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Theorem shsel3 22809
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 22808 . 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 22705 . . . . . . . . . . 11  |-  ( ( A  e.  SH  /\  x  e.  A )  ->  x  e.  ~H )
4 shel 22705 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  z  e.  B )  ->  z  e.  ~H )
5 hvaddsubval 22527 . . . . . . . . . . 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 2489 . . . . . . 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 10059 . . . . . . . . . . 11  |-  -u 1  e.  CC
11 shmulcl 22712 . . . . . . . . . . 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 6081 . . . . . . . . . 10  |-  ( y  =  ( -u 1  .h  z )  ->  (
x  -h  y )  =  ( x  -h  ( -u 1  .h  z
) ) )
1615eqeq2d 2446 . . . . . . . . 9  |-  ( y  =  ( -u 1  .h  z )  ->  ( C  =  ( x  -h  y )  <->  C  =  ( x  -h  ( -u 1  .h  z ) ) ) )
1716rspcev 3044 . . . . . . . 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 2822 . . . 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 22705 . . . . . . . . . . 11  |-  ( ( B  e.  SH  /\  y  e.  B )  ->  y  e.  ~H )
24 hvsubval 22511 . . . . . . . . . . 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 2489 . . . . . . 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 22712 . . . . . . . . . . 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 6081 . . . . . . . . . 10  |-  ( z  =  ( -u 1  .h  y )  ->  (
x  +h  z )  =  ( x  +h  ( -u 1  .h  y
) ) )
3433eqeq2d 2446 . . . . . . . . 9  |-  ( z  =  ( -u 1  .h  y )  ->  ( C  =  ( x  +h  z )  <->  C  =  ( x  +h  ( -u 1  .h  y ) ) ) )
3534rspcev 3044 . . . . . . . 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 2822 . . . 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 2714 . 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 1652    e. wcel 1725   E.wrex 2698  (class class class)co 6073   CCcc 8980   1c1 8983   -ucneg 9284   ~Hchil 22414    +h cva 22415    .h csm 22416    -h cmv 22420   SHcsh 22423    +H cph 22426
This theorem is referenced by:  pjimai  23671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-hilex 22494  ax-hfvadd 22495  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvmulass 22502  ax-hvdistr2 22504  ax-hvmul0 22505
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285  df-neg 9286  df-grpo 21771  df-ablo 21862  df-hvsub 22466  df-sh 22701  df-shs 22802
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