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Theorem intnatN 36545
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 36498 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
213ad2ant1 1129 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ AtLat)
32ad2antrr 724 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ AtLat)
4 eqid 2823 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
5 intnat.a . . . . . 6 𝐴 = (Atoms‘𝐾)
64, 5atn0 36446 . . . . 5 ((𝐾 ∈ AtLat ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 𝑌) ≠ (0.‘𝐾))
73, 6sylancom 590 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 𝑌) ≠ (0.‘𝐾))
87ex 415 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → ((𝑋 𝑌) ∈ 𝐴 → (𝑋 𝑌) ≠ (0.‘𝐾)))
9 simpll1 1208 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ HL)
109hllatd 36502 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ Lat)
11 simpll2 1209 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑋𝐵)
12 simpll3 1210 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑌𝐵)
13 intnat.b . . . . . . . 8 𝐵 = (Base‘𝐾)
14 intnat.m . . . . . . . 8 = (meet‘𝐾)
1513, 14latmcom 17687 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
1610, 11, 12, 15syl3anc 1367 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑋 𝑌) = (𝑌 𝑋))
17 simplr 767 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → ¬ 𝑌 𝑋)
189, 1syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ AtLat)
19 simpr 487 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑌𝐴)
20 intnat.l . . . . . . . . 9 = (le‘𝐾)
2113, 20, 14, 4, 5atnle 36455 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑌𝐴𝑋𝐵) → (¬ 𝑌 𝑋 ↔ (𝑌 𝑋) = (0.‘𝐾)))
2218, 19, 11, 21syl3anc 1367 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (¬ 𝑌 𝑋 ↔ (𝑌 𝑋) = (0.‘𝐾)))
2317, 22mpbid 234 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑌 𝑋) = (0.‘𝐾))
2416, 23eqtrd 2858 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑋 𝑌) = (0.‘𝐾))
2524ex 415 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → (𝑌𝐴 → (𝑋 𝑌) = (0.‘𝐾)))
2625necon3ad 3031 . . 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 457 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (¬ 𝑌 𝑋 ∧ (𝑋 𝑌) ∈ 𝐴)) → ¬ 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wne 3018   class class class wbr 5068  cfv 6357  (class class class)co 7158  Basecbs 16485  lecple 16574  meetcmee 17557  0.cp0 17649  Latclat 17657  Atomscatm 36401  AtLatcal 36402  HLchlt 36488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-proset 17540  df-poset 17558  df-plt 17570  df-lub 17586  df-glb 17587  df-join 17588  df-meet 17589  df-p0 17651  df-lat 17658  df-covers 36404  df-ats 36405  df-atl 36436  df-cvlat 36460  df-hlat 36489
This theorem is referenced by: (None)
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