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Theorem shuni 21993
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 21914 . . . . . . 7  |-  ( ( H  e.  SH  /\  A  e.  H  /\  C  e.  H )  ->  ( A  -h  C
)  e.  H )
51, 2, 3, 4syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( A  -h  C
)  e.  H )
6 shuni.8 . . . . . . . 8  |-  ( ph  ->  ( A  +h  B
)  =  ( C  +h  D ) )
7 shel 21904 . . . . . . . . . 10  |-  ( ( H  e.  SH  /\  A  e.  H )  ->  A  e.  ~H )
81, 2, 7syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  A  e.  ~H )
9 shuni.2 . . . . . . . . . 10  |-  ( ph  ->  K  e.  SH )
10 shuni.5 . . . . . . . . . 10  |-  ( ph  ->  B  e.  K )
11 shel 21904 . . . . . . . . . 10  |-  ( ( K  e.  SH  /\  B  e.  K )  ->  B  e.  ~H )
129, 10, 11syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  B  e.  ~H )
13 shel 21904 . . . . . . . . . 10  |-  ( ( H  e.  SH  /\  C  e.  H )  ->  C  e.  ~H )
141, 3, 13syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  C  e.  ~H )
15 shuni.7 . . . . . . . . . 10  |-  ( ph  ->  D  e.  K )
16 shel 21904 . . . . . . . . . 10  |-  ( ( K  e.  SH  /\  D  e.  K )  ->  D  e.  ~H )
179, 15, 16syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  D  e.  ~H )
18 hvaddsub4 21771 . . . . . . . . 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 1183 . . . . . . . 8  |-  ( ph  ->  ( ( A  +h  B )  =  ( C  +h  D )  <-> 
( A  -h  C
)  =  ( D  -h  B ) ) )
206, 19mpbid 201 . . . . . . 7  |-  ( ph  ->  ( A  -h  C
)  =  ( D  -h  B ) )
21 shsubcl 21914 . . . . . . . 8  |-  ( ( K  e.  SH  /\  D  e.  K  /\  B  e.  K )  ->  ( D  -h  B
)  e.  K )
229, 15, 10, 21syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( D  -h  B
)  e.  K )
2320, 22eqeltrd 2432 . . . . . 6  |-  ( ph  ->  ( A  -h  C
)  e.  K )
24 elin 3434 . . . . . 6  |-  ( ( A  -h  C )  e.  ( H  i^i  K )  <->  ( ( A  -h  C )  e.  H  /\  ( A  -h  C )  e.  K ) )
255, 23, 24sylanbrc 645 . . . . 5  |-  ( ph  ->  ( A  -h  C
)  e.  ( H  i^i  K ) )
26 shuni.3 . . . . 5  |-  ( ph  ->  ( H  i^i  K
)  =  0H )
2725, 26eleqtrd 2434 . . . 4  |-  ( ph  ->  ( A  -h  C
)  e.  0H )
28 elch0 21947 . . . 4  |-  ( ( A  -h  C )  e.  0H  <->  ( A  -h  C )  =  0h )
2927, 28sylib 188 . . 3  |-  ( ph  ->  ( A  -h  C
)  =  0h )
30 hvsubeq0 21761 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  C )  =  0h  <->  A  =  C ) )
318, 14, 30syl2anc 642 . . 3  |-  ( ph  ->  ( ( A  -h  C )  =  0h  <->  A  =  C ) )
3229, 31mpbid 201 . 2  |-  ( ph  ->  A  =  C )
3320, 29eqtr3d 2392 . . . 4  |-  ( ph  ->  ( D  -h  B
)  =  0h )
34 hvsubeq0 21761 . . . . 5  |-  ( ( D  e.  ~H  /\  B  e.  ~H )  ->  ( ( D  -h  B )  =  0h  <->  D  =  B ) )
3517, 12, 34syl2anc 642 . . . 4  |-  ( ph  ->  ( ( D  -h  B )  =  0h  <->  D  =  B ) )
3633, 35mpbid 201 . . 3  |-  ( ph  ->  D  =  B )
3736eqcomd 2363 . 2  |-  ( ph  ->  B  =  D )
3832, 37jca 518 1  |-  ( ph  ->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    i^i cin 3227  (class class class)co 5945   ~Hchil 21613    +h cva 21614   0hc0v 21618    -h cmv 21619   SHcsh 21622   0Hc0h 21629
This theorem is referenced by:  chocunii  21994  pjhthmo  21995  chscllem3  22332
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-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-pre-mulgt0 8904  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-hvdistr1 21702  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-rmo 2627  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-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-hvsub 21665  df-sh 21900  df-ch0 21946
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