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Theorem cdjreui 25787
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 24670 . . . 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 2883 . . . . 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 2456 . . . . . . 7  |-  ( ( C  =  ( x  +h  y )  /\  C  =  ( z  +h  w ) )  -> 
( x  +h  y
)  =  ( z  +h  w ) )
71sheli 24567 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  x  e.  ~H )
82sheli 24567 . . . . . . . . . . . 12  |-  ( y  e.  B  ->  y  e.  ~H )
97, 8anim12i 566 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
101sheli 24567 . . . . . . . . . . . 12  |-  ( z  e.  A  ->  z  e.  ~H )
112sheli 24567 . . . . . . . . . . . 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 24431 . . . . . . . . . . 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 24574 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  SH  /\  w  e.  B  /\  y  e.  B )  ->  ( w  -h  y
)  e.  B )
182, 17mp3an1 1301 . . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . . . 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 24574 . . . . . . . . . . . . . 14  |-  ( ( A  e.  SH  /\  x  e.  A  /\  z  e.  A )  ->  ( x  -h  z
)  e.  A )
241, 23mp3an1 1301 . . . . . . . . . . . . 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 3534 . . . . . . . . . . . 12  |-  ( ( x  -h  z )  e.  ( A  i^i  B )  <->  ( ( x  -h  z )  e.  A  /\  ( x  -h  z )  e.  B ) )
29 eleq2 2499 . . . . . . . . . . . 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 24608 . . . . . . . . . . . 12  |-  ( ( x  -h  z )  e.  0H  <->  ( x  -h  z )  =  0h )
34 hvsubeq0 24421 . . . . . . . . . . . 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 2843 . . . . 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 2803 . . 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 6093 . . . . . 6  |-  ( x  =  z  ->  (
x  +h  y )  =  ( z  +h  y ) )
4645eqeq2d 2449 . . . . 5  |-  ( x  =  z  ->  ( C  =  ( x  +h  y )  <->  C  =  ( z  +h  y
) ) )
4746rexbidv 2731 . . . 4  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. y  e.  B  C  =  ( z  +h  y
) ) )
48 oveq2 6094 . . . . . 6  |-  ( y  =  w  ->  (
z  +h  y )  =  ( z  +h  w ) )
4948eqeq2d 2449 . . . . 5  |-  ( y  =  w  ->  ( C  =  ( z  +h  y )  <->  C  =  ( z  +h  w
) ) )
5049cbvrexv 2943 . . . 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 3148 . 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 1369    e. wcel 1756   A.wral 2710   E.wrex 2711   E!wreu 2712    i^i cin 3322  (class class class)co 6086   ~Hchil 24272    +h cva 24273   0hc0v 24277    -h cmv 24278   SHcsh 24281    +H cph 24284   0Hc0h 24288
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-hilex 24352  ax-hfvadd 24353  ax-hvcom 24354  ax-hvass 24355  ax-hv0cl 24356  ax-hvaddid 24357  ax-hfvmul 24358  ax-hvmulid 24359  ax-hvmulass 24360  ax-hvdistr1 24361  ax-hvdistr2 24362  ax-hvmul0 24363
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-grpo 23629  df-ablo 23720  df-hvsub 24324  df-sh 24560  df-ch0 24607  df-shs 24662
This theorem is referenced by:  cdj3lem2  25790
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