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Theorem chocunii 21826
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 21753 . . . 4  |-  H  e.  SH
32a1i 12 . . 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 21809 . . . 4  |-  ( H  e.  SH  ->  ( _|_ `  H )  e.  SH )
52, 4mp1i 13 . . 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 21821 . . . 4  |-  ( H  e.  SH  ->  ( H  i^i  ( _|_ `  H
) )  =  0H )
72, 6mp1i 13 . . 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 737 . . 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 738 . . 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 739 . . 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 740 . . 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 2274 . . . 4  |-  ( ( R  =  ( A  +h  B )  /\  R  =  ( C  +h  D ) )  -> 
( A  +h  B
)  =  ( C  +h  D ) )
1312adantl 454 . . 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 21825 . 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 425 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 6    /\ wa 360    = wceq 1619    e. wcel 1621    i^i cin 3112   ` cfv 4659  (class class class)co 5778    +h cva 21446   SHcsh 21454   CHcch 21455   _|_cort 21456   0Hc0h 21461
This theorem is referenced by:  pjcompi  22215
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-hilex 21525  ax-hfvadd 21526  ax-hvcom 21527  ax-hvass 21528  ax-hv0cl 21529  ax-hvaddid 21530  ax-hfvmul 21531  ax-hvmulid 21532  ax-hvmulass 21533  ax-hvdistr1 21534  ax-hvdistr2 21535  ax-hvmul0 21536  ax-hfi 21604  ax-his2 21608  ax-his3 21609  ax-his4 21610
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-po 4272  df-so 4273  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-iota 6211  df-riota 6258  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-hvsub 21497  df-sh 21732  df-ch 21747  df-oc 21777  df-ch0 21778
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