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Theorem intnatN 39390
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 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (¬ 𝑌 𝑋 ∧ (𝑋 𝑌) ∈ 𝐴)) → ¬ 𝑌𝐴)

Proof of Theorem intnatN
StepHypRef Expression
1 hlatl 39342 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
213ad2ant1 1132 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ AtLat)
32ad2antrr 726 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ AtLat)
4 eqid 2735 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
5 intnat.a . . . . . 6 𝐴 = (Atoms‘𝐾)
64, 5atn0 39290 . . . . 5 ((𝐾 ∈ AtLat ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 𝑌) ≠ (0.‘𝐾))
73, 6sylancom 588 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 𝑌) ≠ (0.‘𝐾))
87ex 412 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → ((𝑋 𝑌) ∈ 𝐴 → (𝑋 𝑌) ≠ (0.‘𝐾)))
9 simpll1 1211 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ HL)
109hllatd 39346 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ Lat)
11 simpll2 1212 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑋𝐵)
12 simpll3 1213 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑌𝐵)
13 intnat.b . . . . . . . 8 𝐵 = (Base‘𝐾)
14 intnat.m . . . . . . . 8 = (meet‘𝐾)
1513, 14latmcom 18521 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
1610, 11, 12, 15syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑋 𝑌) = (𝑌 𝑋))
17 simplr 769 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → ¬ 𝑌 𝑋)
189, 1syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ AtLat)
19 simpr 484 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑌𝐴)
20 intnat.l . . . . . . . . 9 = (le‘𝐾)
2113, 20, 14, 4, 5atnle 39299 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑌𝐴𝑋𝐵) → (¬ 𝑌 𝑋 ↔ (𝑌 𝑋) = (0.‘𝐾)))
2218, 19, 11, 21syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (¬ 𝑌 𝑋 ↔ (𝑌 𝑋) = (0.‘𝐾)))
2317, 22mpbid 232 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑌 𝑋) = (0.‘𝐾))
2416, 23eqtrd 2775 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑋 𝑌) = (0.‘𝐾))
2524ex 412 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → (𝑌𝐴 → (𝑋 𝑌) = (0.‘𝐾)))
2625necon3ad 2951 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → ((𝑋 𝑌) ≠ (0.‘𝐾) → ¬ 𝑌𝐴))
278, 26syld 47 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → ((𝑋 𝑌) ∈ 𝐴 → ¬ 𝑌𝐴))
2827impr 454 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (¬ 𝑌 𝑋 ∧ (𝑋 𝑌) ∈ 𝐴)) → ¬ 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  meetcmee 18370  0.cp0 18481  Latclat 18489  Atomscatm 39245  AtLatcal 39246  HLchlt 39332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333
This theorem is referenced by: (None)
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