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Theorem atlelt 39421
Description: Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)
Hypotheses
Ref Expression
atlelt.b 𝐵 = (Base‘𝐾)
atlelt.l = (le‘𝐾)
atlelt.s < = (lt‘𝐾)
atlelt.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atlelt ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 < 𝑋)

Proof of Theorem atlelt
StepHypRef Expression
1 simp3r 1201 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄 < 𝑋)
2 breq1 5151 . . 3 (𝑃 = 𝑄 → (𝑃 < 𝑋𝑄 < 𝑋))
31, 2syl5ibrcom 247 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 = 𝑄𝑃 < 𝑋))
4 simp1 1135 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ HL)
5 simp21 1205 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃𝐴)
6 simp22 1206 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄𝐴)
7 atlelt.s . . . . 5 < = (lt‘𝐾)
8 eqid 2735 . . . . 5 (join‘𝐾) = (join‘𝐾)
9 atlelt.a . . . . 5 𝐴 = (Atoms‘𝐾)
107, 8, 9atlt 39420 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃𝑄))
114, 5, 6, 10syl3anc 1370 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃𝑄))
12 simp3l 1200 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 𝑋)
13 simp23 1207 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑋𝐵)
144, 6, 133jca 1127 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵))
15 atlelt.l . . . . . . 7 = (le‘𝐾)
1615, 7pltle 18391 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵) → (𝑄 < 𝑋𝑄 𝑋))
1714, 1, 16sylc 65 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄 𝑋)
18 hllat 39345 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
19183ad2ant1 1132 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ Lat)
20 atlelt.b . . . . . . . 8 𝐵 = (Base‘𝐾)
2120, 9atbase 39271 . . . . . . 7 (𝑃𝐴𝑃𝐵)
225, 21syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃𝐵)
2320, 9atbase 39271 . . . . . . 7 (𝑄𝐴𝑄𝐵)
246, 23syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄𝐵)
2520, 15, 8latjle12 18508 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃(join‘𝐾)𝑄) 𝑋))
2619, 22, 24, 13, 25syl13anc 1371 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃(join‘𝐾)𝑄) 𝑋))
2712, 17, 26mpbi2and 712 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) 𝑋)
28 hlpos 39348 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Poset)
29283ad2ant1 1132 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ Poset)
3020, 8latjcl 18497 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
3119, 22, 24, 30syl3anc 1370 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
3220, 15, 7pltletr 18401 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑃𝐵 ∧ (𝑃(join‘𝐾)𝑄) ∈ 𝐵𝑋𝐵)) → ((𝑃 < (𝑃(join‘𝐾)𝑄) ∧ (𝑃(join‘𝐾)𝑄) 𝑋) → 𝑃 < 𝑋))
3329, 22, 31, 13, 32syl13anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → ((𝑃 < (𝑃(join‘𝐾)𝑄) ∧ (𝑃(join‘𝐾)𝑄) 𝑋) → 𝑃 < 𝑋))
3427, 33mpan2d 694 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) → 𝑃 < 𝑋))
3511, 34sylbird 260 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃𝑄𝑃 < 𝑋))
363, 35pm2.61dne 3026 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 < 𝑋)
Colors of variables: wff setvar class
Syntax hints:  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  Posetcpo 18365  ltcplt 18366  joincjn 18369  Latclat 18489  Atomscatm 39245  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-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333
This theorem is referenced by:  1cvratlt  39457
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