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Theorem cdjreui 25981
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 24864 . . . 4  |-  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y
) )
43biimpi 194 . . 3  |-  ( C  e.  ( A  +H  B )  ->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y
) )
5 reeanv 2987 . . . . 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 2478 . . . . . . 7  |-  ( ( C  =  ( x  +h  y )  /\  C  =  ( z  +h  w ) )  -> 
( x  +h  y
)  =  ( z  +h  w ) )
71sheli 24761 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  x  e.  ~H )
82sheli 24761 . . . . . . . . . . . 12  |-  ( y  e.  B  ->  y  e.  ~H )
97, 8anim12i 566 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
101sheli 24761 . . . . . . . . . . . 12  |-  ( z  e.  A  ->  z  e.  ~H )
112sheli 24761 . . . . . . . . . . . 12  |-  ( w  e.  B  ->  w  e.  ~H )
1210, 11anim12i 566 . . . . . . . . . . 11  |-  ( ( z  e.  A  /\  w  e.  B )  ->  ( z  e.  ~H  /\  w  e.  ~H )
)
13 hvaddsub4 24625 . . . . . . . . . . 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 477 . . . . . . . . . 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 822 . . . . . . . . 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 713 . . . . . . . 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 24768 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  SH  /\  w  e.  B  /\  y  e.  B )  ->  ( w  -h  y
)  e.  B )
182, 17mp3an1 1302 . . . . . . . . . . . . . . 15  |-  ( ( w  e.  B  /\  y  e.  B )  ->  ( w  -h  y
)  e.  B )
1918ancoms 453 . . . . . . . . . . . . . 14  |-  ( ( y  e.  B  /\  w  e.  B )  ->  ( w  -h  y
)  e.  B )
20 eleq1 2523 . . . . . . . . . . . . . 14  |-  ( ( x  -h  z )  =  ( w  -h  y )  ->  (
( x  -h  z
)  e.  B  <->  ( w  -h  y )  e.  B
) )
2119, 20syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( ( y  e.  B  /\  w  e.  B )  ->  ( ( x  -h  z )  =  ( w  -h  y )  ->  ( x  -h  z )  e.  B
) )
2221adantl 466 . . . . . . . . . . . 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 24768 . . . . . . . . . . . . . 14  |-  ( ( A  e.  SH  /\  x  e.  A  /\  z  e.  A )  ->  ( x  -h  z
)  e.  A )
241, 23mp3an1 1302 . . . . . . . . . . . . 13  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( x  -h  z
)  e.  A )
2524adantr 465 . . . . . . . . . . . 12  |-  ( ( ( x  e.  A  /\  z  e.  A
)  /\  ( y  e.  B  /\  w  e.  B ) )  -> 
( x  -h  z
)  e.  A )
2622, 25jctild 543 . . . . . . . . . . 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 713 . . . . . . . . . 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 3640 . . . . . . . . . . . 12  |-  ( ( x  -h  z )  e.  ( A  i^i  B )  <->  ( ( x  -h  z )  e.  A  /\  ( x  -h  z )  e.  B ) )
29 eleq2 2524 . . . . . . . . . . . 12  |-  ( ( A  i^i  B )  =  0H  ->  (
( x  -h  z
)  e.  ( A  i^i  B )  <->  ( x  -h  z )  e.  0H ) )
3028, 29syl5bbr 259 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  =  0H  ->  (
( ( x  -h  z )  e.  A  /\  ( x  -h  z
)  e.  B )  <-> 
( x  -h  z
)  e.  0H ) )
3130ad2antrr 725 . . . . . . . . . 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 214 . . . . . . . . 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 24802 . . . . . . . . . . . 12  |-  ( ( x  -h  z )  e.  0H  <->  ( x  -h  z )  =  0h )
34 hvsubeq0 24615 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  =  0h  <->  x  =  z ) )
3533, 34syl5bb 257 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  e.  0H  <->  x  =  z ) )
367, 10, 35syl2an 477 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( ( x  -h  z )  e.  0H  <->  x  =  z ) )
3736ad2antlr 726 . . . . . . . . 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 214 . . . . . . . 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 215 . . . . . . 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 32 . . . . . 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 2947 . . . . 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 218 . . . 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 2907 . . 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 566 . 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 6200 . . . . . 6  |-  ( x  =  z  ->  (
x  +h  y )  =  ( z  +h  y ) )
4645eqeq2d 2465 . . . . 5  |-  ( x  =  z  ->  ( C  =  ( x  +h  y )  <->  C  =  ( z  +h  y
) ) )
4746rexbidv 2855 . . . 4  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. y  e.  B  C  =  ( z  +h  y
) ) )
48 oveq2 6201 . . . . . 6  |-  ( y  =  w  ->  (
z  +h  y )  =  ( z  +h  w ) )
4948eqeq2d 2465 . . . . 5  |-  ( y  =  w  ->  ( C  =  ( z  +h  y )  <->  C  =  ( z  +h  w
) ) )
5049cbvrexv 3047 . . . 4  |-  ( E. y  e.  B  C  =  ( z  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) )
5147, 50syl6bb 261 . . 3  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) ) )
5251reu4 3253 . 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 212 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   E!wreu 2797    i^i cin 3428  (class class class)co 6193   ~Hchil 24466    +h cva 24467   0hc0v 24471    -h cmv 24472   SHcsh 24475    +H cph 24478   0Hc0h 24482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-hilex 24546  ax-hfvadd 24547  ax-hvcom 24548  ax-hvass 24549  ax-hv0cl 24550  ax-hvaddid 24551  ax-hfvmul 24552  ax-hvmulid 24553  ax-hvmulass 24554  ax-hvdistr1 24555  ax-hvdistr2 24556  ax-hvmul0 24557
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-grpo 23823  df-ablo 23914  df-hvsub 24518  df-sh 24754  df-ch0 24801  df-shs 24856
This theorem is referenced by:  cdj3lem2  25984
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