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Theorem cdjreui 22972
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
StepHypRef Expression
1 cdjreu.1 . . . . 5  |-  A  e.  SH
2 cdjreu.2 . . . . 5  |-  B  e.  SH
31, 2shseli 21855 . . . 4  |-  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y
) )
43biimpi 188 . . 3  |-  ( C  e.  ( A  +H  B )  ->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y
) )
5 reeanv 2682 . . . . 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 2276 . . . . . . 7  |-  ( ( C  =  ( x  +h  y )  /\  C  =  ( z  +h  w ) )  -> 
( x  +h  y
)  =  ( z  +h  w ) )
71sheli 21753 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  x  e.  ~H )
82sheli 21753 . . . . . . . . . . . 12  |-  ( y  e.  B  ->  y  e.  ~H )
97, 8anim12i 551 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
101sheli 21753 . . . . . . . . . . . 12  |-  ( z  e.  A  ->  z  e.  ~H )
112sheli 21753 . . . . . . . . . . . 12  |-  ( w  e.  B  ->  w  e.  ~H )
1210, 11anim12i 551 . . . . . . . . . . 11  |-  ( ( z  e.  A  /\  w  e.  B )  ->  ( z  e.  ~H  /\  w  e.  ~H )
)
13 hvaddsub4 21617 . . . . . . . . . . 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 465 . . . . . . . . . 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 802 . . . . . . . . 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 697 . . . . . . . 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 21760 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  SH  /\  w  e.  B  /\  y  e.  B )  ->  ( w  -h  y
)  e.  B )
182, 17mp3an1 1269 . . . . . . . . . . . . . . 15  |-  ( ( w  e.  B  /\  y  e.  B )  ->  ( w  -h  y
)  e.  B )
1918ancoms 441 . . . . . . . . . . . . . 14  |-  ( ( y  e.  B  /\  w  e.  B )  ->  ( w  -h  y
)  e.  B )
20 eleq1 2318 . . . . . . . . . . . . . 14  |-  ( ( x  -h  z )  =  ( w  -h  y )  ->  (
( x  -h  z
)  e.  B  <->  ( w  -h  y )  e.  B
) )
2119, 20syl5ibrcom 215 . . . . . . . . . . . . 13  |-  ( ( y  e.  B  /\  w  e.  B )  ->  ( ( x  -h  z )  =  ( w  -h  y )  ->  ( x  -h  z )  e.  B
) )
2221adantl 454 . . . . . . . . . . . 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 21760 . . . . . . . . . . . . . 14  |-  ( ( A  e.  SH  /\  x  e.  A  /\  z  e.  A )  ->  ( x  -h  z
)  e.  A )
241, 23mp3an1 1269 . . . . . . . . . . . . 13  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( x  -h  z
)  e.  A )
2524adantr 453 . . . . . . . . . . . 12  |-  ( ( ( x  e.  A  /\  z  e.  A
)  /\  ( y  e.  B  /\  w  e.  B ) )  -> 
( x  -h  z
)  e.  A )
2622, 25jctild 529 . . . . . . . . . . 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 697 . . . . . . . . . 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 3333 . . . . . . . . . . . 12  |-  ( ( x  -h  z )  e.  ( A  i^i  B )  <->  ( ( x  -h  z )  e.  A  /\  ( x  -h  z )  e.  B ) )
29 eleq2 2319 . . . . . . . . . . . 12  |-  ( ( A  i^i  B )  =  0H  ->  (
( x  -h  z
)  e.  ( A  i^i  B )  <->  ( x  -h  z )  e.  0H ) )
3028, 29syl5bbr 252 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  =  0H  ->  (
( ( x  -h  z )  e.  A  /\  ( x  -h  z
)  e.  B )  <-> 
( x  -h  z
)  e.  0H ) )
3130ad2antrr 709 . . . . . . . . . 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 207 . . . . . . . . 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 21793 . . . . . . . . . . . 12  |-  ( ( x  -h  z )  e.  0H  <->  ( x  -h  z )  =  0h )
34 hvsubeq0 21607 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  =  0h  <->  x  =  z ) )
3533, 34syl5bb 250 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  e.  0H  <->  x  =  z ) )
367, 10, 35syl2an 465 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( ( x  -h  z )  e.  0H  <->  x  =  z ) )
3736ad2antlr 710 . . . . . . . . 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 207 . . . . . . . 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 208 . . . . . . 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 30 . . . . . 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 2649 . . . . 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 211 . . . 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 2610 . . 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 551 . 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 5799 . . . . . 6  |-  ( x  =  z  ->  (
x  +h  y )  =  ( z  +h  y ) )
4645eqeq2d 2269 . . . . 5  |-  ( x  =  z  ->  ( C  =  ( x  +h  y )  <->  C  =  ( z  +h  y
) ) )
4746rexbidv 2539 . . . 4  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. y  e.  B  C  =  ( z  +h  y
) ) )
48 oveq2 5800 . . . . . 6  |-  ( y  =  w  ->  (
z  +h  y )  =  ( z  +h  w ) )
4948eqeq2d 2269 . . . . 5  |-  ( y  =  w  ->  ( C  =  ( z  +h  y )  <->  C  =  ( z  +h  w
) ) )
5049cbvrexv 2740 . . . 4  |-  ( E. y  e.  B  C  =  ( z  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) )
5147, 50syl6bb 254 . . 3  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) ) )
5251reu4 2934 . 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 205 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518   E.wrex 2519   E!wreu 2520    i^i cin 3126  (class class class)co 5792   ~Hchil 21459    +h cva 21460   0hc0v 21464    -h cmv 21465   SHcsh 21468    +H cph 21471   0Hc0h 21475
This theorem is referenced by:  cdj3lem2  22975
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-hilex 21539  ax-hfvadd 21540  ax-hvcom 21541  ax-hvass 21542  ax-hv0cl 21543  ax-hvaddid 21544  ax-hfvmul 21545  ax-hvmulid 21546  ax-hvmulass 21547  ax-hvdistr1 21548  ax-hvdistr2 21549  ax-hvmul0 21550
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-grpo 20818  df-ablo 20909  df-hvsub 21511  df-sh 21746  df-ch0 21792  df-shs 21847
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