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Theorem shsel3 21896
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 21895 . 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 19 . . . . . . . 8  |-  ( C  =  ( x  +h  z )  ->  C  =  ( x  +h  z ) )
3 shel 21792 . . . . . . . . . . 11  |-  ( ( A  e.  SH  /\  x  e.  A )  ->  x  e.  ~H )
4 shel 21792 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  z  e.  B )  ->  z  e.  ~H )
5 hvaddsubval 21614 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  +h  z
)  =  ( x  -h  ( -u 1  .h  z ) ) )
63, 4, 5syl2an 463 . . . . . . . . . 10  |-  ( ( ( A  e.  SH  /\  x  e.  A )  /\  ( B  e.  SH  /\  z  e.  B ) )  -> 
( x  +h  z
)  =  ( x  -h  ( -u 1  .h  z ) ) )
76an4s 799 . . . . . . . . 9  |-  ( ( ( A  e.  SH  /\  B  e.  SH )  /\  ( x  e.  A  /\  z  e.  B ) )  -> 
( x  +h  z
)  =  ( x  -h  ( -u 1  .h  z ) ) )
87anassrs 629 . . . . . . . 8  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  z  e.  B )  ->  (
x  +h  z )  =  ( x  -h  ( -u 1  .h  z
) ) )
92, 8sylan9eqr 2339 . . . . . . 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 9815 . . . . . . . . . . 11  |-  -u 1  e.  CC
11 shmulcl 21799 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  -u 1  e.  CC  /\  z  e.  B )  ->  ( -u 1  .h  z )  e.  B
)
1210, 11mp3an2 1265 . . . . . . . . . 10  |-  ( ( B  e.  SH  /\  z  e.  B )  ->  ( -u 1  .h  z )  e.  B
)
1312adantll 694 . . . . . . . . 9  |-  ( ( ( A  e.  SH  /\  B  e.  SH )  /\  z  e.  B
)  ->  ( -u 1  .h  z )  e.  B
)
1413adantlr 695 . . . . . . . 8  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  z  e.  B )  ->  ( -u 1  .h  z )  e.  B )
15 oveq2 5868 . . . . . . . . . 10  |-  ( y  =  ( -u 1  .h  z )  ->  (
x  -h  y )  =  ( x  -h  ( -u 1  .h  z
) ) )
1615eqeq2d 2296 . . . . . . . . 9  |-  ( y  =  ( -u 1  .h  z )  ->  ( C  =  ( x  -h  y )  <->  C  =  ( x  -h  ( -u 1  .h  z ) ) ) )
1716rspcev 2886 . . . . . . . 8  |-  ( ( ( -u 1  .h  z )  e.  B  /\  C  =  (
x  -h  ( -u
1  .h  z ) ) )  ->  E. y  e.  B  C  =  ( x  -h  y
) )
1814, 17sylan 457 . . . . . . 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 456 . . . . . 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 423 . . . . 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 2669 . . . 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 19 . . . . . . . 8  |-  ( C  =  ( x  -h  y )  ->  C  =  ( x  -h  y ) )
23 shel 21792 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  y  e.  B )  ->  y  e.  ~H )
24 hvsubval 21598 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  =  ( x  +h  ( -u 1  .h  y ) ) )
253, 23, 24syl2an 463 . . . . . . . . . 10  |-  ( ( ( A  e.  SH  /\  x  e.  A )  /\  ( B  e.  SH  /\  y  e.  B ) )  -> 
( x  -h  y
)  =  ( x  +h  ( -u 1  .h  y ) ) )
2625an4s 799 . . . . . . . . 9  |-  ( ( ( A  e.  SH  /\  B  e.  SH )  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  -h  y
)  =  ( x  +h  ( -u 1  .h  y ) ) )
2726anassrs 629 . . . . . . . 8  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  y  e.  B )  ->  (
x  -h  y )  =  ( x  +h  ( -u 1  .h  y
) ) )
2822, 27sylan9eqr 2339 . . . . . . 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 21799 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  -u 1  e.  CC  /\  y  e.  B )  ->  ( -u 1  .h  y )  e.  B
)
3010, 29mp3an2 1265 . . . . . . . . . 10  |-  ( ( B  e.  SH  /\  y  e.  B )  ->  ( -u 1  .h  y )  e.  B
)
3130adantll 694 . . . . . . . . 9  |-  ( ( ( A  e.  SH  /\  B  e.  SH )  /\  y  e.  B
)  ->  ( -u 1  .h  y )  e.  B
)
3231adantlr 695 . . . . . . . 8  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH )  /\  x  e.  A )  /\  y  e.  B )  ->  ( -u 1  .h  y )  e.  B )
33 oveq2 5868 . . . . . . . . . 10  |-  ( z  =  ( -u 1  .h  y )  ->  (
x  +h  z )  =  ( x  +h  ( -u 1  .h  y
) ) )
3433eqeq2d 2296 . . . . . . . . 9  |-  ( z  =  ( -u 1  .h  y )  ->  ( C  =  ( x  +h  z )  <->  C  =  ( x  +h  ( -u 1  .h  y ) ) ) )
3534rspcev 2886 . . . . . . . 8  |-  ( ( ( -u 1  .h  y )  e.  B  /\  C  =  (
x  +h  ( -u
1  .h  y ) ) )  ->  E. z  e.  B  C  =  ( x  +h  z
) )
3632, 35sylan 457 . . . . . . 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 456 . . . . . 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 423 . . . . 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 2669 . . . 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 183 . . 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 2562 . 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 244 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 176    /\ wa 358    = wceq 1625    e. wcel 1686   E.wrex 2546  (class class class)co 5860   CCcc 8737   1c1 8740   -ucneg 9040   ~Hchil 21501    +h cva 21502    .h csm 21503    -h cmv 21507   SHcsh 21510    +H cph 21513
This theorem is referenced by:  pjimai  22758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-hilex 21581  ax-hfvadd 21582  ax-hvcom 21583  ax-hvass 21584  ax-hv0cl 21585  ax-hvaddid 21586  ax-hfvmul 21587  ax-hvmulid 21588  ax-hvmulass 21589  ax-hvdistr2 21591  ax-hvmul0 21592
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-ltxr 8874  df-sub 9041  df-neg 9042  df-grpo 20860  df-ablo 20951  df-hvsub 21553  df-sh 21788  df-shs 21889
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