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Theorem shuni 22807
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 22728 . . . . . . 7  |-  ( ( H  e.  SH  /\  A  e.  H  /\  C  e.  H )  ->  ( A  -h  C
)  e.  H )
51, 2, 3, 4syl3anc 1185 . . . . . 6  |-  ( ph  ->  ( A  -h  C
)  e.  H )
6 shuni.8 . . . . . . . 8  |-  ( ph  ->  ( A  +h  B
)  =  ( C  +h  D ) )
7 shel 22718 . . . . . . . . . 10  |-  ( ( H  e.  SH  /\  A  e.  H )  ->  A  e.  ~H )
81, 2, 7syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  A  e.  ~H )
9 shuni.2 . . . . . . . . . 10  |-  ( ph  ->  K  e.  SH )
10 shuni.5 . . . . . . . . . 10  |-  ( ph  ->  B  e.  K )
11 shel 22718 . . . . . . . . . 10  |-  ( ( K  e.  SH  /\  B  e.  K )  ->  B  e.  ~H )
129, 10, 11syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  B  e.  ~H )
13 shel 22718 . . . . . . . . . 10  |-  ( ( H  e.  SH  /\  C  e.  H )  ->  C  e.  ~H )
141, 3, 13syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  C  e.  ~H )
15 shuni.7 . . . . . . . . . 10  |-  ( ph  ->  D  e.  K )
16 shel 22718 . . . . . . . . . 10  |-  ( ( K  e.  SH  /\  D  e.  K )  ->  D  e.  ~H )
179, 15, 16syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  D  e.  ~H )
18 hvaddsub4 22585 . . . . . . . . 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 1186 . . . . . . . 8  |-  ( ph  ->  ( ( A  +h  B )  =  ( C  +h  D )  <-> 
( A  -h  C
)  =  ( D  -h  B ) ) )
206, 19mpbid 203 . . . . . . 7  |-  ( ph  ->  ( A  -h  C
)  =  ( D  -h  B ) )
21 shsubcl 22728 . . . . . . . 8  |-  ( ( K  e.  SH  /\  D  e.  K  /\  B  e.  K )  ->  ( D  -h  B
)  e.  K )
229, 15, 10, 21syl3anc 1185 . . . . . . 7  |-  ( ph  ->  ( D  -h  B
)  e.  K )
2320, 22eqeltrd 2512 . . . . . 6  |-  ( ph  ->  ( A  -h  C
)  e.  K )
24 elin 3532 . . . . . 6  |-  ( ( A  -h  C )  e.  ( H  i^i  K )  <->  ( ( A  -h  C )  e.  H  /\  ( A  -h  C )  e.  K ) )
255, 23, 24sylanbrc 647 . . . . 5  |-  ( ph  ->  ( A  -h  C
)  e.  ( H  i^i  K ) )
26 shuni.3 . . . . 5  |-  ( ph  ->  ( H  i^i  K
)  =  0H )
2725, 26eleqtrd 2514 . . . 4  |-  ( ph  ->  ( A  -h  C
)  e.  0H )
28 elch0 22761 . . . 4  |-  ( ( A  -h  C )  e.  0H  <->  ( A  -h  C )  =  0h )
2927, 28sylib 190 . . 3  |-  ( ph  ->  ( A  -h  C
)  =  0h )
30 hvsubeq0 22575 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  C )  =  0h  <->  A  =  C ) )
318, 14, 30syl2anc 644 . . 3  |-  ( ph  ->  ( ( A  -h  C )  =  0h  <->  A  =  C ) )
3229, 31mpbid 203 . 2  |-  ( ph  ->  A  =  C )
3320, 29eqtr3d 2472 . . . 4  |-  ( ph  ->  ( D  -h  B
)  =  0h )
34 hvsubeq0 22575 . . . . 5  |-  ( ( D  e.  ~H  /\  B  e.  ~H )  ->  ( ( D  -h  B )  =  0h  <->  D  =  B ) )
3517, 12, 34syl2anc 644 . . . 4  |-  ( ph  ->  ( ( D  -h  B )  =  0h  <->  D  =  B ) )
3633, 35mpbid 203 . . 3  |-  ( ph  ->  D  =  B )
3736eqcomd 2443 . 2  |-  ( ph  ->  B  =  D )
3832, 37jca 520 1  |-  ( ph  ->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321  (class class class)co 6084   ~Hchil 22427    +h cva 22428   0hc0v 22432    -h cmv 22433   SHcsh 22436   0Hc0h 22443
This theorem is referenced by:  chocunii  22808  pjhthmo  22809  chscllem3  23146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-hilex 22507  ax-hfvadd 22508  ax-hvcom 22509  ax-hvass 22510  ax-hv0cl 22511  ax-hvaddid 22512  ax-hfvmul 22513  ax-hvmulid 22514  ax-hvmulass 22515  ax-hvdistr1 22516  ax-hvdistr2 22517  ax-hvmul0 22518
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-hvsub 22479  df-sh 22714  df-ch0 22760
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