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Theorem lhpm0atN 38538
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 1197 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ (𝑋 ∧ π‘Š) = 0 )
2 simpl 484 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
3 simpr1 1195 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ 𝑋 ∈ 𝐡)
4 simpr2 1196 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ 𝑋 β‰  0 )
5 hllat 37871 . . . . . . . . . . 11 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
65ad2antrr 725 . . . . . . . . . 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 38507 . . . . . . . . . . 11 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
109ad2antlr 726 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ π‘Š ∈ 𝐡)
11 eqid 2733 . . . . . . . . . . 11 (leβ€˜πΎ) = (leβ€˜πΎ)
12 lhpm0at.m . . . . . . . . . . 11 ∧ = (meetβ€˜πΎ)
137, 11, 12latleeqm1 18361 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Š ∈ 𝐡) β†’ (𝑋(leβ€˜πΎ)π‘Š ↔ (𝑋 ∧ π‘Š) = 𝑋))
146, 3, 10, 13syl3anc 1372 . . . . . . . . 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 478 . . . . . . . 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 1218 . . . . . . . 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 2775 . . . . . . 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 414 . . . . . 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 2953 . . . . 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 2733 . . . . 5 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
227, 11, 12, 21, 8lhpmcvr 38532 . . . 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 836 . . 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 5130 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ∧ (𝑋 ∧ π‘Š) = 0 )) β†’ 0 ( β‹– β€˜πΎ)𝑋)
25 simpll 766 . . 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 37795 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∈ 𝐴 ↔ 0 ( β‹– β€˜πΎ)𝑋))
2925, 3, 28syl2anc 585 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  meetcmee 18206  0.cp0 18317  Latclat 18325   β‹– ccvr 37770  Atomscatm 37771  HLchlt 37858  LHypclh 38493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-p1 18320  df-lat 18326  df-clat 18393  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859  df-lhyp 38497
This theorem is referenced by: (None)
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