Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrat Structured version   Visualization version   GIF version

Theorem cvrat 36438
Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 30090 analog.) (Contributed by NM, 22-Nov-2011.)
Hypotheses
Ref Expression
cvrat.b 𝐵 = (Base‘𝐾)
cvrat.s < = (lt‘𝐾)
cvrat.j = (join‘𝐾)
cvrat.z 0 = (0.‘𝐾)
cvrat.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cvrat ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋0𝑋 < (𝑃 𝑄)) → 𝑋𝐴))

Proof of Theorem cvrat
StepHypRef Expression
1 cvrat.b . . . 4 𝐵 = (Base‘𝐾)
2 cvrat.s . . . 4 < = (lt‘𝐾)
3 cvrat.j . . . 4 = (join‘𝐾)
4 cvrat.z . . . 4 0 = (0.‘𝐾)
5 cvrat.a . . . 4 𝐴 = (Atoms‘𝐾)
61, 2, 3, 4, 5cvratlem 36437 . . 3 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑋0𝑋 < (𝑃 𝑄))) → (¬ 𝑃(le‘𝐾)𝑋𝑋𝐴))
7 hllat 36379 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Lat)
87adantr 481 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ Lat)
9 simpr2 1187 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐴)
101, 5atbase 36305 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
119, 10syl 17 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐵)
12 simpr3 1188 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
131, 5atbase 36305 . . . . . . . . 9 (𝑄𝐴𝑄𝐵)
1412, 13syl 17 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐵)
151, 3latjcom 17657 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
168, 11, 14, 15syl3anc 1363 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
1716breq2d 5069 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 < (𝑃 𝑄) ↔ 𝑋 < (𝑄 𝑃)))
1817anbi2d 628 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋0𝑋 < (𝑃 𝑄)) ↔ (𝑋0𝑋 < (𝑄 𝑃))))
19 simpl 483 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ HL)
20 simpr1 1186 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑋𝐵)
211, 2, 3, 4, 5cvratlem 36437 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑄𝐴𝑃𝐴)) ∧ (𝑋0𝑋 < (𝑄 𝑃))) → (¬ 𝑄(le‘𝐾)𝑋𝑋𝐴))
2221ex 413 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑄𝐴𝑃𝐴)) → ((𝑋0𝑋 < (𝑄 𝑃)) → (¬ 𝑄(le‘𝐾)𝑋𝑋𝐴)))
2319, 20, 12, 9, 22syl13anc 1364 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋0𝑋 < (𝑄 𝑃)) → (¬ 𝑄(le‘𝐾)𝑋𝑋𝐴)))
2418, 23sylbid 241 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋0𝑋 < (𝑃 𝑄)) → (¬ 𝑄(le‘𝐾)𝑋𝑋𝐴)))
2524imp 407 . . 3 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑋0𝑋 < (𝑃 𝑄))) → (¬ 𝑄(le‘𝐾)𝑋𝑋𝐴))
26 hlpos 36382 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Poset)
2726adantr 481 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ Poset)
281, 3latjcl 17649 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
298, 11, 14, 28syl3anc 1363 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) ∈ 𝐵)
30 eqid 2818 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
311, 30, 2pltnle 17564 . . . . . . . . 9 (((𝐾 ∈ Poset ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) ∧ 𝑋 < (𝑃 𝑄)) → ¬ (𝑃 𝑄)(le‘𝐾)𝑋)
3231ex 413 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 < (𝑃 𝑄) → ¬ (𝑃 𝑄)(le‘𝐾)𝑋))
3327, 20, 29, 32syl3anc 1363 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 < (𝑃 𝑄) → ¬ (𝑃 𝑄)(le‘𝐾)𝑋))
341, 30, 3latjle12 17660 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃(le‘𝐾)𝑋𝑄(le‘𝐾)𝑋) ↔ (𝑃 𝑄)(le‘𝐾)𝑋))
358, 11, 14, 20, 34syl13anc 1364 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃(le‘𝐾)𝑋𝑄(le‘𝐾)𝑋) ↔ (𝑃 𝑄)(le‘𝐾)𝑋))
3635biimpd 230 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃(le‘𝐾)𝑋𝑄(le‘𝐾)𝑋) → (𝑃 𝑄)(le‘𝐾)𝑋))
3733, 36nsyld 159 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 < (𝑃 𝑄) → ¬ (𝑃(le‘𝐾)𝑋𝑄(le‘𝐾)𝑋)))
38 ianor 975 . . . . . 6 (¬ (𝑃(le‘𝐾)𝑋𝑄(le‘𝐾)𝑋) ↔ (¬ 𝑃(le‘𝐾)𝑋 ∨ ¬ 𝑄(le‘𝐾)𝑋))
3937, 38syl6ib 252 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 < (𝑃 𝑄) → (¬ 𝑃(le‘𝐾)𝑋 ∨ ¬ 𝑄(le‘𝐾)𝑋)))
4039imp 407 . . . 4 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑋 < (𝑃 𝑄)) → (¬ 𝑃(le‘𝐾)𝑋 ∨ ¬ 𝑄(le‘𝐾)𝑋))
4140adantrl 712 . . 3 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑋0𝑋 < (𝑃 𝑄))) → (¬ 𝑃(le‘𝐾)𝑋 ∨ ¬ 𝑄(le‘𝐾)𝑋))
426, 25, 41mpjaod 854 . 2 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑋0𝑋 < (𝑃 𝑄))) → 𝑋𝐴)
4342ex 413 1 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋0𝑋 < (𝑃 𝑄)) → 𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  Posetcpo 17538  ltcplt 17539  joincjn 17542  0.cp0 17635  Latclat 17643  Atomscatm 36279  HLchlt 36366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  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 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367
This theorem is referenced by:  cvrat2  36445
  Copyright terms: Public domain W3C validator