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Theorem atlelt 35394
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 1259 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄 < 𝑋)
2 breq1 4812 . . 3 (𝑃 = 𝑄 → (𝑃 < 𝑋𝑄 < 𝑋))
31, 2syl5ibrcom 238 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 = 𝑄𝑃 < 𝑋))
4 simp1 1166 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ HL)
5 simp21 1263 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃𝐴)
6 simp22 1264 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄𝐴)
7 atlelt.s . . . . 5 < = (lt‘𝐾)
8 eqid 2765 . . . . 5 (join‘𝐾) = (join‘𝐾)
9 atlelt.a . . . . 5 𝐴 = (Atoms‘𝐾)
107, 8, 9atlt 35393 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃𝑄))
114, 5, 6, 10syl3anc 1490 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃𝑄))
12 simp3l 1258 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 𝑋)
13 simp23 1265 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑋𝐵)
144, 6, 133jca 1158 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵))
15 atlelt.l . . . . . . 7 = (le‘𝐾)
1615, 7pltle 17227 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵) → (𝑄 < 𝑋𝑄 𝑋))
1714, 1, 16sylc 65 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄 𝑋)
18 hllat 35319 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
19183ad2ant1 1163 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ Lat)
20 atlelt.b . . . . . . . 8 𝐵 = (Base‘𝐾)
2120, 9atbase 35245 . . . . . . 7 (𝑃𝐴𝑃𝐵)
225, 21syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃𝐵)
2320, 9atbase 35245 . . . . . . 7 (𝑄𝐴𝑄𝐵)
246, 23syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄𝐵)
2520, 15, 8latjle12 17328 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃(join‘𝐾)𝑄) 𝑋))
2619, 22, 24, 13, 25syl13anc 1491 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃(join‘𝐾)𝑄) 𝑋))
2712, 17, 26mpbi2and 703 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) 𝑋)
28 hlpos 35322 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Poset)
29283ad2ant1 1163 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ Poset)
3020, 8latjcl 17317 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
3119, 22, 24, 30syl3anc 1490 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
3220, 15, 7pltletr 17237 . . . . 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 1491 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → ((𝑃 < (𝑃(join‘𝐾)𝑄) ∧ (𝑃(join‘𝐾)𝑄) 𝑋) → 𝑃 < 𝑋))
3427, 33mpan2d 685 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) → 𝑃 < 𝑋))
3511, 34sylbird 251 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃𝑄𝑃 < 𝑋))
363, 35pm2.61dne 3023 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 < 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16130  lecple 16221  Posetcpo 17206  ltcplt 17207  joincjn 17210  Latclat 17311  Atomscatm 35219  HLchlt 35306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-proset 17194  df-poset 17212  df-plt 17224  df-lub 17240  df-glb 17241  df-join 17242  df-meet 17243  df-p0 17305  df-lat 17312  df-clat 17374  df-oposet 35132  df-ol 35134  df-oml 35135  df-covers 35222  df-ats 35223  df-atl 35254  df-cvlat 35278  df-hlat 35307
This theorem is referenced by:  1cvratlt  35430
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