HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  shuni Structured version   Unicode version

Theorem shuni 22794
Description: Two subspaces with trivial intersection have a unique decomposition of the elements of the subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
shuni.1  |-  ( ph  ->  H  e.  SH )
shuni.2  |-  ( ph  ->  K  e.  SH )
shuni.3  |-  ( ph  ->  ( H  i^i  K
)  =  0H )
shuni.4  |-  ( ph  ->  A  e.  H )
shuni.5  |-  ( ph  ->  B  e.  K )
shuni.6  |-  ( ph  ->  C  e.  H )
shuni.7  |-  ( ph  ->  D  e.  K )
shuni.8  |-  ( ph  ->  ( A  +h  B
)  =  ( C  +h  D ) )
Assertion
Ref Expression
shuni  |-  ( ph  ->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem shuni
StepHypRef Expression
1 shuni.1 . . . . . . 7  |-  ( ph  ->  H  e.  SH )
2 shuni.4 . . . . . . 7  |-  ( ph  ->  A  e.  H )
3 shuni.6 . . . . . . 7  |-  ( ph  ->  C  e.  H )
4 shsubcl 22715 . . . . . . 7  |-  ( ( H  e.  SH  /\  A  e.  H  /\  C  e.  H )  ->  ( A  -h  C
)  e.  H )
51, 2, 3, 4syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( A  -h  C
)  e.  H )
6 shuni.8 . . . . . . . 8  |-  ( ph  ->  ( A  +h  B
)  =  ( C  +h  D ) )
7 shel 22705 . . . . . . . . . 10  |-  ( ( H  e.  SH  /\  A  e.  H )  ->  A  e.  ~H )
81, 2, 7syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  A  e.  ~H )
9 shuni.2 . . . . . . . . . 10  |-  ( ph  ->  K  e.  SH )
10 shuni.5 . . . . . . . . . 10  |-  ( ph  ->  B  e.  K )
11 shel 22705 . . . . . . . . . 10  |-  ( ( K  e.  SH  /\  B  e.  K )  ->  B  e.  ~H )
129, 10, 11syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  B  e.  ~H )
13 shel 22705 . . . . . . . . . 10  |-  ( ( H  e.  SH  /\  C  e.  H )  ->  C  e.  ~H )
141, 3, 13syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  C  e.  ~H )
15 shuni.7 . . . . . . . . . 10  |-  ( ph  ->  D  e.  K )
16 shel 22705 . . . . . . . . . 10  |-  ( ( K  e.  SH  /\  D  e.  K )  ->  D  e.  ~H )
179, 15, 16syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  D  e.  ~H )
18 hvaddsub4 22572 . . . . . . . . 9  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  +h  B )  =  ( C  +h  D
)  <->  ( A  -h  C )  =  ( D  -h  B ) ) )
198, 12, 14, 17, 18syl22anc 1185 . . . . . . . 8  |-  ( ph  ->  ( ( A  +h  B )  =  ( C  +h  D )  <-> 
( A  -h  C
)  =  ( D  -h  B ) ) )
206, 19mpbid 202 . . . . . . 7  |-  ( ph  ->  ( A  -h  C
)  =  ( D  -h  B ) )
21 shsubcl 22715 . . . . . . . 8  |-  ( ( K  e.  SH  /\  D  e.  K  /\  B  e.  K )  ->  ( D  -h  B
)  e.  K )
229, 15, 10, 21syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( D  -h  B
)  e.  K )
2320, 22eqeltrd 2509 . . . . . 6  |-  ( ph  ->  ( A  -h  C
)  e.  K )
24 elin 3522 . . . . . 6  |-  ( ( A  -h  C )  e.  ( H  i^i  K )  <->  ( ( A  -h  C )  e.  H  /\  ( A  -h  C )  e.  K ) )
255, 23, 24sylanbrc 646 . . . . 5  |-  ( ph  ->  ( A  -h  C
)  e.  ( H  i^i  K ) )
26 shuni.3 . . . . 5  |-  ( ph  ->  ( H  i^i  K
)  =  0H )
2725, 26eleqtrd 2511 . . . 4  |-  ( ph  ->  ( A  -h  C
)  e.  0H )
28 elch0 22748 . . . 4  |-  ( ( A  -h  C )  e.  0H  <->  ( A  -h  C )  =  0h )
2927, 28sylib 189 . . 3  |-  ( ph  ->  ( A  -h  C
)  =  0h )
30 hvsubeq0 22562 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  C )  =  0h  <->  A  =  C ) )
318, 14, 30syl2anc 643 . . 3  |-  ( ph  ->  ( ( A  -h  C )  =  0h  <->  A  =  C ) )
3229, 31mpbid 202 . 2  |-  ( ph  ->  A  =  C )
3320, 29eqtr3d 2469 . . . 4  |-  ( ph  ->  ( D  -h  B
)  =  0h )
34 hvsubeq0 22562 . . . . 5  |-  ( ( D  e.  ~H  /\  B  e.  ~H )  ->  ( ( D  -h  B )  =  0h  <->  D  =  B ) )
3517, 12, 34syl2anc 643 . . . 4  |-  ( ph  ->  ( ( D  -h  B )  =  0h  <->  D  =  B ) )
3633, 35mpbid 202 . . 3  |-  ( ph  ->  D  =  B )
3736eqcomd 2440 . 2  |-  ( ph  ->  B  =  D )
3832, 37jca 519 1  |-  ( ph  ->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3311  (class class class)co 6073   ~Hchil 22414    +h cva 22415   0hc0v 22419    -h cmv 22420   SHcsh 22423   0Hc0h 22430
This theorem is referenced by:  chocunii  22795  pjhthmo  22796  chscllem3  23133
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-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-pre-mulgt0 9059  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-hvdistr1 22503  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-rmo 2705  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-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-hvsub 22466  df-sh 22701  df-ch0 22747
  Copyright terms: Public domain W3C validator