HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chocunii Unicode version

Theorem chocunii 21896
Description: Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of [Beran] p. 102 (uniqueness part). (Contributed by NM, 23-Oct-1999.) (Proof shortened by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
Hypothesis
Ref Expression
chocuni.1  |-  H  e. 
CH
Assertion
Ref Expression
chocunii  |-  ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H
) ) )  -> 
( ( R  =  ( A  +h  B
)  /\  R  =  ( C  +h  D
) )  ->  ( A  =  C  /\  B  =  D )
) )

Proof of Theorem chocunii
StepHypRef Expression
1 chocuni.1 . . . . 5  |-  H  e. 
CH
21chshii 21823 . . . 4  |-  H  e.  SH
32a1i 10 . . 3  |-  ( ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H
) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  /\  ( R  =  ( A  +h  B
)  /\  R  =  ( C  +h  D
) ) )  ->  H  e.  SH )
4 shocsh 21879 . . . 4  |-  ( H  e.  SH  ->  ( _|_ `  H )  e.  SH )
52, 4mp1i 11 . . 3  |-  ( ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H
) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  /\  ( R  =  ( A  +h  B
)  /\  R  =  ( C  +h  D
) ) )  -> 
( _|_ `  H
)  e.  SH )
6 ocin 21891 . . . 4  |-  ( H  e.  SH  ->  ( H  i^i  ( _|_ `  H
) )  =  0H )
72, 6mp1i 11 . . 3  |-  ( ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H
) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  /\  ( R  =  ( A  +h  B
)  /\  R  =  ( C  +h  D
) ) )  -> 
( H  i^i  ( _|_ `  H ) )  =  0H )
8 simplll 734 . . 3  |-  ( ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H
) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  /\  ( R  =  ( A  +h  B
)  /\  R  =  ( C  +h  D
) ) )  ->  A  e.  H )
9 simpllr 735 . . 3  |-  ( ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H
) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  /\  ( R  =  ( A  +h  B
)  /\  R  =  ( C  +h  D
) ) )  ->  B  e.  ( _|_ `  H ) )
10 simplrl 736 . . 3  |-  ( ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H
) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  /\  ( R  =  ( A  +h  B
)  /\  R  =  ( C  +h  D
) ) )  ->  C  e.  H )
11 simplrr 737 . . 3  |-  ( ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H
) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  /\  ( R  =  ( A  +h  B
)  /\  R  =  ( C  +h  D
) ) )  ->  D  e.  ( _|_ `  H ) )
12 eqtr2 2314 . . . 4  |-  ( ( R  =  ( A  +h  B )  /\  R  =  ( C  +h  D ) )  -> 
( A  +h  B
)  =  ( C  +h  D ) )
1312adantl 452 . . 3  |-  ( ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H
) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  /\  ( R  =  ( A  +h  B
)  /\  R  =  ( C  +h  D
) ) )  -> 
( A  +h  B
)  =  ( C  +h  D ) )
143, 5, 7, 8, 9, 10, 11, 13shuni 21895 . 2  |-  ( ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H
) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  /\  ( R  =  ( A  +h  B
)  /\  R  =  ( C  +h  D
) ) )  -> 
( A  =  C  /\  B  =  D ) )
1514ex 423 1  |-  ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H
) ) )  -> 
( ( R  =  ( A  +h  B
)  /\  R  =  ( C  +h  D
) )  ->  ( A  =  C  /\  B  =  D )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164   ` cfv 5271  (class class class)co 5874    +h cva 21516   SHcsh 21524   CHcch 21525   _|_cort 21526   0Hc0h 21531
This theorem is referenced by:  pjcompi  22267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his2 21678  ax-his3 21679  ax-his4 21680
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-hvsub 21567  df-sh 21802  df-ch 21817  df-oc 21847  df-ch0 21848
  Copyright terms: Public domain W3C validator