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Theorem shsel3 22008
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 22007 . 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 21904 . . . . . . . . . . 11  |-  ( ( A  e.  SH  /\  x  e.  A )  ->  x  e.  ~H )
4 shel 21904 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  z  e.  B )  ->  z  e.  ~H )
5 hvaddsubval 21726 . . . . . . . . . . 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 2412 . . . . . . 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 9903 . . . . . . . . . . 11  |-  -u 1  e.  CC
11 shmulcl 21911 . . . . . . . . . . 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 5953 . . . . . . . . . 10  |-  ( y  =  ( -u 1  .h  z )  ->  (
x  -h  y )  =  ( x  -h  ( -u 1  .h  z
) ) )
1615eqeq2d 2369 . . . . . . . . 9  |-  ( y  =  ( -u 1  .h  z )  ->  ( C  =  ( x  -h  y )  <->  C  =  ( x  -h  ( -u 1  .h  z ) ) ) )
1716rspcev 2960 . . . . . . . 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 2743 . . . 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 21904 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  y  e.  B )  ->  y  e.  ~H )
24 hvsubval 21710 . . . . . . . . . . 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 2412 . . . . . . 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 21911 . . . . . . . . . . 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 5953 . . . . . . . . . 10  |-  ( z  =  ( -u 1  .h  y )  ->  (
x  +h  z )  =  ( x  +h  ( -u 1  .h  y
) ) )
3433eqeq2d 2369 . . . . . . . . 9  |-  ( z  =  ( -u 1  .h  y )  ->  ( C  =  ( x  +h  z )  <->  C  =  ( x  +h  ( -u 1  .h  y ) ) ) )
3534rspcev 2960 . . . . . . . 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 2743 . . . 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 2636 . 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 1642    e. wcel 1710   E.wrex 2620  (class class class)co 5945   CCcc 8825   1c1 8828   -ucneg 9128   ~Hchil 21613    +h cva 21614    .h csm 21615    -h cmv 21619   SHcsh 21622    +H cph 21625
This theorem is referenced by:  pjimai  22870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-hilex 21693  ax-hfvadd 21694  ax-hvcom 21695  ax-hvass 21696  ax-hv0cl 21697  ax-hvaddid 21698  ax-hfvmul 21699  ax-hvmulid 21700  ax-hvmulass 21701  ax-hvdistr2 21703  ax-hvmul0 21704
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-ltxr 8962  df-sub 9129  df-neg 9130  df-grpo 20970  df-ablo 21061  df-hvsub 21665  df-sh 21900  df-shs 22001
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