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Theorem atlelt 38614
Description: Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)
Hypotheses
Ref Expression
atlelt.b 𝐡 = (Baseβ€˜πΎ)
atlelt.l ≀ = (leβ€˜πΎ)
atlelt.s < = (ltβ€˜πΎ)
atlelt.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
atlelt ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝑃 < 𝑋)
Proof of Theorem atlelt
StepHypRef Expression
1 simp3r 1200 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝑄 < 𝑋)
2 breq1 5152 . . 3 (𝑃 = 𝑄 β†’ (𝑃 < 𝑋 ↔ 𝑄 < 𝑋))
31, 2syl5ibrcom 246 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ (𝑃 = 𝑄 β†’ 𝑃 < 𝑋))
4 simp1 1134 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝐾 ∈ HL)
5 simp21 1204 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝑃 ∈ 𝐴)
6 simp22 1205 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝑄 ∈ 𝐴)
7 atlelt.s . . . . 5 < = (ltβ€˜πΎ)
8 eqid 2730 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
9 atlelt.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
107, 8, 9atlt 38613 . . . 4 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 < (𝑃(joinβ€˜πΎ)𝑄) ↔ 𝑃 β‰  𝑄))
114, 5, 6, 10syl3anc 1369 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ (𝑃 < (𝑃(joinβ€˜πΎ)𝑄) ↔ 𝑃 β‰  𝑄))
12 simp3l 1199 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝑃 ≀ 𝑋)
13 simp23 1206 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝑋 ∈ 𝐡)
144, 6, 133jca 1126 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ (𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡))
15 atlelt.l . . . . . . 7 ≀ = (leβ€˜πΎ)
1615, 7pltle 18292 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (𝑄 < 𝑋 β†’ 𝑄 ≀ 𝑋))
1714, 1, 16sylc 65 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝑄 ≀ 𝑋)
18 hllat 38538 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
19183ad2ant1 1131 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝐾 ∈ Lat)
20 atlelt.b . . . . . . . 8 𝐡 = (Baseβ€˜πΎ)
2120, 9atbase 38464 . . . . . . 7 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
225, 21syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝑃 ∈ 𝐡)
2320, 9atbase 38464 . . . . . . 7 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
246, 23syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝑄 ∈ 𝐡)
2520, 15, 8latjle12 18409 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋) ↔ (𝑃(joinβ€˜πΎ)𝑄) ≀ 𝑋))
2619, 22, 24, 13, 25syl13anc 1370 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ ((𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋) ↔ (𝑃(joinβ€˜πΎ)𝑄) ≀ 𝑋))
2712, 17, 26mpbi2and 708 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ (𝑃(joinβ€˜πΎ)𝑄) ≀ 𝑋)
28 hlpos 38541 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
29283ad2ant1 1131 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝐾 ∈ Poset)
3020, 8latjcl 18398 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃(joinβ€˜πΎ)𝑄) ∈ 𝐡)
3119, 22, 24, 30syl3anc 1369 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ (𝑃(joinβ€˜πΎ)𝑄) ∈ 𝐡)
3220, 15, 7pltletr 18302 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑃 ∈ 𝐡 ∧ (𝑃(joinβ€˜πΎ)𝑄) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑃 < (𝑃(joinβ€˜πΎ)𝑄) ∧ (𝑃(joinβ€˜πΎ)𝑄) ≀ 𝑋) β†’ 𝑃 < 𝑋))
3329, 22, 31, 13, 32syl13anc 1370 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ ((𝑃 < (𝑃(joinβ€˜πΎ)𝑄) ∧ (𝑃(joinβ€˜πΎ)𝑄) ≀ 𝑋) β†’ 𝑃 < 𝑋))
3427, 33mpan2d 690 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ (𝑃 < (𝑃(joinβ€˜πΎ)𝑄) β†’ 𝑃 < 𝑋))
3511, 34sylbird 259 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ (𝑃 β‰  𝑄 β†’ 𝑃 < 𝑋))
363, 35pm2.61dne 3026 1 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝑃 < 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7413  Basecbs 17150  lecple 17210  Posetcpo 18266  ltcplt 18267  joincjn 18270  Latclat 18390  Atomscatm 38438  HLchlt 38525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-proset 18254  df-poset 18272  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-lat 18391  df-clat 18458  df-oposet 38351  df-ol 38353  df-oml 38354  df-covers 38441  df-ats 38442  df-atl 38473  df-cvlat 38497  df-hlat 38526
This theorem is referenced by:  1cvratlt  38650
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