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Theorem lhpocnle 39190
Description: The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)
Hypotheses
Ref Expression
lhpocnle.l ≀ = (leβ€˜πΎ)
lhpocnle.o βŠ₯ = (ocβ€˜πΎ)
lhpocnle.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
lhpocnle ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ Β¬ ( βŠ₯ β€˜π‘Š) ≀ π‘Š)

Proof of Theorem lhpocnle
StepHypRef Expression
1 hlatl 38533 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
21adantr 479 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐾 ∈ AtLat)
3 simpr 483 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘Š ∈ 𝐻)
4 eqid 2730 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
5 lhpocnle.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
64, 5lhpbase 39172 . . . . . 6 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
7 lhpocnle.o . . . . . . 7 βŠ₯ = (ocβ€˜πΎ)
8 eqid 2730 . . . . . . 7 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
94, 7, 8, 5lhpoc 39188 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ (π‘Š ∈ 𝐻 ↔ ( βŠ₯ β€˜π‘Š) ∈ (Atomsβ€˜πΎ)))
106, 9sylan2 591 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (π‘Š ∈ 𝐻 ↔ ( βŠ₯ β€˜π‘Š) ∈ (Atomsβ€˜πΎ)))
113, 10mpbid 231 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( βŠ₯ β€˜π‘Š) ∈ (Atomsβ€˜πΎ))
12 eqid 2730 . . . . 5 (0.β€˜πΎ) = (0.β€˜πΎ)
1312, 8atn0 38481 . . . 4 ((𝐾 ∈ AtLat ∧ ( βŠ₯ β€˜π‘Š) ∈ (Atomsβ€˜πΎ)) β†’ ( βŠ₯ β€˜π‘Š) β‰  (0.β€˜πΎ))
142, 11, 13syl2anc 582 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( βŠ₯ β€˜π‘Š) β‰  (0.β€˜πΎ))
1514neneqd 2943 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ Β¬ ( βŠ₯ β€˜π‘Š) = (0.β€˜πΎ))
16 simpr 483 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ ( βŠ₯ β€˜π‘Š) ≀ π‘Š)
17 hllat 38536 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1817ad2antrr 722 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ 𝐾 ∈ Lat)
19 hlop 38535 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
2019ad2antrr 722 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ 𝐾 ∈ OP)
216ad2antlr 723 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ π‘Š ∈ (Baseβ€˜πΎ))
224, 7opoccl 38367 . . . . . . 7 ((𝐾 ∈ OP ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ( βŠ₯ β€˜π‘Š) ∈ (Baseβ€˜πΎ))
2320, 21, 22syl2anc 582 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ ( βŠ₯ β€˜π‘Š) ∈ (Baseβ€˜πΎ))
24 lhpocnle.l . . . . . . 7 ≀ = (leβ€˜πΎ)
254, 24latref 18398 . . . . . 6 ((𝐾 ∈ Lat ∧ ( βŠ₯ β€˜π‘Š) ∈ (Baseβ€˜πΎ)) β†’ ( βŠ₯ β€˜π‘Š) ≀ ( βŠ₯ β€˜π‘Š))
2618, 23, 25syl2anc 582 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ ( βŠ₯ β€˜π‘Š) ≀ ( βŠ₯ β€˜π‘Š))
27 eqid 2730 . . . . . . 7 (meetβ€˜πΎ) = (meetβ€˜πΎ)
284, 24, 27latlem12 18423 . . . . . 6 ((𝐾 ∈ Lat ∧ (( βŠ₯ β€˜π‘Š) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ) ∧ ( βŠ₯ β€˜π‘Š) ∈ (Baseβ€˜πΎ))) β†’ ((( βŠ₯ β€˜π‘Š) ≀ π‘Š ∧ ( βŠ₯ β€˜π‘Š) ≀ ( βŠ₯ β€˜π‘Š)) ↔ ( βŠ₯ β€˜π‘Š) ≀ (π‘Š(meetβ€˜πΎ)( βŠ₯ β€˜π‘Š))))
2918, 23, 21, 23, 28syl13anc 1370 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ ((( βŠ₯ β€˜π‘Š) ≀ π‘Š ∧ ( βŠ₯ β€˜π‘Š) ≀ ( βŠ₯ β€˜π‘Š)) ↔ ( βŠ₯ β€˜π‘Š) ≀ (π‘Š(meetβ€˜πΎ)( βŠ₯ β€˜π‘Š))))
3016, 26, 29mpbi2and 708 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ ( βŠ₯ β€˜π‘Š) ≀ (π‘Š(meetβ€˜πΎ)( βŠ₯ β€˜π‘Š)))
314, 7, 27, 12opnoncon 38381 . . . . 5 ((𝐾 ∈ OP ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ (π‘Š(meetβ€˜πΎ)( βŠ₯ β€˜π‘Š)) = (0.β€˜πΎ))
3220, 21, 31syl2anc 582 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ (π‘Š(meetβ€˜πΎ)( βŠ₯ β€˜π‘Š)) = (0.β€˜πΎ))
3330, 32breqtrd 5173 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ ( βŠ₯ β€˜π‘Š) ≀ (0.β€˜πΎ))
344, 24, 12ople0 38360 . . . 4 ((𝐾 ∈ OP ∧ ( βŠ₯ β€˜π‘Š) ∈ (Baseβ€˜πΎ)) β†’ (( βŠ₯ β€˜π‘Š) ≀ (0.β€˜πΎ) ↔ ( βŠ₯ β€˜π‘Š) = (0.β€˜πΎ)))
3520, 23, 34syl2anc 582 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ (( βŠ₯ β€˜π‘Š) ≀ (0.β€˜πΎ) ↔ ( βŠ₯ β€˜π‘Š) = (0.β€˜πΎ)))
3633, 35mpbid 231 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘Š) ≀ π‘Š) β†’ ( βŠ₯ β€˜π‘Š) = (0.β€˜πΎ))
3715, 36mtand 812 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ Β¬ ( βŠ₯ β€˜π‘Š) ≀ π‘Š)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  lecple 17208  occoc 17209  meetcmee 18269  0.cp0 18380  Latclat 18388  OPcops 38345  Atomscatm 38436  AtLatcal 38437  HLchlt 38523  LHypclh 39158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-meet 18306  df-p0 18382  df-p1 18383  df-lat 18389  df-oposet 38349  df-ol 38351  df-oml 38352  df-covers 38439  df-ats 38440  df-atl 38471  df-cvlat 38495  df-hlat 38524  df-lhyp 39162
This theorem is referenced by:  lhpocnel  39192
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