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Theorem pjhthmo 26948
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 832 . . . 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 2957 . . . . . 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 1046 . . . . . . . . . 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 1047 . . . . . . . . . 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 1048 . . . . . . . . . 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 769 . . . . . . . . . 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 771 . . . . . . . . . 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 770 . . . . . . . . . 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 772 . . . . . . . . . 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 773 . . . . . . . . . . 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 774 . . . . . . . . . . 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 2486 . . . . . . . . . 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 26946 . . . . . . . . 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 461 . . . . . . . 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 609 . . . . . . 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 2884 . . . . . 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 222 . . . . 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 607 . . . 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 222 . . 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 1773 . 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 2516 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
22 oveq1 6295 . . . . . . 7  |-  ( x  =  z  ->  (
x  +h  y )  =  ( z  +h  y ) )
2322eqeq2d 2460 . . . . . 6  |-  ( x  =  z  ->  ( C  =  ( x  +h  y )  <->  C  =  ( z  +h  y
) ) )
2423rexbidv 2900 . . . . 5  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. y  e.  B  C  =  ( z  +h  y
) ) )
25 oveq2 6296 . . . . . . 7  |-  ( y  =  w  ->  (
z  +h  y )  =  ( z  +h  w ) )
2625eqeq2d 2460 . . . . . 6  |-  ( y  =  w  ->  ( C  =  ( z  +h  y )  <->  C  =  ( z  +h  w
) ) )
2726cbvrexv 3019 . . . . 5  |-  ( E. y  e.  B  C  =  ( z  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) )
2824, 27syl6bb 265 . . . 4  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) ) )
2921, 28anbi12d 716 . . 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 2345 . 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 216 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 371    /\ w3a 984   A.wal 1441    = wceq 1443    e. wcel 1886   E*wmo 2299   E.wrex 2737    i^i cin 3402  (class class class)co 6288    +h cva 26566   SHcsh 26574   0Hc0h 26581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-hilex 26645  ax-hfvadd 26646  ax-hvcom 26647  ax-hvass 26648  ax-hv0cl 26649  ax-hvaddid 26650  ax-hfvmul 26651  ax-hvmulid 26652  ax-hvmulass 26653  ax-hvdistr1 26654  ax-hvdistr2 26655  ax-hvmul0 26656
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-hvsub 26617  df-sh 26853  df-ch0 26899
This theorem is referenced by:  pjhtheu  27040  pjpreeq  27044
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