21305 . . . 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 2283 . . . . . . 7  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
174, 16, 5, 1sspsval 21307 . . . . . 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 5874 . . 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 2315 . 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 21200 . . . 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 2283 . . . . . . 7  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2726, 4, 1sspba 21303 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
2827sseld 3179 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( A  e.  Y  ->  A  e.  ( BaseSet `  U
) ) )
2927sseld 3179 . . . . 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 21200 . . . . 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 2325 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   -ucneg 9038   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   -vcnsb 21145   SubSpcss 21297
This theorem is referenced by:  sspm  21310  sspz  21311  sspimsval  21316  sspph  21433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ssp 21298
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