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Theorem lhpm0atN 39542
Description: If the meet of a lattice hyperplane with a nonzero element is zero, the element is an atom. (Contributed by NM, 28-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lhpm0at.b 𝐡 = (Baseβ€˜πΎ)
lhpm0at.m ∧ = (meetβ€˜πΎ)
lhpm0at.o 0 = (0.β€˜πΎ)
lhpm0at.a 𝐴 = (Atomsβ€˜πΎ)
lhpm0at.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
lhpm0atN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ 𝑋 ∈ 𝐴)
Proof of Theorem lhpm0atN
StepHypRef Expression
1 simpr3 1193 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ (𝑋 ∧ π‘Š) = 0 )
2 simpl 481 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
3 simpr1 1191 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ 𝑋 ∈ 𝐡)
4 simpr2 1192 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ 𝑋 β‰  0 )
5 hllat 38875 . . . . . . . . . . 11 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
65ad2antrr 724 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ 𝐾 ∈ Lat)
7 lhpm0at.b . . . . . . . . . . . 12 𝐡 = (Baseβ€˜πΎ)
8 lhpm0at.h . . . . . . . . . . . 12 𝐻 = (LHypβ€˜πΎ)
97, 8lhpbase 39511 . . . . . . . . . . 11 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
109ad2antlr 725 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ π‘Š ∈ 𝐡)
11 eqid 2728 . . . . . . . . . . 11 (leβ€˜πΎ) = (leβ€˜πΎ)
12 lhpm0at.m . . . . . . . . . . 11 ∧ = (meetβ€˜πΎ)
137, 11, 12latleeqm1 18468 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Š ∈ 𝐡) β†’ (𝑋(leβ€˜πΎ)π‘Š ↔ (𝑋 ∧ π‘Š) = 𝑋))
146, 3, 10, 13syl3anc 1368 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ (𝑋(leβ€˜πΎ)π‘Š ↔ (𝑋 ∧ π‘Š) = 𝑋))
1514biimpa 475 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) ∧ 𝑋(leβ€˜πΎ)π‘Š) β†’ (𝑋 ∧ π‘Š) = 𝑋)
16 simplr3 1214 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) ∧ 𝑋(leβ€˜πΎ)π‘Š) β†’ (𝑋 ∧ π‘Š) = 0 )
1715, 16eqtr3d 2770 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) ∧ 𝑋(leβ€˜πΎ)π‘Š) β†’ 𝑋 = 0 )
1817ex 411 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ (𝑋(leβ€˜πΎ)π‘Š β†’ 𝑋 = 0 ))
1918necon3ad 2950 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ (𝑋 β‰  0 β†’ Β¬ 𝑋(leβ€˜πΎ)π‘Š))
204, 19mpd 15 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ Β¬ 𝑋(leβ€˜πΎ)π‘Š)
21 eqid 2728 . . . . 5 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
227, 11, 12, 21, 8lhpmcvr 39536 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋(leβ€˜πΎ)π‘Š)) β†’ (𝑋 ∧ π‘Š)( β‹– β€˜πΎ)𝑋)
232, 3, 20, 22syl12anc 835 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ (𝑋 ∧ π‘Š)( β‹– β€˜πΎ)𝑋)
241, 23eqbrtrrd 5176 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ 0 ( β‹– β€˜πΎ)𝑋)
25 simpll 765 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ 𝐾 ∈ HL)
26 lhpm0at.o . . . 4 0 = (0.β€˜πΎ)
27 lhpm0at.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
287, 26, 21, 27isat2 38799 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∈ 𝐴 ↔ 0 ( β‹– β€˜πΎ)𝑋))
2925, 3, 28syl2anc 582 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ (𝑋 ∈ 𝐴 ↔ 0 ( β‹– β€˜πΎ)𝑋))
3024, 29mpbird 256 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ 𝑋 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17189  lecple 17249  meetcmee 18313  0.cp0 18424  Latclat 18432   β‹– ccvr 38774  Atomscatm 38775  HLchlt 38862  LHypclh 39497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-proset 18296  df-poset 18314  df-plt 18331  df-lub 18347  df-glb 18348  df-join 18349  df-meet 18350  df-p0 18426  df-p1 18427  df-lat 18433  df-clat 18500  df-oposet 38688  df-ol 38690  df-oml 38691  df-covers 38778  df-ats 38779  df-atl 38810  df-cvlat 38834  df-hlat 38863  df-lhyp 39501
This theorem is referenced by: (None)
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