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Theorem lhpm0atN 39413
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 482 . . . 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 38746 . . . . . . . . . . 11 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
65ad2antrr 723 . . . . . . . . . 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 39382 . . . . . . . . . . 11 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
109ad2antlr 724 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ π‘Š ∈ 𝐡)
11 eqid 2726 . . . . . . . . . . 11 (leβ€˜πΎ) = (leβ€˜πΎ)
12 lhpm0at.m . . . . . . . . . . 11 ∧ = (meetβ€˜πΎ)
137, 11, 12latleeqm1 18432 . . . . . . . . . 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 476 . . . . . . . 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 2768 . . . . . . 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 412 . . . . . 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 2947 . . . . 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 2726 . . . . 5 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
227, 11, 12, 21, 8lhpmcvr 39407 . . . 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 834 . . 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 5165 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ 0 ( β‹– β€˜πΎ)𝑋)
25 simpll 764 . . 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 38670 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∈ 𝐴 ↔ 0 ( β‹– β€˜πΎ)𝑋))
2925, 3, 28syl2anc 583 . 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 257 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  meetcmee 18277  0.cp0 18388  Latclat 18396   β‹– ccvr 38645  Atomscatm 38646  HLchlt 38733  LHypclh 39368
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-lhyp 39372
This theorem is referenced by: (None)
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