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Theorem intnatN 35420
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 35373 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
213ad2ant1 1164 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ AtLat)
32ad2antrr 718 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ AtLat)
4 eqid 2797 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
5 intnat.a . . . . . 6 𝐴 = (Atoms‘𝐾)
64, 5atn0 35321 . . . . 5 ((𝐾 ∈ AtLat ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 𝑌) ≠ (0.‘𝐾))
73, 6sylancom 583 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 𝑌) ≠ (0.‘𝐾))
87ex 402 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → ((𝑋 𝑌) ∈ 𝐴 → (𝑋 𝑌) ≠ (0.‘𝐾)))
9 simpll1 1270 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ HL)
109hllatd 35377 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ Lat)
11 simpll2 1272 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑋𝐵)
12 simpll3 1274 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑌𝐵)
13 intnat.b . . . . . . . 8 𝐵 = (Base‘𝐾)
14 intnat.m . . . . . . . 8 = (meet‘𝐾)
1513, 14latmcom 17387 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
1610, 11, 12, 15syl3anc 1491 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑋 𝑌) = (𝑌 𝑋))
17 simplr 786 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → ¬ 𝑌 𝑋)
189, 1syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ AtLat)
19 simpr 478 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑌𝐴)
20 intnat.l . . . . . . . . 9 = (le‘𝐾)
2113, 20, 14, 4, 5atnle 35330 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑌𝐴𝑋𝐵) → (¬ 𝑌 𝑋 ↔ (𝑌 𝑋) = (0.‘𝐾)))
2218, 19, 11, 21syl3anc 1491 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (¬ 𝑌 𝑋 ↔ (𝑌 𝑋) = (0.‘𝐾)))
2317, 22mpbid 224 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑌 𝑋) = (0.‘𝐾))
2416, 23eqtrd 2831 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑋 𝑌) = (0.‘𝐾))
2524ex 402 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → (𝑌𝐴 → (𝑋 𝑌) = (0.‘𝐾)))
2625necon3ad 2982 . . 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 447 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (¬ 𝑌 𝑋 ∧ (𝑋 𝑌) ∈ 𝐴)) → ¬ 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2969   class class class wbr 4841  cfv 6099  (class class class)co 6876  Basecbs 16181  lecple 16271  meetcmee 17257  0.cp0 17349  Latclat 17357  Atomscatm 35276  AtLatcal 35277  HLchlt 35363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-proset 17240  df-poset 17258  df-plt 17270  df-lub 17286  df-glb 17287  df-join 17288  df-meet 17289  df-p0 17351  df-lat 17358  df-covers 35279  df-ats 35280  df-atl 35311  df-cvlat 35335  df-hlat 35364
This theorem is referenced by: (None)
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