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Theorem 2atlt 36562
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 36412 . . 3 (𝑃𝐴𝑃𝐵)
4 eqid 2819 . . . 4 (le‘𝐾) = (le‘𝐾)
5 2atomslt.s . . . 4 < = (lt‘𝐾)
6 eqid 2819 . . . 4 (join‘𝐾) = (join‘𝐾)
71, 4, 5, 6, 2hlrelat 36525 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐵𝑋𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞𝐴 (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋))
83, 7syl3anl2 1407 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞𝐴 (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋))
9 simp3l 1195 . . . . . . . 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 1260 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝐾 ∈ HL)
11 simp1l2 1261 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃𝐴)
12 simp2 1131 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞𝐴)
13 eqid 2819 . . . . . . . . . 10 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
145, 6, 2, 13atltcvr 36558 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑃𝐴𝑞𝐴)) → (𝑃 < (𝑃(join‘𝐾)𝑞) ↔ 𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
1510, 11, 11, 12, 14syl13anc 1366 . . . . . . . 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 234 . . . . . . 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 36540 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑞𝐴) → (𝑃𝑞𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
1810, 11, 12, 17syl3anc 1365 . . . . . . 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 259 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃𝑞)
2019necomd 3069 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞𝑃)
215, 6, 2atlt 36560 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑞𝐴𝑃𝐴) → (𝑞 < (𝑞(join‘𝐾)𝑃) ↔ 𝑞𝑃))
2210, 12, 11, 21syl3anc 1365 . . . . . . . 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 259 . . . . . . 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 36487 . . . . . . . 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 36412 . . . . . . . . 9 (𝑞𝐴𝑞𝐵)
27263ad2ant2 1128 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞𝐵)
281, 6latjcom 17661 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑞𝐵) → (𝑃(join‘𝐾)𝑞) = (𝑞(join‘𝐾)𝑃))
2924, 25, 27, 28syl3anc 1365 . . . . . . 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 5085 . . . . . 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 1196 . . . . . 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 36489 . . . . . . . 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 17653 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑞𝐵) → (𝑃(join‘𝐾)𝑞) ∈ 𝐵)
3524, 25, 27, 34syl3anc 1365 . . . . . . 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 1262 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑋𝐵)
371, 4, 5pltletr 17573 . . . . . . 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 1366 . . . . . 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 697 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 < 𝑋)
4020, 39jca 514 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑞𝑃𝑞 < 𝑋))
41403exp 1113 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑃 < 𝑋) → (𝑞𝐴 → ((𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → (𝑞𝑃𝑞 < 𝑋))))
4241reximdvai 3270 . 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 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wne 3014  wrex 3137   class class class wbr 5057  cfv 6348  (class class class)co 7148  Basecbs 16475  lecple 16564  Posetcpo 17542  ltcplt 17543  joincjn 17546  Latclat 17647  ccvr 36385  Atomscatm 36386  HLchlt 36473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36299  df-ol 36301  df-oml 36302  df-covers 36389  df-ats 36390  df-atl 36421  df-cvlat 36445  df-hlat 36474
This theorem is referenced by:  cdlemb  36917  lhpexle1  37131
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