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Theorem atlelt 33742
 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 1083 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄 < 𝑋)
2 breq1 4586 . . 3 (𝑃 = 𝑄 → (𝑃 < 𝑋𝑄 < 𝑋))
31, 2syl5ibrcom 236 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 = 𝑄𝑃 < 𝑋))
4 simp1 1054 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ HL)
5 simp21 1087 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃𝐴)
6 simp22 1088 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄𝐴)
7 atlelt.s . . . . 5 < = (lt‘𝐾)
8 eqid 2610 . . . . 5 (join‘𝐾) = (join‘𝐾)
9 atlelt.a . . . . 5 𝐴 = (Atoms‘𝐾)
107, 8, 9atlt 33741 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃𝑄))
114, 5, 6, 10syl3anc 1318 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃𝑄))
12 simp3l 1082 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 𝑋)
13 simp23 1089 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑋𝐵)
144, 6, 133jca 1235 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵))
15 atlelt.l . . . . . . 7 = (le‘𝐾)
1615, 7pltle 16784 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵) → (𝑄 < 𝑋𝑄 𝑋))
1714, 1, 16sylc 63 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄 𝑋)
18 hllat 33668 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
19183ad2ant1 1075 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ Lat)
20 atlelt.b . . . . . . . 8 𝐵 = (Base‘𝐾)
2120, 9atbase 33594 . . . . . . 7 (𝑃𝐴𝑃𝐵)
225, 21syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃𝐵)
2320, 9atbase 33594 . . . . . . 7 (𝑄𝐴𝑄𝐵)
246, 23syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄𝐵)
2520, 15, 8latjle12 16885 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃(join‘𝐾)𝑄) 𝑋))
2619, 22, 24, 13, 25syl13anc 1320 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃(join‘𝐾)𝑄) 𝑋))
2712, 17, 26mpbi2and 958 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) 𝑋)
28 hlpos 33670 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Poset)
29283ad2ant1 1075 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ Poset)
3020, 8latjcl 16874 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
3119, 22, 24, 30syl3anc 1318 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
3220, 15, 7pltletr 16794 . . . . 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 1320 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → ((𝑃 < (𝑃(join‘𝐾)𝑄) ∧ (𝑃(join‘𝐾)𝑄) 𝑋) → 𝑃 < 𝑋))
3427, 33mpan2d 706 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) → 𝑃 < 𝑋))
3511, 34sylbird 249 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃𝑄𝑃 < 𝑋))
363, 35pm2.61dne 2868 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 < 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  Posetcpo 16763  ltcplt 16764  joincjn 16767  Latclat 16868  Atomscatm 33568  HLchlt 33655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656 This theorem is referenced by:  1cvratlt  33778
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