Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trlat Unicode version

Theorem trlat 30427
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 955 . . 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 983 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  F  e.  T
)
3 simp2 956 . . 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 2358 . . . 4  |-  ( join `  K )  =  (
join `  K )
6 eqid 2358 . . . 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 30421 . . 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 1182 . 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 981 . . . 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 30398 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
151, 2, 13, 14syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( F `  P )  e.  A
)
16 simp3r 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( F `  P )  =/=  P
)
1716necomd 2604 . . 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 30301 . . 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 1186 . 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 2432 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   lecple 13312   joincjn 14177   meetcmee 14178   Atomscatm 29522   HLchlt 29609   LHypclh 30242   LTrncltrn 30359   trLctrl 30416
This theorem is referenced by:  trlator0  30429  trlnidat  30431  trlnle  30444  trlval3  30445  trlval4  30446  cdlemc5  30453  cdlemg17dALTN  30922  cdlemg27a  30950  cdlemg31b0N  30952  cdlemg27b  30954  cdlemg31c  30957  cdlemg35  30971  dia2dimlem1  31323  dia2dimlem2  31324  dia2dimlem3  31325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-map 6862  df-poset 14179  df-plt 14191  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-p0 14244  df-p1 14245  df-lat 14251  df-clat 14313  df-oposet 29435  df-ol 29437  df-oml 29438  df-covers 29525  df-ats 29526  df-atl 29557  df-cvlat 29581  df-hlat 29610  df-lhyp 30246  df-laut 30247  df-ldil 30362  df-ltrn 30363  df-trl 30417
  Copyright terms: Public domain W3C validator