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Theorem cdjreui 23927
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 22810 . . . 4  |-  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y
) )
43biimpi 187 . . 3  |-  ( C  e.  ( A  +H  B )  ->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y
) )
5 reeanv 2867 . . . . 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 2453 . . . . . . 7  |-  ( ( C  =  ( x  +h  y )  /\  C  =  ( z  +h  w ) )  -> 
( x  +h  y
)  =  ( z  +h  w ) )
71sheli 22708 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  x  e.  ~H )
82sheli 22708 . . . . . . . . . . . 12  |-  ( y  e.  B  ->  y  e.  ~H )
97, 8anim12i 550 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
101sheli 22708 . . . . . . . . . . . 12  |-  ( z  e.  A  ->  z  e.  ~H )
112sheli 22708 . . . . . . . . . . . 12  |-  ( w  e.  B  ->  w  e.  ~H )
1210, 11anim12i 550 . . . . . . . . . . 11  |-  ( ( z  e.  A  /\  w  e.  B )  ->  ( z  e.  ~H  /\  w  e.  ~H )
)
13 hvaddsub4 22572 . . . . . . . . . . 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 464 . . . . . . . . . 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 800 . . . . . . . . 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 695 . . . . . . . 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 22715 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  SH  /\  w  e.  B  /\  y  e.  B )  ->  ( w  -h  y
)  e.  B )
182, 17mp3an1 1266 . . . . . . . . . . . . . . 15  |-  ( ( w  e.  B  /\  y  e.  B )  ->  ( w  -h  y
)  e.  B )
1918ancoms 440 . . . . . . . . . . . . . 14  |-  ( ( y  e.  B  /\  w  e.  B )  ->  ( w  -h  y
)  e.  B )
20 eleq1 2495 . . . . . . . . . . . . . 14  |-  ( ( x  -h  z )  =  ( w  -h  y )  ->  (
( x  -h  z
)  e.  B  <->  ( w  -h  y )  e.  B
) )
2119, 20syl5ibrcom 214 . . . . . . . . . . . . 13  |-  ( ( y  e.  B  /\  w  e.  B )  ->  ( ( x  -h  z )  =  ( w  -h  y )  ->  ( x  -h  z )  e.  B
) )
2221adantl 453 . . . . . . . . . . . 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 22715 . . . . . . . . . . . . . 14  |-  ( ( A  e.  SH  /\  x  e.  A  /\  z  e.  A )  ->  ( x  -h  z
)  e.  A )
241, 23mp3an1 1266 . . . . . . . . . . . . 13  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( x  -h  z
)  e.  A )
2524adantr 452 . . . . . . . . . . . 12  |-  ( ( ( x  e.  A  /\  z  e.  A
)  /\  ( y  e.  B  /\  w  e.  B ) )  -> 
( x  -h  z
)  e.  A )
2622, 25jctild 528 . . . . . . . . . . 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 695 . . . . . . . . . 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 3522 . . . . . . . . . . . 12  |-  ( ( x  -h  z )  e.  ( A  i^i  B )  <->  ( ( x  -h  z )  e.  A  /\  ( x  -h  z )  e.  B ) )
29 eleq2 2496 . . . . . . . . . . . 12  |-  ( ( A  i^i  B )  =  0H  ->  (
( x  -h  z
)  e.  ( A  i^i  B )  <->  ( x  -h  z )  e.  0H ) )
3028, 29syl5bbr 251 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  =  0H  ->  (
( ( x  -h  z )  e.  A  /\  ( x  -h  z
)  e.  B )  <-> 
( x  -h  z
)  e.  0H ) )
3130ad2antrr 707 . . . . . . . . . 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 206 . . . . . . . . 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 22748 . . . . . . . . . . . 12  |-  ( ( x  -h  z )  e.  0H  <->  ( x  -h  z )  =  0h )
34 hvsubeq0 22562 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  =  0h  <->  x  =  z ) )
3533, 34syl5bb 249 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  e.  0H  <->  x  =  z ) )
367, 10, 35syl2an 464 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( ( x  -h  z )  e.  0H  <->  x  =  z ) )
3736ad2antlr 708 . . . . . . . . 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 206 . . . . . . . 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 207 . . . . . . 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 2829 . . . . 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 210 . . . 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 2790 . . 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 550 . 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 6080 . . . . . 6  |-  ( x  =  z  ->  (
x  +h  y )  =  ( z  +h  y ) )
4645eqeq2d 2446 . . . . 5  |-  ( x  =  z  ->  ( C  =  ( x  +h  y )  <->  C  =  ( z  +h  y
) ) )
4746rexbidv 2718 . . . 4  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. y  e.  B  C  =  ( z  +h  y
) ) )
48 oveq2 6081 . . . . . 6  |-  ( y  =  w  ->  (
z  +h  y )  =  ( z  +h  w ) )
4948eqeq2d 2446 . . . . 5  |-  ( y  =  w  ->  ( C  =  ( z  +h  y )  <->  C  =  ( z  +h  w
) ) )
5049cbvrexv 2925 . . . 4  |-  ( E. y  e.  B  C  =  ( z  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) )
5147, 50syl6bb 253 . . 3  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) ) )
5251reu4 3120 . 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 204 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   E!wreu 2699    i^i cin 3311  (class class class)co 6073   ~Hchil 22414    +h cva 22415   0hc0v 22419    -h cmv 22420   SHcsh 22423    +H cph 22426   0Hc0h 22430
This theorem is referenced by:  cdj3lem2  23930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-hilex 22494  ax-hfvadd 22495  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvmulass 22502  ax-hvdistr1 22503  ax-hvdistr2 22504  ax-hvmul0 22505
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-grpo 21771  df-ablo 21862  df-hvsub 22466  df-sh 22701  df-ch0 22747  df-shs 22802
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