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Theorem 2atlt 35513
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 35363 . . 3 (𝑃𝐴𝑃𝐵)
4 eqid 2824 . . . 4 (le‘𝐾) = (le‘𝐾)
5 2atomslt.s . . . 4 < = (lt‘𝐾)
6 eqid 2824 . . . 4 (join‘𝐾) = (join‘𝐾)
71, 4, 5, 6, 2hlrelat 35476 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐵𝑋𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞𝐴 (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋))
83, 7syl3anl2 1540 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞𝐴 (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋))
9 simp3l 1264 . . . . . . . 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 1371 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝐾 ∈ HL)
11 simp1l2 1372 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃𝐴)
12 simp2 1173 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞𝐴)
13 eqid 2824 . . . . . . . . . 10 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
145, 6, 2, 13atltcvr 35509 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑃𝐴𝑞𝐴)) → (𝑃 < (𝑃(join‘𝐾)𝑞) ↔ 𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
1510, 11, 11, 12, 14syl13anc 1497 . . . . . . . 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 224 . . . . . . 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 35491 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑞𝐴) → (𝑃𝑞𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
1810, 11, 12, 17syl3anc 1496 . . . . . . 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 249 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃𝑞)
2019necomd 3053 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞𝑃)
215, 6, 2atlt 35511 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑞𝐴𝑃𝐴) → (𝑞 < (𝑞(join‘𝐾)𝑃) ↔ 𝑞𝑃))
2210, 12, 11, 21syl3anc 1496 . . . . . . . 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 249 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 < (𝑞(join‘𝐾)𝑃))
2410hllatd 35438 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝐾 ∈ Lat)
2511, 3syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃𝐵)
261, 2atbase 35363 . . . . . . . . 9 (𝑞𝐴𝑞𝐵)
27263ad2ant2 1170 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞𝐵)
281, 6latjcom 17411 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑞𝐵) → (𝑃(join‘𝐾)𝑞) = (𝑞(join‘𝐾)𝑃))
2924, 25, 27, 28syl3anc 1496 . . . . . . 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))
3023, 29breqtrrd 4900 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 < (𝑃(join‘𝐾)𝑞))
31 simp3r 1265 . . . . . 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)
32 hlpos 35440 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3310, 32syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝐾 ∈ Poset)
341, 6latjcl 17403 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑞𝐵) → (𝑃(join‘𝐾)𝑞) ∈ 𝐵)
3524, 25, 27, 34syl3anc 1496 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑃(join‘𝐾)𝑞) ∈ 𝐵)
36 simp1l3 1373 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑋𝐵)
371, 4, 5pltletr 17323 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑞𝐵 ∧ (𝑃(join‘𝐾)𝑞) ∈ 𝐵𝑋𝐵)) → ((𝑞 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑞 < 𝑋))
3833, 27, 35, 36, 37syl13anc 1497 . . . . . 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))
3930, 31, 38mp2and 692 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 < 𝑋)
4020, 39jca 509 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑞𝑃𝑞 < 𝑋))
41403exp 1154 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → (𝑞𝐴 → ((𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → (𝑞𝑃𝑞 < 𝑋))))
4241reximdvai 3222 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → (∃𝑞𝐴 (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → ∃𝑞𝐴 (𝑞𝑃𝑞 < 𝑋)))
438, 42mpd 15 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞𝐴 (𝑞𝑃𝑞 < 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wne 2998  wrex 3117   class class class wbr 4872  cfv 6122  (class class class)co 6904  Basecbs 16221  lecple 16311  Posetcpo 17292  ltcplt 17293  joincjn 17296  Latclat 17397  ccvr 35336  Atomscatm 35337  HLchlt 35424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-oprab 6908  df-proset 17280  df-poset 17298  df-plt 17310  df-lub 17326  df-glb 17327  df-join 17328  df-meet 17329  df-p0 17391  df-lat 17398  df-clat 17460  df-oposet 35250  df-ol 35252  df-oml 35253  df-covers 35340  df-ats 35341  df-atl 35372  df-cvlat 35396  df-hlat 35425
This theorem is referenced by:  cdlemb  35868  lhpexle1  36082
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