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Theorem cdleme9tN 29713
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 29709 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 936    = wceq 1624    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   HLchlt 28807   LHypclh 29440
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444
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