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Theorem cdleme9tN 29597
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114.  X and  F represent t1 and f(t) respectively. In their notation, we prove f(t)  \/ t1 = q  \/ t1. (Contributed by NM, 8-Oct-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme9t.l  |-  .<_  =  ( le `  K )
cdleme9t.j  |-  .\/  =  ( join `  K )
cdleme9t.m  |-  ./\  =  ( meet `  K )
cdleme9t.a  |-  A  =  ( Atoms `  K )
cdleme9t.h  |-  H  =  ( LHyp `  K
)
cdleme9t.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme9t.g  |-  F  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
cdleme9t.x  |-  X  =  ( ( P  .\/  T )  ./\  W )
Assertion
Ref Expression
cdleme9tN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  ( F  .\/  X )  =  ( Q  .\/  X
) )

Proof of Theorem cdleme9tN
StepHypRef Expression
1 cdleme9t.l . 2  |-  .<_  =  ( le `  K )
2 cdleme9t.j . 2  |-  .\/  =  ( join `  K )
3 cdleme9t.m . 2  |-  ./\  =  ( meet `  K )
4 cdleme9t.a . 2  |-  A  =  ( Atoms `  K )
5 cdleme9t.h . 2  |-  H  =  ( LHyp `  K
)
6 cdleme9t.u . 2  |-  U  =  ( ( P  .\/  Q )  ./\  W )
7 cdleme9t.g . 2  |-  F  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
8 cdleme9t.x . 2  |-  X  =  ( ( P  .\/  T )  ./\  W )
91, 2, 3, 4, 5, 6, 7, 8cdleme9 29593 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  ( F  .\/  X )  =  ( Q  .\/  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   lecple 13163   joincjn 14026   meetcmee 14027   Atomscatm 28604   HLchlt 28691   LHypclh 29324
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-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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-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-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  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-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328
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