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Theorem shsel3 22774
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 22773 . 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 22670 . . . . . . . . . . 11  |-  ( ( A  e.  SH  /\  x  e.  A )  ->  x  e.  ~H )
4 shel 22670 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  z  e.  B )  ->  z  e.  ~H )
5 hvaddsubval 22492 . . . . . . . . . . 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 2462 . . . . . . 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 10027 . . . . . . . . . . 11  |-  -u 1  e.  CC
11 shmulcl 22677 . . . . . . . . . . 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 6052 . . . . . . . . . 10  |-  ( y  =  ( -u 1  .h  z )  ->  (
x  -h  y )  =  ( x  -h  ( -u 1  .h  z
) ) )
1615eqeq2d 2419 . . . . . . . . 9  |-  ( y  =  ( -u 1  .h  z )  ->  ( C  =  ( x  -h  y )  <->  C  =  ( x  -h  ( -u 1  .h  z ) ) ) )
1716rspcev 3016 . . . . . . . 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 2794 . . . 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 22670 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  y  e.  B )  ->  y  e.  ~H )
24 hvsubval 22476 . . . . . . . . . . 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 2462 . . . . . . 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 22677 . . . . . . . . . . 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 6052 . . . . . . . . . 10  |-  ( z  =  ( -u 1  .h  y )  ->  (
x  +h  z )  =  ( x  +h  ( -u 1  .h  y
) ) )
3433eqeq2d 2419 . . . . . . . . 9  |-  ( z  =  ( -u 1  .h  y )  ->  ( C  =  ( x  +h  z )  <->  C  =  ( x  +h  ( -u 1  .h  y ) ) ) )
3534rspcev 3016 . . . . . . . 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 2794 . . . 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 2687 . 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 1721   E.wrex 2671  (class class class)co 6044   CCcc 8948   1c1 8951   -ucneg 9252   ~Hchil 22379    +h cva 22380    .h csm 22381    -h cmv 22385   SHcsh 22388    +H cph 22391
This theorem is referenced by:  pjimai  23636
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-hilex 22459  ax-hfvadd 22460  ax-hvcom 22461  ax-hvass 22462  ax-hv0cl 22463  ax-hvaddid 22464  ax-hfvmul 22465  ax-hvmulid 22466  ax-hvmulass 22467  ax-hvdistr2 22469  ax-hvmul0 22470
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-riota 6512  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-ltxr 9085  df-sub 9253  df-neg 9254  df-grpo 21736  df-ablo 21827  df-hvsub 22431  df-sh 22666  df-shs 22767
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