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Theorem shuni 24703
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 24623 . . . . . . 7  |-  ( ( H  e.  SH  /\  A  e.  H  /\  C  e.  H )  ->  ( A  -h  C
)  e.  H )
51, 2, 3, 4syl3anc 1218 . . . . . 6  |-  ( ph  ->  ( A  -h  C
)  e.  H )
6 shuni.8 . . . . . . . 8  |-  ( ph  ->  ( A  +h  B
)  =  ( C  +h  D ) )
7 shel 24613 . . . . . . . . . 10  |-  ( ( H  e.  SH  /\  A  e.  H )  ->  A  e.  ~H )
81, 2, 7syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  A  e.  ~H )
9 shuni.2 . . . . . . . . . 10  |-  ( ph  ->  K  e.  SH )
10 shuni.5 . . . . . . . . . 10  |-  ( ph  ->  B  e.  K )
11 shel 24613 . . . . . . . . . 10  |-  ( ( K  e.  SH  /\  B  e.  K )  ->  B  e.  ~H )
129, 10, 11syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  B  e.  ~H )
13 shel 24613 . . . . . . . . . 10  |-  ( ( H  e.  SH  /\  C  e.  H )  ->  C  e.  ~H )
141, 3, 13syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  C  e.  ~H )
15 shuni.7 . . . . . . . . . 10  |-  ( ph  ->  D  e.  K )
16 shel 24613 . . . . . . . . . 10  |-  ( ( K  e.  SH  /\  D  e.  K )  ->  D  e.  ~H )
179, 15, 16syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  D  e.  ~H )
18 hvaddsub4 24480 . . . . . . . . 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 1219 . . . . . . . 8  |-  ( ph  ->  ( ( A  +h  B )  =  ( C  +h  D )  <-> 
( A  -h  C
)  =  ( D  -h  B ) ) )
206, 19mpbid 210 . . . . . . 7  |-  ( ph  ->  ( A  -h  C
)  =  ( D  -h  B ) )
21 shsubcl 24623 . . . . . . . 8  |-  ( ( K  e.  SH  /\  D  e.  K  /\  B  e.  K )  ->  ( D  -h  B
)  e.  K )
229, 15, 10, 21syl3anc 1218 . . . . . . 7  |-  ( ph  ->  ( D  -h  B
)  e.  K )
2320, 22eqeltrd 2517 . . . . . 6  |-  ( ph  ->  ( A  -h  C
)  e.  K )
245, 23elind 3540 . . . . 5  |-  ( ph  ->  ( A  -h  C
)  e.  ( H  i^i  K ) )
25 shuni.3 . . . . 5  |-  ( ph  ->  ( H  i^i  K
)  =  0H )
2624, 25eleqtrd 2519 . . . 4  |-  ( ph  ->  ( A  -h  C
)  e.  0H )
27 elch0 24657 . . . 4  |-  ( ( A  -h  C )  e.  0H  <->  ( A  -h  C )  =  0h )
2826, 27sylib 196 . . 3  |-  ( ph  ->  ( A  -h  C
)  =  0h )
29 hvsubeq0 24470 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  C )  =  0h  <->  A  =  C ) )
308, 14, 29syl2anc 661 . . 3  |-  ( ph  ->  ( ( A  -h  C )  =  0h  <->  A  =  C ) )
3128, 30mpbid 210 . 2  |-  ( ph  ->  A  =  C )
3220, 28eqtr3d 2477 . . . 4  |-  ( ph  ->  ( D  -h  B
)  =  0h )
33 hvsubeq0 24470 . . . . 5  |-  ( ( D  e.  ~H  /\  B  e.  ~H )  ->  ( ( D  -h  B )  =  0h  <->  D  =  B ) )
3417, 12, 33syl2anc 661 . . . 4  |-  ( ph  ->  ( ( D  -h  B )  =  0h  <->  D  =  B ) )
3532, 34mpbid 210 . . 3  |-  ( ph  ->  D  =  B )
3635eqcomd 2448 . 2  |-  ( ph  ->  B  =  D )
3731, 36jca 532 1  |-  ( ph  ->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3327  (class class class)co 6091   ~Hchil 24321    +h cva 24322   0hc0v 24326    -h cmv 24327   SHcsh 24330   0Hc0h 24337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-hilex 24401  ax-hfvadd 24402  ax-hvcom 24403  ax-hvass 24404  ax-hv0cl 24405  ax-hvaddid 24406  ax-hfvmul 24407  ax-hvmulid 24408  ax-hvmulass 24409  ax-hvdistr1 24410  ax-hvdistr2 24411  ax-hvmul0 24412
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-hvsub 24373  df-sh 24609  df-ch0 24656
This theorem is referenced by:  chocunii  24704  pjhthmo  24705  chscllem3  25042
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