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Theorem cdjreui 23028
Description: A member of the sum of disjoint subspaces has a unique decomposition. Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 20-May-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdjreu.1  |-  A  e.  SH
cdjreu.2  |-  B  e.  SH
Assertion
Ref Expression
cdjreui  |-  ( ( C  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  ->  E! x  e.  A  E. y  e.  B  C  =  ( x  +h  y ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y

Proof of Theorem cdjreui
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdjreu.1 . . . . 5  |-  A  e.  SH
2 cdjreu.2 . . . . 5  |-  B  e.  SH
31, 2shseli 21911 . . . 4  |-  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y
) )
43biimpi 186 . . 3  |-  ( C  e.  ( A  +H  B )  ->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y
) )
5 reeanv 2720 . . . . 5  |-  ( E. y  e.  B  E. w  e.  B  ( C  =  ( x  +h  y )  /\  C  =  ( z  +h  w ) )  <->  ( E. y  e.  B  C  =  ( x  +h  y )  /\  E. w  e.  B  C  =  ( z  +h  w ) ) )
6 eqtr2 2314 . . . . . . 7  |-  ( ( C  =  ( x  +h  y )  /\  C  =  ( z  +h  w ) )  -> 
( x  +h  y
)  =  ( z  +h  w ) )
71sheli 21809 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  x  e.  ~H )
82sheli 21809 . . . . . . . . . . . 12  |-  ( y  e.  B  ->  y  e.  ~H )
97, 8anim12i 549 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
101sheli 21809 . . . . . . . . . . . 12  |-  ( z  e.  A  ->  z  e.  ~H )
112sheli 21809 . . . . . . . . . . . 12  |-  ( w  e.  B  ->  w  e.  ~H )
1210, 11anim12i 549 . . . . . . . . . . 11  |-  ( ( z  e.  A  /\  w  e.  B )  ->  ( z  e.  ~H  /\  w  e.  ~H )
)
13 hvaddsub4 21673 . . . . . . . . . . 11  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
x  +h  y )  =  ( z  +h  w )  <->  ( x  -h  z )  =  ( w  -h  y ) ) )
149, 12, 13syl2an 463 . . . . . . . . . 10  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  A  /\  w  e.  B ) )  -> 
( ( x  +h  y )  =  ( z  +h  w )  <-> 
( x  -h  z
)  =  ( w  -h  y ) ) )
1514an4s 799 . . . . . . . . 9  |-  ( ( ( x  e.  A  /\  z  e.  A
)  /\  ( y  e.  B  /\  w  e.  B ) )  -> 
( ( x  +h  y )  =  ( z  +h  w )  <-> 
( x  -h  z
)  =  ( w  -h  y ) ) )
1615adantll 694 . . . . . . . 8  |-  ( ( ( ( A  i^i  B )  =  0H  /\  ( x  e.  A  /\  z  e.  A
) )  /\  (
y  e.  B  /\  w  e.  B )
)  ->  ( (
x  +h  y )  =  ( z  +h  w )  <->  ( x  -h  z )  =  ( w  -h  y ) ) )
17 shsubcl 21816 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  SH  /\  w  e.  B  /\  y  e.  B )  ->  ( w  -h  y
)  e.  B )
182, 17mp3an1 1264 . . . . . . . . . . . . . . 15  |-  ( ( w  e.  B  /\  y  e.  B )  ->  ( w  -h  y
)  e.  B )
1918ancoms 439 . . . . . . . . . . . . . 14  |-  ( ( y  e.  B  /\  w  e.  B )  ->  ( w  -h  y
)  e.  B )
20 eleq1 2356 . . . . . . . . . . . . . 14  |-  ( ( x  -h  z )  =  ( w  -h  y )  ->  (
( x  -h  z
)  e.  B  <->  ( w  -h  y )  e.  B
) )
2119, 20syl5ibrcom 213 . . . . . . . . . . . . 13  |-  ( ( y  e.  B  /\  w  e.  B )  ->  ( ( x  -h  z )  =  ( w  -h  y )  ->  ( x  -h  z )  e.  B
) )
2221adantl 452 . . . . . . . . . . . 12  |-  ( ( ( x  e.  A  /\  z  e.  A
)  /\  ( y  e.  B  /\  w  e.  B ) )  -> 
( ( x  -h  z )  =  ( w  -h  y )  ->  ( x  -h  z )  e.  B
) )
23 shsubcl 21816 . . . . . . . . . . . . . 14  |-  ( ( A  e.  SH  /\  x  e.  A  /\  z  e.  A )  ->  ( x  -h  z
)  e.  A )
241, 23mp3an1 1264 . . . . . . . . . . . . 13  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( x  -h  z
)  e.  A )
2524adantr 451 . . . . . . . . . . . 12  |-  ( ( ( x  e.  A  /\  z  e.  A
)  /\  ( y  e.  B  /\  w  e.  B ) )  -> 
( x  -h  z
)  e.  A )
2622, 25jctild 527 . . . . . . . . . . 11  |-  ( ( ( x  e.  A  /\  z  e.  A
)  /\  ( y  e.  B  /\  w  e.  B ) )  -> 
( ( x  -h  z )  =  ( w  -h  y )  ->  ( ( x  -h  z )  e.  A  /\  ( x  -h  z )  e.  B ) ) )
2726adantll 694 . . . . . . . . . 10  |-  ( ( ( ( A  i^i  B )  =  0H  /\  ( x  e.  A  /\  z  e.  A
) )  /\  (
y  e.  B  /\  w  e.  B )
)  ->  ( (
x  -h  z )  =  ( w  -h  y )  ->  (
( x  -h  z
)  e.  A  /\  ( x  -h  z
)  e.  B ) ) )
28 elin 3371 . . . . . . . . . . . 12  |-  ( ( x  -h  z )  e.  ( A  i^i  B )  <->  ( ( x  -h  z )  e.  A  /\  ( x  -h  z )  e.  B ) )
29 eleq2 2357 . . . . . . . . . . . 12  |-  ( ( A  i^i  B )  =  0H  ->  (
( x  -h  z
)  e.  ( A  i^i  B )  <->  ( x  -h  z )  e.  0H ) )
3028, 29syl5bbr 250 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  =  0H  ->  (
( ( x  -h  z )  e.  A  /\  ( x  -h  z
)  e.  B )  <-> 
( x  -h  z
)  e.  0H ) )
3130ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( A  i^i  B )  =  0H  /\  ( x  e.  A  /\  z  e.  A
) )  /\  (
y  e.  B  /\  w  e.  B )
)  ->  ( (
( x  -h  z
)  e.  A  /\  ( x  -h  z
)  e.  B )  <-> 
( x  -h  z
)  e.  0H ) )
3227, 31sylibd 205 . . . . . . . . 9  |-  ( ( ( ( A  i^i  B )  =  0H  /\  ( x  e.  A  /\  z  e.  A
) )  /\  (
y  e.  B  /\  w  e.  B )
)  ->  ( (
x  -h  z )  =  ( w  -h  y )  ->  (
x  -h  z )  e.  0H ) )
33 elch0 21849 . . . . . . . . . . . 12  |-  ( ( x  -h  z )  e.  0H  <->  ( x  -h  z )  =  0h )
34 hvsubeq0 21663 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  =  0h  <->  x  =  z ) )
3533, 34syl5bb 248 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  e.  0H  <->  x  =  z ) )
367, 10, 35syl2an 463 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( ( x  -h  z )  e.  0H  <->  x  =  z ) )
3736ad2antlr 707 . . . . . . . . 9  |-  ( ( ( ( A  i^i  B )  =  0H  /\  ( x  e.  A  /\  z  e.  A
) )  /\  (
y  e.  B  /\  w  e.  B )
)  ->  ( (
x  -h  z )  e.  0H  <->  x  =  z ) )
3832, 37sylibd 205 . . . . . . . 8  |-  ( ( ( ( A  i^i  B )  =  0H  /\  ( x  e.  A  /\  z  e.  A
) )  /\  (
y  e.  B  /\  w  e.  B )
)  ->  ( (
x  -h  z )  =  ( w  -h  y )  ->  x  =  z ) )
3916, 38sylbid 206 . . . . . . 7  |-  ( ( ( ( A  i^i  B )  =  0H  /\  ( x  e.  A  /\  z  e.  A
) )  /\  (
y  e.  B  /\  w  e.  B )
)  ->  ( (
x  +h  y )  =  ( z  +h  w )  ->  x  =  z ) )
406, 39syl5 28 . . . . . 6  |-  ( ( ( ( A  i^i  B )  =  0H  /\  ( x  e.  A  /\  z  e.  A
) )  /\  (
y  e.  B  /\  w  e.  B )
)  ->  ( ( C  =  ( x  +h  y )  /\  C  =  ( z  +h  w ) )  ->  x  =  z )
)
4140rexlimdvva 2687 . . . . 5  |-  ( ( ( A  i^i  B
)  =  0H  /\  ( x  e.  A  /\  z  e.  A
) )  ->  ( E. y  e.  B  E. w  e.  B  ( C  =  (
x  +h  y )  /\  C  =  ( z  +h  w ) )  ->  x  =  z ) )
425, 41syl5bir 209 . . . 4  |-  ( ( ( A  i^i  B
)  =  0H  /\  ( x  e.  A  /\  z  e.  A
) )  ->  (
( E. y  e.  B  C  =  ( x  +h  y )  /\  E. w  e.  B  C  =  ( z  +h  w ) )  ->  x  =  z ) )
4342ralrimivva 2648 . . 3  |-  ( ( A  i^i  B )  =  0H  ->  A. x  e.  A  A. z  e.  A  ( ( E. y  e.  B  C  =  ( x  +h  y )  /\  E. w  e.  B  C  =  ( z  +h  w ) )  ->  x  =  z )
)
444, 43anim12i 549 . 2  |-  ( ( C  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  -> 
( E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y )  /\  A. x  e.  A  A. z  e.  A  ( ( E. y  e.  B  C  =  ( x  +h  y )  /\  E. w  e.  B  C  =  ( z  +h  w ) )  ->  x  =  z )
) )
45 oveq1 5881 . . . . . 6  |-  ( x  =  z  ->  (
x  +h  y )  =  ( z  +h  y ) )
4645eqeq2d 2307 . . . . 5  |-  ( x  =  z  ->  ( C  =  ( x  +h  y )  <->  C  =  ( z  +h  y
) ) )
4746rexbidv 2577 . . . 4  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. y  e.  B  C  =  ( z  +h  y
) ) )
48 oveq2 5882 . . . . . 6  |-  ( y  =  w  ->  (
z  +h  y )  =  ( z  +h  w ) )
4948eqeq2d 2307 . . . . 5  |-  ( y  =  w  ->  ( C  =  ( z  +h  y )  <->  C  =  ( z  +h  w
) ) )
5049cbvrexv 2778 . . . 4  |-  ( E. y  e.  B  C  =  ( z  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) )
5147, 50syl6bb 252 . . 3  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) ) )
5251reu4 2972 . 2  |-  ( E! x  e.  A  E. y  e.  B  C  =  ( x  +h  y )  <->  ( E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y )  /\  A. x  e.  A  A. z  e.  A  (
( E. y  e.  B  C  =  ( x  +h  y )  /\  E. w  e.  B  C  =  ( z  +h  w ) )  ->  x  =  z ) ) )
5344, 52sylibr 203 1  |-  ( ( C  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  ->  E! x  e.  A  E. y  e.  B  C  =  ( x  +h  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558    i^i cin 3164  (class class class)co 5874   ~Hchil 21515    +h cva 21516   0hc0v 21520    -h cmv 21521   SHcsh 21524    +H cph 21527   0Hc0h 21531
This theorem is referenced by:  cdj3lem2  23031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-grpo 20874  df-ablo 20965  df-hvsub 21567  df-sh 21802  df-ch0 21848  df-shs 21903
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