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Theorem pjhthmo 26043
Description: Projection Theorem, uniqueness part. Any two disjoint subspaces yield a unique decomposition of vectors into each subspace. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
pjhthmo  |-  ( ( A  e.  SH  /\  B  e.  SH  /\  ( A  i^i  B )  =  0H )  ->  E* x ( 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 pjhthmo
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 an4 822 . . . 4  |-  ( ( ( x  e.  A  /\  z  e.  A
)  /\  ( E. y  e.  B  C  =  ( x  +h  y )  /\  E. w  e.  B  C  =  ( z  +h  w ) ) )  <-> 
( ( x  e.  A  /\  E. y  e.  B  C  =  ( x  +h  y
) )  /\  (
z  e.  A  /\  E. w  e.  B  C  =  ( z  +h  w ) ) ) )
2 reeanv 3034 . . . . . 6  |-  ( 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 ) ) )
3 simpll1 1035 . . . . . . . . . 10  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 ) ) ) )  ->  A  e.  SH )
4 simpll2 1036 . . . . . . . . . 10  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 ) ) ) )  ->  B  e.  SH )
5 simpll3 1037 . . . . . . . . . 10  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 ) ) ) )  -> 
( A  i^i  B
)  =  0H )
6 simplrl 759 . . . . . . . . . 10  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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  e.  A )
7 simprll 761 . . . . . . . . . 10  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 ) ) ) )  -> 
y  e.  B )
8 simplrr 760 . . . . . . . . . 10  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 ) ) ) )  -> 
z  e.  A )
9 simprlr 762 . . . . . . . . . 10  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 ) ) ) )  ->  w  e.  B )
10 simprrl 763 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 ) ) ) )  ->  C  =  ( x  +h  y ) )
11 simprrr 764 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 ) ) ) )  ->  C  =  ( z  +h  w ) )
1210, 11eqtr3d 2510 . . . . . . . . . 10  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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  +h  y
)  =  ( z  +h  w ) )
133, 4, 5, 6, 7, 8, 9, 12shuni 26041 . . . . . . . . 9  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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  /\  y  =  w ) )
1413simpld 459 . . . . . . . 8  |-  ( ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 )
1514exp32 605 . . . . . . 7  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 )
) )
1615rexlimdvv 2965 . . . . . 6  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 ) )
172, 16syl5bir 218 . . . . 5  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 ) )
1817expimpd 603 . . . 4  |-  ( ( A  e.  SH  /\  B  e.  SH  /\  ( 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 ) )
191, 18syl5bir 218 . . 3  |-  ( ( A  e.  SH  /\  B  e.  SH  /\  ( A  i^i  B )  =  0H )  ->  (
( ( x  e.  A  /\  E. y  e.  B  C  =  ( x  +h  y
) )  /\  (
z  e.  A  /\  E. w  e.  B  C  =  ( z  +h  w ) ) )  ->  x  =  z ) )
2019alrimivv 1696 . 2  |-  ( ( A  e.  SH  /\  B  e.  SH  /\  ( A  i^i  B )  =  0H )  ->  A. x A. z ( ( ( x  e.  A  /\  E. y  e.  B  C  =  ( x  +h  y ) )  /\  ( z  e.  A  /\  E. w  e.  B  C  =  ( z  +h  w ) ) )  ->  x  =  z ) )
21 eleq1 2539 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
22 oveq1 6302 . . . . . . 7  |-  ( x  =  z  ->  (
x  +h  y )  =  ( z  +h  y ) )
2322eqeq2d 2481 . . . . . 6  |-  ( x  =  z  ->  ( C  =  ( x  +h  y )  <->  C  =  ( z  +h  y
) ) )
2423rexbidv 2978 . . . . 5  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. y  e.  B  C  =  ( z  +h  y
) ) )
25 oveq2 6303 . . . . . . 7  |-  ( y  =  w  ->  (
z  +h  y )  =  ( z  +h  w ) )
2625eqeq2d 2481 . . . . . 6  |-  ( y  =  w  ->  ( C  =  ( z  +h  y )  <->  C  =  ( z  +h  w
) ) )
2726cbvrexv 3094 . . . . 5  |-  ( E. y  e.  B  C  =  ( z  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) )
2824, 27syl6bb 261 . . . 4  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) ) )
2921, 28anbi12d 710 . . 3  |-  ( x  =  z  ->  (
( x  e.  A  /\  E. y  e.  B  C  =  ( x  +h  y ) )  <->  ( z  e.  A  /\  E. w  e.  B  C  =  ( z  +h  w
) ) ) )
3029mo4 2339 . 2  |-  ( E* x ( x  e.  A  /\  E. y  e.  B  C  =  ( x  +h  y
) )  <->  A. x A. z ( ( ( x  e.  A  /\  E. y  e.  B  C  =  ( x  +h  y ) )  /\  ( z  e.  A  /\  E. w  e.  B  C  =  ( z  +h  w ) ) )  ->  x  =  z ) )
3120, 30sylibr 212 1  |-  ( ( A  e.  SH  /\  B  e.  SH  /\  ( A  i^i  B )  =  0H )  ->  E* x ( x  e.  A  /\  E. y  e.  B  C  =  ( x  +h  y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767   E*wmo 2276   E.wrex 2818    i^i cin 3480  (class class class)co 6295    +h cva 25660   SHcsh 25668   0Hc0h 25675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-hilex 25739  ax-hfvadd 25740  ax-hvcom 25741  ax-hvass 25742  ax-hv0cl 25743  ax-hvaddid 25744  ax-hfvmul 25745  ax-hvmulid 25746  ax-hvmulass 25747  ax-hvdistr1 25748  ax-hvdistr2 25749  ax-hvmul0 25750
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-hvsub 25711  df-sh 25947  df-ch0 25994
This theorem is referenced by:  pjhtheu  26135  pjpreeq  26139
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