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Theorem chocunii 24709
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 24635 . . . 4  |-  H  e.  SH
32a1i 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 )
4 shocsh 24692 . . . 4  |-  ( H  e.  SH  ->  ( _|_ `  H )  e.  SH )
52, 4mp1i 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 )
6 ocin 24704 . . . 4  |-  ( H  e.  SH  ->  ( H  i^i  ( _|_ `  H
) )  =  0H )
72, 6mp1i 12 . . 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 757 . . 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 758 . . 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 759 . . 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 760 . . 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 2461 . . . 4  |-  ( ( R  =  ( A  +h  B )  /\  R  =  ( C  +h  D ) )  -> 
( A  +h  B
)  =  ( C  +h  D ) )
1312adantl 466 . . 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 24708 . 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 434 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3332   ` cfv 5423  (class class class)co 6096    +h cva 24327   SHcsh 24335   CHcch 24336   _|_cort 24337   0Hc0h 24342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-hilex 24406  ax-hfvadd 24407  ax-hvcom 24408  ax-hvass 24409  ax-hv0cl 24410  ax-hvaddid 24411  ax-hfvmul 24412  ax-hvmulid 24413  ax-hvmulass 24414  ax-hvdistr1 24415  ax-hvdistr2 24416  ax-hvmul0 24417  ax-hfi 24486  ax-his2 24490  ax-his3 24491  ax-his4 24492
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-hvsub 24378  df-sh 24614  df-ch 24629  df-oc 24660  df-ch0 24661
This theorem is referenced by:  pjcompi  25080
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