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Theorem atlelt 35223
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 1245 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄 < 𝑋)
2 breq1 4803 . . 3 (𝑃 = 𝑄 → (𝑃 < 𝑋𝑄 < 𝑋))
31, 2syl5ibrcom 237 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 = 𝑄𝑃 < 𝑋))
4 simp1 1131 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ HL)
5 simp21 1249 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃𝐴)
6 simp22 1250 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄𝐴)
7 atlelt.s . . . . 5 < = (lt‘𝐾)
8 eqid 2756 . . . . 5 (join‘𝐾) = (join‘𝐾)
9 atlelt.a . . . . 5 𝐴 = (Atoms‘𝐾)
107, 8, 9atlt 35222 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃𝑄))
114, 5, 6, 10syl3anc 1477 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃𝑄))
12 simp3l 1244 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 𝑋)
13 simp23 1251 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑋𝐵)
144, 6, 133jca 1123 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵))
15 atlelt.l . . . . . . 7 = (le‘𝐾)
1615, 7pltle 17158 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵) → (𝑄 < 𝑋𝑄 𝑋))
1714, 1, 16sylc 65 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄 𝑋)
18 hllat 35149 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
19183ad2ant1 1128 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ Lat)
20 atlelt.b . . . . . . . 8 𝐵 = (Base‘𝐾)
2120, 9atbase 35075 . . . . . . 7 (𝑃𝐴𝑃𝐵)
225, 21syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃𝐵)
2320, 9atbase 35075 . . . . . . 7 (𝑄𝐴𝑄𝐵)
246, 23syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑄𝐵)
2520, 15, 8latjle12 17259 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃(join‘𝐾)𝑄) 𝑋))
2619, 22, 24, 13, 25syl13anc 1479 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃(join‘𝐾)𝑄) 𝑋))
2712, 17, 26mpbi2and 994 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) 𝑋)
28 hlpos 35151 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Poset)
29283ad2ant1 1128 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝐾 ∈ Poset)
3020, 8latjcl 17248 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
3119, 22, 24, 30syl3anc 1477 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
3220, 15, 7pltletr 17168 . . . . 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 1479 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → ((𝑃 < (𝑃(join‘𝐾)𝑄) ∧ (𝑃(join‘𝐾)𝑄) 𝑋) → 𝑃 < 𝑋))
3427, 33mpan2d 712 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) → 𝑃 < 𝑋))
3511, 34sylbird 250 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → (𝑃𝑄𝑃 < 𝑋))
363, 35pm2.61dne 3014 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋𝑄 < 𝑋)) → 𝑃 < 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1628  wcel 2135  wne 2928   class class class wbr 4800  cfv 6045  (class class class)co 6809  Basecbs 16055  lecple 16146  Posetcpo 17137  ltcplt 17138  joincjn 17141  Latclat 17242  Atomscatm 35049  HLchlt 35136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-preset 17125  df-poset 17143  df-plt 17155  df-lub 17171  df-glb 17172  df-join 17173  df-meet 17174  df-p0 17236  df-lat 17243  df-clat 17305  df-oposet 34962  df-ol 34964  df-oml 34965  df-covers 35052  df-ats 35053  df-atl 35084  df-cvlat 35108  df-hlat 35137
This theorem is referenced by:  1cvratlt  35259
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