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Theorem intnatN 34693
Description: If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
intnat.b |- B = ( Base ` K )
intnat.l |- .<_ = ( le ` K )
intnat.m |- ./\ = ( meet ` K )
intnat.a |- A = ( Atoms ` K )
Assertion
Ref Expression
intnatN |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( -. Y .<_ X /\ ( X ./\ Y ) e. A ) ) -> -. Y e. A )
Proof of Theorem intnatN
StepHypRef Expression
1 hlatl 34647 . . . . . . 7 |- ( K e. HL -> K e. AtLat )
213ad2ant1 1082 . . . . . 6 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> K e. AtLat )
32ad2antrr 762 . . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ ( X ./\ Y ) e. A ) -> K e. AtLat )
4 eqid 2622 . . . . . 6 |- ( 0. ` K ) = ( 0. ` K )
5 intnat.a . . . . . 6 |- A = ( Atoms ` K )
64, 5atn0 34595 . . . . 5 |- ( ( K e. AtLat /\ ( X ./\ Y ) e. A ) -> ( X ./\ Y ) =/= ( 0. ` K ) )
73, 6sylancom 701 . . . 4 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ ( X ./\ Y ) e. A ) -> ( X ./\ Y ) =/= ( 0. ` K ) )
87ex 450 . . 3 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) -> ( ( X ./\ Y ) e. A -> ( X ./\ Y ) =/= ( 0. ` K ) ) )
9 simpll1 1100 . . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ Y e. A ) -> K e. HL )
10 hllat 34650 . . . . . . . 8 |- ( K e. HL -> K e. Lat )
119, 10syl 17 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ Y e. A ) -> K e. Lat )
12 simpll2 1101 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ Y e. A ) -> X e. B )
13 simpll3 1102 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ Y e. A ) -> Y e. B )
14 intnat.b . . . . . . . 8 |- B = ( Base ` K )
15 intnat.m . . . . . . . 8 |- ./\ = ( meet ` K )
1614, 15latmcom 17075 . . . . . . 7 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( Y ./\ X ) )
1711, 12, 13, 16syl3anc 1326 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ Y e. A ) -> ( X ./\ Y ) = ( Y ./\ X ) )
18 simplr 792 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ Y e. A ) -> -. Y .<_ X )
199, 1syl 17 . . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ Y e. A ) -> K e. AtLat )
20 simpr 477 . . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ Y e. A ) -> Y e. A )
21 intnat.l . . . . . . . . 9 |- .<_ = ( le ` K )
2214, 21, 15, 4, 5atnle 34604 . . . . . . . 8 |- ( ( K e. AtLat /\ Y e. A /\ X e. B ) -> ( -. Y .<_ X <-> ( Y ./\ X ) = ( 0. ` K ) ) )
2319, 20, 12, 22syl3anc 1326 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ Y e. A ) -> ( -. Y .<_ X <-> ( Y ./\ X ) = ( 0. ` K ) ) )
2418, 23mpbid 222 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ Y e. A ) -> ( Y ./\ X ) = ( 0. ` K ) )
2517, 24eqtrd 2656 . . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) /\ Y e. A ) -> ( X ./\ Y ) = ( 0. ` K ) )
2625ex 450 . . . 4 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) -> ( Y e. A -> ( X ./\ Y ) = ( 0. ` K ) ) )
2726necon3ad 2807 . . 3 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) -> ( ( X ./\ Y ) =/= ( 0. ` K ) -> -. Y e. A ) )
288, 27syld 47 . 2 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ -. Y .<_ X ) -> ( ( X ./\ Y ) e. A -> -. Y e. A ) )
2928impr 649 1 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( -. Y .<_ X /\ ( X ./\ Y ) e. A ) ) -> -. Y e. A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> 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
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-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638
This theorem is referenced by: (None)
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