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Theorem chocunii 22644
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 22571 . . . 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 22627 . . . 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 22639 . . . 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 735 . . 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 736 . . 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 737 . . 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 738 . . 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 2398 . . . 4  |-  ( ( R  =  ( A  +h  B )  /\  R  =  ( C  +h  D ) )  -> 
( A  +h  B
)  =  ( C  +h  D ) )
1312adantl 453 . . 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 22643 . 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 424 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 359    = wceq 1649    e. wcel 1717    i^i cin 3255   ` cfv 5387  (class class class)co 6013    +h cva 22264   SHcsh 22272   CHcch 22273   _|_cort 22274   0Hc0h 22279
This theorem is referenced by:  pjcompi  23015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-hilex 22343  ax-hfvadd 22344  ax-hvcom 22345  ax-hvass 22346  ax-hv0cl 22347  ax-hvaddid 22348  ax-hfvmul 22349  ax-hvmulid 22350  ax-hvmulass 22351  ax-hvdistr1 22352  ax-hvdistr2 22353  ax-hvmul0 22354  ax-hfi 22422  ax-his2 22426  ax-his3 22427  ax-his4 22428
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-hvsub 22315  df-sh 22550  df-ch 22565  df-oc 22595  df-ch0 22596
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