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Theorem 2llnmeqat 34857
Description: An atom equals the intersection of two majorizing lines. (Contributed by NM, 3-Apr-2013.)
Hypotheses
Ref Expression
2llnmeqat.l |- .<_ = ( le ` K )
2llnmeqat.m |- ./\ = ( meet ` K )
2llnmeqat.a |- A = ( Atoms ` K )
2llnmeqat.n |- N = ( LLines ` K )
Assertion
Ref Expression
2llnmeqat |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> P = ( X ./\ Y ) )
Proof of Theorem 2llnmeqat
StepHypRef Expression
1 simp3r 1090 . 2 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> P .<_ ( X ./\ Y ) )
2 hlatl 34647 . . . 4 |- ( K e. HL -> K e. AtLat )
323ad2ant1 1082 . . 3 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> K e. AtLat )
4 simp23 1096 . . 3 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> P e. A )
5 simp1 1061 . . . 4 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> K e. HL )
6 simp21 1094 . . . 4 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> X e. N )
7 simp22 1095 . . . 4 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> Y e. N )
8 simp3l 1089 . . . 4 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> X =/= Y )
9 hllat 34650 . . . . . . . 8 |- ( K e. HL -> K e. Lat )
1093ad2ant1 1082 . . . . . . 7 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> K e. Lat )
11 eqid 2622 . . . . . . . . 9 |- ( Base ` K ) = ( Base ` K )
12 2llnmeqat.a . . . . . . . . 9 |- A = ( Atoms ` K )
1311, 12atbase 34576 . . . . . . . 8 |- ( P e. A -> P e. ( Base ` K ) )
144, 13syl 17 . . . . . . 7 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> P e. ( Base ` K ) )
15 2llnmeqat.n . . . . . . . . 9 |- N = ( LLines ` K )
1611, 15llnbase 34795 . . . . . . . 8 |- ( X e. N -> X e. ( Base ` K ) )
176, 16syl 17 . . . . . . 7 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> X e. ( Base ` K ) )
1811, 15llnbase 34795 . . . . . . . 8 |- ( Y e. N -> Y e. ( Base ` K ) )
197, 18syl 17 . . . . . . 7 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> Y e. ( Base ` K ) )
20 2llnmeqat.l . . . . . . . 8 |- .<_ = ( le ` K )
21 2llnmeqat.m . . . . . . . 8 |- ./\ = ( meet ` K )
2211, 20, 21latlem12 17078 . . . . . . 7 |- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( P .<_ X /\ P .<_ Y ) <-> P .<_ ( X ./\ Y ) ) )
2310, 14, 17, 19, 22syl13anc 1328 . . . . . 6 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> ( ( P .<_ X /\ P .<_ Y ) <-> P .<_ ( X ./\ Y ) ) )
241, 23mpbird 247 . . . . 5 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> ( P .<_ X /\ P .<_ Y ) )
25 eqid 2622 . . . . . 6 |- ( 0. ` K ) = ( 0. ` K )
2620, 21, 25, 12, 152llnm4 34856 . . . . 5 |- ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> ( X ./\ Y ) =/= ( 0. ` K ) )
275, 4, 6, 7, 24, 26syl131anc 1339 . . . 4 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> ( X ./\ Y ) =/= ( 0. ` K ) )
2821, 25, 12, 152llnmat 34810 . . . 4 |- ( ( ( K e. HL /\ X e. N /\ Y e. N ) /\ ( X =/= Y /\ ( X ./\ Y ) =/= ( 0. ` K ) ) ) -> ( X ./\ Y ) e. A )
295, 6, 7, 8, 27, 28syl32anc 1334 . . 3 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> ( X ./\ Y ) e. A )
3020, 12atcmp 34598 . . 3 |- ( ( K e. AtLat /\ P e. A /\ ( X ./\ Y ) e. A ) -> ( P .<_ ( X ./\ Y ) <-> P = ( X ./\ Y ) ) )
313, 4, 29, 30syl3anc 1326 . 2 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> ( P .<_ ( X ./\ Y ) <-> P = ( X ./\ Y ) ) )
321, 31mpbid 222 1 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ P e. A ) /\ ( X =/= Y /\ P .<_ ( X ./\ Y ) ) ) -> P = ( X ./\ Y ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   meetcmee 16945   0.cp0 17037   Latclat 17045   Atomscatm 34550   AtLatcal 34551   HLchlt 34637   LLinesclln 34777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784
This theorem is referenced by:  cdlemeg46req  35817
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