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Theorem pjhthmo 21883
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 797 . . . 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 2709 . . . . . 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 994 . . . . . . . . . 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 995 . . . . . . . . . 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 996 . . . . . . . . . 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 736 . . . . . . . . . 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 738 . . . . . . . . . 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 737 . . . . . . . . . 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 739 . . . . . . . . . 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 740 . . . . . . . . . . 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 741 . . . . . . . . . . 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 2319 . . . . . . . . . 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 21881 . . . . . . . . 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 445 . . . . . . . 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 588 . . . . . . 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 2675 . . . . . 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 209 . . . . 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 586 . . . 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 209 . . 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 1620 . 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 2345 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
22 oveq1 5867 . . . . . . 7  |-  ( x  =  z  ->  (
x  +h  y )  =  ( z  +h  y ) )
2322eqeq2d 2296 . . . . . 6  |-  ( x  =  z  ->  ( C  =  ( x  +h  y )  <->  C  =  ( z  +h  y
) ) )
2423rexbidv 2566 . . . . 5  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. y  e.  B  C  =  ( z  +h  y
) ) )
25 oveq2 5868 . . . . . . 7  |-  ( y  =  w  ->  (
z  +h  y )  =  ( z  +h  w ) )
2625eqeq2d 2296 . . . . . 6  |-  ( y  =  w  ->  ( C  =  ( z  +h  y )  <->  C  =  ( z  +h  w
) ) )
2726cbvrexv 2767 . . . . 5  |-  ( E. y  e.  B  C  =  ( z  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) )
2824, 27syl6bb 252 . . . 4  |-  ( x  =  z  ->  ( E. y  e.  B  C  =  ( x  +h  y )  <->  E. w  e.  B  C  =  ( z  +h  w
) ) )
2921, 28anbi12d 691 . . 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 2178 . 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 203 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1529    = wceq 1625    e. wcel 1686   E*wmo 2146   E.wrex 2546    i^i cin 3153  (class class class)co 5860    +h cva 21502   SHcsh 21510   0Hc0h 21517
This theorem is referenced by:  pjhtheu  21975  pjpreeq  21979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-hilex 21581  ax-hfvadd 21582  ax-hvcom 21583  ax-hvass 21584  ax-hv0cl 21585  ax-hvaddid 21586  ax-hfvmul 21587  ax-hvmulid 21588  ax-hvmulass 21589  ax-hvdistr1 21590  ax-hvdistr2 21591  ax-hvmul0 21592
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-hvsub 21553  df-sh 21788  df-ch0 21834
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