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Theorem chocunii 21710
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 21637 . . . 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 21693 . . . 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 21705 . . . 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 2271 . . . 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 21709 . 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 3077   ` cfv 4592  (class class class)co 5710    +h cva 21330   SHcsh 21338   CHcch 21339   _|_cort 21340   0Hc0h 21345
This theorem is referenced by:  pjcompi  22099
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-hilex 21409  ax-hfvadd 21410  ax-hvcom 21411  ax-hvass 21412  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvmulass 21417  ax-hvdistr1 21418  ax-hvdistr2 21419  ax-hvmul0 21420  ax-hfi 21488  ax-his2 21492  ax-his3 21493  ax-his4 21494
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-hvsub 21381  df-sh 21616  df-ch 21631  df-oc 21661  df-ch0 21662
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