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Theorem shsel3 22665
Description: Membership in the subspace sum of two Hilbert subspaces, using vector subtraction. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
Assertion
Ref Expression
shsel3  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  -h  y ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y

Proof of Theorem shsel3
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 shsel 22664 . 2  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. z  e.  B  C  =  ( x  +h  z ) ) )
2 id 20 . . . . . . . 8  |-  ( C  =  ( x  +h  z )  ->  C  =  ( x  +h  z ) )
3 shel 22561 . . . . . . . . . . 11  |-  ( ( A  e.  SH  /\  x  e.  A )  ->  x  e.  ~H )
4 shel 22561 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  z  e.  B )  ->  z  e.  ~H )
5 hvaddsubval 22383 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  +h  z
)  =  ( x  -h  ( -u 1  .h  z ) ) )
63, 4, 5syl2an 464 . . . . . . . . . 10  |-  ( ( ( A  e.  SH  /\  x  e.  A )  /\  ( B  e.  SH  /\  z  e.  B ) )  -> 
( x  +h  z
)  =  ( x  -h  ( -u 1  .h  z ) ) )
76an4s 800 . . . . . . . . 9  |-  ( ( ( A  e.  SH  /\  B  e.  SH )  /\  ( x  e.  A  /\  z  e.  B ) )  -> 
( x  +h  z
)  =  ( x  -h  ( -u 1  .h  z ) ) )
87anassrs 630 . . . . . . . 8  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  z  e.  B )  ->  (
x  +h  z )  =  ( x  -h  ( -u 1  .h  z
) ) )
92, 8sylan9eqr 2441 . . . . . . 7  |-  ( ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  z  e.  B )  /\  C  =  ( x  +h  z ) )  ->  C  =  ( x  -h  ( -u 1  .h  z ) ) )
10 neg1cn 9999 . . . . . . . . . . 11  |-  -u 1  e.  CC
11 shmulcl 22568 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  -u 1  e.  CC  /\  z  e.  B )  ->  ( -u 1  .h  z )  e.  B
)
1210, 11mp3an2 1267 . . . . . . . . . 10  |-  ( ( B  e.  SH  /\  z  e.  B )  ->  ( -u 1  .h  z )  e.  B
)
1312adantll 695 . . . . . . . . 9  |-  ( ( ( A  e.  SH  /\  B  e.  SH )  /\  z  e.  B
)  ->  ( -u 1  .h  z )  e.  B
)
1413adantlr 696 . . . . . . . 8  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  z  e.  B )  ->  ( -u 1  .h  z )  e.  B )
15 oveq2 6028 . . . . . . . . . 10  |-  ( y  =  ( -u 1  .h  z )  ->  (
x  -h  y )  =  ( x  -h  ( -u 1  .h  z
) ) )
1615eqeq2d 2398 . . . . . . . . 9  |-  ( y  =  ( -u 1  .h  z )  ->  ( C  =  ( x  -h  y )  <->  C  =  ( x  -h  ( -u 1  .h  z ) ) ) )
1716rspcev 2995 . . . . . . . 8  |-  ( ( ( -u 1  .h  z )  e.  B  /\  C  =  (
x  -h  ( -u
1  .h  z ) ) )  ->  E. y  e.  B  C  =  ( x  -h  y
) )
1814, 17sylan 458 . . . . . . 7  |-  ( ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  z  e.  B )  /\  C  =  ( x  -h  ( -u 1  .h  z
) ) )  ->  E. y  e.  B  C  =  ( x  -h  y ) )
199, 18syldan 457 . . . . . 6  |-  ( ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  z  e.  B )  /\  C  =  ( x  +h  z ) )  ->  E. y  e.  B  C  =  ( x  -h  y ) )
2019ex 424 . . . . 5  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  z  e.  B )  ->  ( C  =  ( x  +h  z )  ->  E. y  e.  B  C  =  ( x  -h  y
) ) )
2120rexlimdva 2773 . . . 4  |-  ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A
)  ->  ( E. z  e.  B  C  =  ( x  +h  z )  ->  E. y  e.  B  C  =  ( x  -h  y
) ) )
22 id 20 . . . . . . . 8  |-  ( C  =  ( x  -h  y )  ->  C  =  ( x  -h  y ) )
23 shel 22561 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  y  e.  B )  ->  y  e.  ~H )
24 hvsubval 22367 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  =  ( x  +h  ( -u 1  .h  y ) ) )
253, 23, 24syl2an 464 . . . . . . . . . 10  |-  ( ( ( A  e.  SH  /\  x  e.  A )  /\  ( B  e.  SH  /\  y  e.  B ) )  -> 
( x  -h  y
)  =  ( x  +h  ( -u 1  .h  y ) ) )
2625an4s 800 . . . . . . . . 9  |-  ( ( ( A  e.  SH  /\  B  e.  SH )  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  -h  y
)  =  ( x  +h  ( -u 1  .h  y ) ) )
2726anassrs 630 . . . . . . . 8  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  y  e.  B )  ->  (
x  -h  y )  =  ( x  +h  ( -u 1  .h  y
) ) )
2822, 27sylan9eqr 2441 . . . . . . 7  |-  ( ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  y  e.  B )  /\  C  =  ( x  -h  y ) )  ->  C  =  ( x  +h  ( -u 1  .h  y ) ) )
29 shmulcl 22568 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  -u 1  e.  CC  /\  y  e.  B )  ->  ( -u 1  .h  y )  e.  B
)
3010, 29mp3an2 1267 . . . . . . . . . 10  |-  ( ( B  e.  SH  /\  y  e.  B )  ->  ( -u 1  .h  y )  e.  B
)
3130adantll 695 . . . . . . . . 9  |-  ( ( ( A  e.  SH  /\  B  e.  SH )  /\  y  e.  B
)  ->  ( -u 1  .h  y )  e.  B
)
3231adantlr 696 . . . . . . . 8  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  y  e.  B )  ->  ( -u 1  .h  y )  e.  B )
33 oveq2 6028 . . . . . . . . . 10  |-  ( z  =  ( -u 1  .h  y )  ->  (
x  +h  z )  =  ( x  +h  ( -u 1  .h  y
) ) )
3433eqeq2d 2398 . . . . . . . . 9  |-  ( z  =  ( -u 1  .h  y )  ->  ( C  =  ( x  +h  z )  <->  C  =  ( x  +h  ( -u 1  .h  y ) ) ) )
3534rspcev 2995 . . . . . . . 8  |-  ( ( ( -u 1  .h  y )  e.  B  /\  C  =  (
x  +h  ( -u
1  .h  y ) ) )  ->  E. z  e.  B  C  =  ( x  +h  z
) )
3632, 35sylan 458 . . . . . . 7  |-  ( ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  y  e.  B )  /\  C  =  ( x  +h  ( -u 1  .h  y
) ) )  ->  E. z  e.  B  C  =  ( x  +h  z ) )
3728, 36syldan 457 . . . . . 6  |-  ( ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  y  e.  B )  /\  C  =  ( x  -h  y ) )  ->  E. z  e.  B  C  =  ( x  +h  z ) )
3837ex 424 . . . . 5  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  y  e.  B )  ->  ( C  =  ( x  -h  y )  ->  E. z  e.  B  C  =  ( x  +h  z
) ) )
3938rexlimdva 2773 . . . 4  |-  ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A
)  ->  ( E. y  e.  B  C  =  ( x  -h  y )  ->  E. z  e.  B  C  =  ( x  +h  z
) ) )
4021, 39impbid 184 . . 3  |-  ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A
)  ->  ( E. z  e.  B  C  =  ( x  +h  z )  <->  E. y  e.  B  C  =  ( x  -h  y
) ) )
4140rexbidva 2666 . 2  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( E. x  e.  A  E. z  e.  B  C  =  ( x  +h  z )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  -h  y ) ) )
421, 41bitrd 245 1  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  -h  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2650  (class class class)co 6020   CCcc 8921   1c1 8924   -ucneg 9224   ~Hchil 22270    +h cva 22271    .h csm 22272    -h cmv 22276   SHcsh 22279    +H cph 22282
This theorem is referenced by:  pjimai  23527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-hilex 22350  ax-hfvadd 22351  ax-hvcom 22352  ax-hvass 22353  ax-hv0cl 22354  ax-hvaddid 22355  ax-hfvmul 22356  ax-hvmulid 22357  ax-hvmulass 22358  ax-hvdistr2 22360  ax-hvmul0 22361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-sub 9225  df-neg 9226  df-grpo 21627  df-ablo 21718  df-hvsub 22322  df-sh 22557  df-shs 22658
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