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Theorem pjhthmo 26940
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 831 . . . 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 2996 . . . . . 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 1044 . . . . . . . . . 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 1045 . . . . . . . . . 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 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  i^i  B
)  =  0H )
6 simplrl 768 . . . . . . . . . 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 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 ) ) ) )  -> 
y  e.  B )
8 simplrr 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 ) ) ) )  -> 
z  e.  A )
9 simprlr 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 ) ) ) )  ->  w  e.  B )
10 simprrl 772 . . . . . . . . . . 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 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  =  ( z  +h  w ) )
1210, 11eqtr3d 2465 . . . . . . . . . 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 26938 . . . . . . . . 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 460 . . . . . . . 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 608 . . . . . . 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 2923 . . . . . 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 221 . . . . 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 606 . . . 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 221 . . 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 1764 . 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 2494 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
22 oveq1 6308 . . . . . . 7  |-  ( x  =  z  ->  (
x  +h  y )  =  ( z  +h  y ) )
2322eqeq2d 2436 . . . . . 6  |-  ( x  =  z  ->  ( C  =  ( x  +h  y )  <->  C  =  ( z  +h  y
) ) )
2423rexbidv 2939 . . . . 5  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. y  e.  B  C  =  ( z  +h  y
) ) )
25 oveq2 6309 . . . . . . 7  |-  ( y  =  w  ->  (
z  +h  y )  =  ( z  +h  w ) )
2625eqeq2d 2436 . . . . . 6  |-  ( y  =  w  ->  ( C  =  ( z  +h  y )  <->  C  =  ( z  +h  w
) ) )
2726cbvrexv 3056 . . . . 5  |-  ( E. y  e.  B  C  =  ( z  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) )
2824, 27syl6bb 264 . . . 4  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) ) )
2921, 28anbi12d 715 . . 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 2313 . 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 215 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 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1868   E*wmo 2266   E.wrex 2776    i^i cin 3435  (class class class)co 6301    +h cva 26558   SHcsh 26566   0Hc0h 26573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-hilex 26637  ax-hfvadd 26638  ax-hvcom 26639  ax-hvass 26640  ax-hv0cl 26641  ax-hvaddid 26642  ax-hfvmul 26643  ax-hvmulid 26644  ax-hvmulass 26645  ax-hvdistr1 26646  ax-hvdistr2 26647  ax-hvmul0 26648
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-po 4770  df-so 4771  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-hvsub 26609  df-sh 26845  df-ch0 26891
This theorem is referenced by:  pjhtheu  27032  pjpreeq  27036
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