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Theorem trlat 34842
Description: If an atom differs from its translation, the trace is an atom. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
Hypotheses
Ref Expression
trlat.l  |-  .<_  =  ( le `  K )
trlat.a  |-  A  =  ( Atoms `  K )
trlat.h  |-  H  =  ( LHyp `  K
)
trlat.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlat.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlat  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)

Proof of Theorem trlat
StepHypRef Expression
1 simp1 991 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp3l 1019 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  F  e.  T
)
3 simp2 992 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 trlat.l . . . 4  |-  .<_  =  ( le `  K )
5 eqid 2462 . . . 4  |-  ( join `  K )  =  (
join `  K )
6 eqid 2462 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
7 trlat.a . . . 4  |-  A  =  ( Atoms `  K )
8 trlat.h . . . 4  |-  H  =  ( LHyp `  K
)
9 trlat.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
10 trlat.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
114, 5, 6, 7, 8, 9, 10trlval2 34836 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P ( join `  K ) ( F `
 P ) ) ( meet `  K
) W ) )
121, 2, 3, 11syl3anc 1223 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  =  ( ( P ( join `  K ) ( F `
 P ) ) ( meet `  K
) W ) )
13 simp2l 1017 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  P  e.  A
)
144, 7, 8, 9ltrnat 34813 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
151, 2, 13, 14syl3anc 1223 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( F `  P )  e.  A
)
16 simp3r 1020 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( F `  P )  =/=  P
)
1716necomd 2733 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  P  =/=  ( F `  P )
)
184, 5, 6, 7, 8lhpat 34716 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  e.  A  /\  P  =/=  ( F `  P ) ) )  ->  ( ( P ( join `  K
) ( F `  P ) ) (
meet `  K ) W )  e.  A
)
191, 3, 15, 17, 18syl112anc 1227 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( ( P ( join `  K
) ( F `  P ) ) (
meet `  K ) W )  e.  A
)
2012, 19eqeltrd 2550 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   lecple 14553   joincjn 15422   meetcmee 15423   Atomscatm 33937   HLchlt 34024   LHypclh 34657   LTrncltrn 34774   trLctrl 34831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-p1 15518  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-lhyp 34661  df-laut 34662  df-ldil 34777  df-ltrn 34778  df-trl 34832
This theorem is referenced by:  trlator0  34844  trlnidat  34846  trlnle  34859  trlval3  34860  trlval4  34861  cdlemc5  34868  cdlemg17dALTN  35337  cdlemg27a  35365  cdlemg31b0N  35367  cdlemg27b  35369  cdlemg31c  35372  cdlemg35  35386  dia2dimlem1  35738  dia2dimlem2  35739  dia2dimlem3  35740
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