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Theorem 2atlt 33543
Description: Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)
Hypotheses
Ref Expression
2atomslt.b 𝐵 = (Base‘𝐾)
2atomslt.s < = (lt‘𝐾)
2atomslt.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
2atlt (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞𝐴 (𝑞𝑃𝑞 < 𝑋))
Distinct variable groups:   𝐴,𝑞   𝐵,𝑞   𝐾,𝑞   𝑃,𝑞   < ,𝑞   𝑋,𝑞

Proof of Theorem 2atlt
StepHypRef Expression
1 2atomslt.b . . . 4 𝐵 = (Base‘𝐾)
2 2atomslt.a . . . 4 𝐴 = (Atoms‘𝐾)
31, 2atbase 33394 . . 3 (𝑃𝐴𝑃𝐵)
4 eqid 2606 . . . 4 (le‘𝐾) = (le‘𝐾)
5 2atomslt.s . . . 4 < = (lt‘𝐾)
6 eqid 2606 . . . 4 (join‘𝐾) = (join‘𝐾)
71, 4, 5, 6, 2hlrelat 33506 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐵𝑋𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞𝐴 (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋))
83, 7syl3anl2 1366 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞𝐴 (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋))
9 simp3l 1081 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃 < (𝑃(join‘𝐾)𝑞))
10 simp1l1 1146 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝐾 ∈ HL)
11 simp1l2 1147 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃𝐴)
12 simp2 1054 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞𝐴)
13 eqid 2606 . . . . . . . . . 10 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
145, 6, 2, 13atltcvr 33539 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑃𝐴𝑞𝐴)) → (𝑃 < (𝑃(join‘𝐾)𝑞) ↔ 𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
1510, 11, 11, 12, 14syl13anc 1319 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑞) ↔ 𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
169, 15mpbid 220 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞))
176, 13, 2atcvr1 33521 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑞𝐴) → (𝑃𝑞𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
1810, 11, 12, 17syl3anc 1317 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑃𝑞𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
1916, 18mpbird 245 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃𝑞)
2019necomd 2833 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞𝑃)
215, 6, 2atlt 33541 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑞𝐴𝑃𝐴) → (𝑞 < (𝑞(join‘𝐾)𝑃) ↔ 𝑞𝑃))
2210, 12, 11, 21syl3anc 1317 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑞 < (𝑞(join‘𝐾)𝑃) ↔ 𝑞𝑃))
2320, 22mpbird 245 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 < (𝑞(join‘𝐾)𝑃))
24 hllat 33468 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2510, 24syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝐾 ∈ Lat)
2611, 3syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃𝐵)
271, 2atbase 33394 . . . . . . . . 9 (𝑞𝐴𝑞𝐵)
28273ad2ant2 1075 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞𝐵)
291, 6latjcom 16825 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑞𝐵) → (𝑃(join‘𝐾)𝑞) = (𝑞(join‘𝐾)𝑃))
3025, 26, 28, 29syl3anc 1317 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑃(join‘𝐾)𝑞) = (𝑞(join‘𝐾)𝑃))
3123, 30breqtrrd 4602 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 < (𝑃(join‘𝐾)𝑞))
32 simp3r 1082 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)
33 hlpos 33470 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3410, 33syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝐾 ∈ Poset)
351, 6latjcl 16817 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑞𝐵) → (𝑃(join‘𝐾)𝑞) ∈ 𝐵)
3625, 26, 28, 35syl3anc 1317 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑃(join‘𝐾)𝑞) ∈ 𝐵)
37 simp1l3 1148 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑋𝐵)
381, 4, 5pltletr 16737 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑞𝐵 ∧ (𝑃(join‘𝐾)𝑞) ∈ 𝐵𝑋𝐵)) → ((𝑞 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑞 < 𝑋))
3934, 28, 36, 37, 38syl13anc 1319 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → ((𝑞 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑞 < 𝑋))
4031, 32, 39mp2and 710 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 < 𝑋)
4120, 40jca 552 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑞𝑃𝑞 < 𝑋))
42413exp 1255 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → (𝑞𝐴 → ((𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → (𝑞𝑃𝑞 < 𝑋))))
4342reximdvai 2994 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → (∃𝑞𝐴 (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → ∃𝑞𝐴 (𝑞𝑃𝑞 < 𝑋)))
448, 43mpd 15 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞𝐴 (𝑞𝑃𝑞 < 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2776  wrex 2893   class class class wbr 4574  cfv 5787  (class class class)co 6524  Basecbs 15638  lecple 15718  Posetcpo 16706  ltcplt 16707  joincjn 16710  Latclat 16811  ccvr 33367  Atomscatm 33368  HLchlt 33455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-preset 16694  df-poset 16712  df-plt 16724  df-lub 16740  df-glb 16741  df-join 16742  df-meet 16743  df-p0 16805  df-lat 16812  df-clat 16874  df-oposet 33281  df-ol 33283  df-oml 33284  df-covers 33371  df-ats 33372  df-atl 33403  df-cvlat 33427  df-hlat 33456
This theorem is referenced by:  cdlemb  33898  lhpexle1  34112
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