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Theorem pjhthmo 24705
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 820 . . . 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 2888 . . . . . 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 1027 . . . . . . . . . 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 1028 . . . . . . . . . 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 1029 . . . . . . . . . 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 2477 . . . . . . . . . 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 24703 . . . . . . . . 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 2847 . . . . . 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 1686 . 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 2503 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
22 oveq1 6098 . . . . . . 7  |-  ( x  =  z  ->  (
x  +h  y )  =  ( z  +h  y ) )
2322eqeq2d 2454 . . . . . 6  |-  ( x  =  z  ->  ( C  =  ( x  +h  y )  <->  C  =  ( z  +h  y
) ) )
2423rexbidv 2736 . . . . 5  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. y  e.  B  C  =  ( z  +h  y
) ) )
25 oveq2 6099 . . . . . . 7  |-  ( y  =  w  ->  (
z  +h  y )  =  ( z  +h  w ) )
2625eqeq2d 2454 . . . . . 6  |-  ( y  =  w  ->  ( C  =  ( z  +h  y )  <->  C  =  ( z  +h  w
) ) )
2726cbvrexv 2948 . . . . 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 2317 . 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 965   A.wal 1367    = wceq 1369    e. wcel 1756   E*wmo 2254   E.wrex 2716    i^i cin 3327  (class class class)co 6091    +h cva 24322   SHcsh 24330   0Hc0h 24337
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 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-hilex 24401  ax-hfvadd 24402  ax-hvcom 24403  ax-hvass 24404  ax-hv0cl 24405  ax-hvaddid 24406  ax-hfvmul 24407  ax-hvmulid 24408  ax-hvmulass 24409  ax-hvdistr1 24410  ax-hvdistr2 24411  ax-hvmul0 24412
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-hvsub 24373  df-sh 24609  df-ch0 24656
This theorem is referenced by:  pjhtheu  24797  pjpreeq  24801
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