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Theorem atlelt 36460
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 1196 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄 < 𝑋)
2 breq1 5066 . . 3 (𝑃 = 𝑄 → (𝑃 < 𝑋𝑄 < 𝑋))
31, 2syl5ibrcom 248 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 = 𝑄𝑃 < 𝑋))
4 simp1 1130 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ HL)
5 simp21 1200 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃𝐴)
6 simp22 1201 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄𝐴)
7 atlelt.s . . . . 5 < = (lt‘𝐾)
8 eqid 2826 . . . . 5 (join‘𝐾) = (join‘𝐾)
9 atlelt.a . . . . 5 𝐴 = (Atoms‘𝐾)
107, 8, 9atlt 36459 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃𝑄))
114, 5, 6, 10syl3anc 1365 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃𝑄))
12 simp3l 1195 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 𝑋)
13 simp23 1202 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑋𝐵)
144, 6, 133jca 1122 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵))
15 atlelt.l . . . . . . 7 = (le‘𝐾)
1615, 7pltle 17566 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵) → (𝑄 < 𝑋𝑄 𝑋))
1714, 1, 16sylc 65 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄 𝑋)
18 hllat 36385 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
19183ad2ant1 1127 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ Lat)
20 atlelt.b . . . . . . . 8 𝐵 = (Base‘𝐾)
2120, 9atbase 36311 . . . . . . 7 (𝑃𝐴𝑃𝐵)
225, 21syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃𝐵)
2320, 9atbase 36311 . . . . . . 7 (𝑄𝐴𝑄𝐵)
246, 23syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄𝐵)
2520, 15, 8latjle12 17667 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃(join‘𝐾)𝑄) 𝑋))
2619, 22, 24, 13, 25syl13anc 1366 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃(join‘𝐾)𝑄) 𝑋))
2712, 17, 26mpbi2and 708 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) 𝑋)
28 hlpos 36388 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Poset)
29283ad2ant1 1127 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ Poset)
3020, 8latjcl 17656 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
3119, 22, 24, 30syl3anc 1365 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
3220, 15, 7pltletr 17576 . . . . 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 1366 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → ((𝑃 < (𝑃(join‘𝐾)𝑄) ∧ (𝑃(join‘𝐾)𝑄) 𝑋) → 𝑃 < 𝑋))
3427, 33mpan2d 690 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) → 𝑃 < 𝑋))
3511, 34sylbird 261 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃𝑄𝑃 < 𝑋))
363, 35pm2.61dne 3108 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 < 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021   class class class wbr 5063  cfv 6354  (class class class)co 7150  Basecbs 16478  lecple 16567  Posetcpo 17545  ltcplt 17546  joincjn 17549  Latclat 17650  Atomscatm 36285  HLchlt 36372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-lat 17651  df-clat 17713  df-oposet 36198  df-ol 36200  df-oml 36201  df-covers 36288  df-ats 36289  df-atl 36320  df-cvlat 36344  df-hlat 36373
This theorem is referenced by:  1cvratlt  36496
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