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Theorem shuni 22763
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 22684 . . . . . . 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 22674 . . . . . . . . . 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 22674 . . . . . . . . . 10  |-  ( ( K  e.  SH  /\  B  e.  K )  ->  B  e.  ~H )
129, 10, 11syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  B  e.  ~H )
13 shel 22674 . . . . . . . . . 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 22674 . . . . . . . . . 10  |-  ( ( K  e.  SH  /\  D  e.  K )  ->  D  e.  ~H )
179, 15, 16syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  D  e.  ~H )
18 hvaddsub4 22541 . . . . . . . . 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 22684 . . . . . . . 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 2486 . . . . . 6  |-  ( ph  ->  ( A  -h  C
)  e.  K )
24 elin 3498 . . . . . 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 2488 . . . 4  |-  ( ph  ->  ( A  -h  C
)  e.  0H )
28 elch0 22717 . . . 4  |-  ( ( A  -h  C )  e.  0H  <->  ( A  -h  C )  =  0h )
2927, 28sylib 189 . . 3  |-  ( ph  ->  ( A  -h  C
)  =  0h )
30 hvsubeq0 22531 . . . 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 2446 . . . 4  |-  ( ph  ->  ( D  -h  B
)  =  0h )
34 hvsubeq0 22531 . . . . 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 2417 . 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 1649    e. wcel 1721    i^i cin 3287  (class class class)co 6048   ~Hchil 22383    +h cva 22384   0hc0v 22388    -h cmv 22389   SHcsh 22392   0Hc0h 22399
This theorem is referenced by:  chocunii  22764  pjhthmo  22765  chscllem3  23102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-hilex 22463  ax-hfvadd 22464  ax-hvcom 22465  ax-hvass 22466  ax-hv0cl 22467  ax-hvaddid 22468  ax-hfvmul 22469  ax-hvmulid 22470  ax-hvmulass 22471  ax-hvdistr1 22472  ax-hvdistr2 22473  ax-hvmul0 22474
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-hvsub 22435  df-sh 22670  df-ch0 22716
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