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Theorem cvrat 38797
Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 32134 analog.) (Contributed by NM, 22-Nov-2011.)
Hypotheses
Ref Expression
cvrat.b 𝐡 = (Baseβ€˜πΎ)
cvrat.s < = (ltβ€˜πΎ)
cvrat.j ∨ = (joinβ€˜πΎ)
cvrat.z 0 = (0.β€˜πΎ)
cvrat.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
cvrat ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) β†’ 𝑋 ∈ 𝐴))
Proof of Theorem cvrat
StepHypRef Expression
1 cvrat.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 cvrat.s . . . 4 < = (ltβ€˜πΎ)
3 cvrat.j . . . 4 ∨ = (joinβ€˜πΎ)
4 cvrat.z . . . 4 0 = (0.β€˜πΎ)
5 cvrat.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
61, 2, 3, 4, 5cvratlem 38796 . . 3 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) β†’ (Β¬ 𝑃(leβ€˜πΎ)𝑋 β†’ 𝑋 ∈ 𝐴))
7 hllat 38737 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
87adantr 480 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
9 simpr2 1192 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
101, 5atbase 38663 . . . . . . . . 9 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
119, 10syl 17 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐡)
12 simpr3 1193 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
131, 5atbase 38663 . . . . . . . . 9 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1412, 13syl 17 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
151, 3latjcom 18408 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
168, 11, 14, 15syl3anc 1368 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
1716breq2d 5151 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 < (𝑃 ∨ 𝑄) ↔ 𝑋 < (𝑄 ∨ 𝑃)))
1817anbi2d 628 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) ↔ (𝑋 β‰  0 ∧ 𝑋 < (𝑄 ∨ 𝑃))))
19 simpl 482 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
20 simpr1 1191 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
211, 2, 3, 4, 5cvratlem 38796 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴)) ∧ (𝑋 β‰  0 ∧ 𝑋 < (𝑄 ∨ 𝑃))) β†’ (Β¬ 𝑄(leβ€˜πΎ)𝑋 β†’ 𝑋 ∈ 𝐴))
2221ex 412 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑋 < (𝑄 ∨ 𝑃)) β†’ (Β¬ 𝑄(leβ€˜πΎ)𝑋 β†’ 𝑋 ∈ 𝐴)))
2319, 20, 12, 9, 22syl13anc 1369 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑋 < (𝑄 ∨ 𝑃)) β†’ (Β¬ 𝑄(leβ€˜πΎ)𝑋 β†’ 𝑋 ∈ 𝐴)))
2418, 23sylbid 239 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) β†’ (Β¬ 𝑄(leβ€˜πΎ)𝑋 β†’ 𝑋 ∈ 𝐴)))
2524imp 406 . . 3 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) β†’ (Β¬ 𝑄(leβ€˜πΎ)𝑋 β†’ 𝑋 ∈ 𝐴))
26 hlpos 38740 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
2726adantr 480 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Poset)
281, 3latjcl 18400 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
298, 11, 14, 28syl3anc 1368 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
30 eqid 2724 . . . . . . . . . 10 (leβ€˜πΎ) = (leβ€˜πΎ)
311, 30, 2pltnle 18299 . . . . . . . . 9 (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) ∧ 𝑋 < (𝑃 ∨ 𝑄)) β†’ Β¬ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋)
3231ex 412 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋 < (𝑃 ∨ 𝑄) β†’ Β¬ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋))
3327, 20, 29, 32syl3anc 1368 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 < (𝑃 ∨ 𝑄) β†’ Β¬ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋))
341, 30, 3latjle12 18411 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑃(leβ€˜πΎ)𝑋 ∧ 𝑄(leβ€˜πΎ)𝑋) ↔ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋))
358, 11, 14, 20, 34syl13anc 1369 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃(leβ€˜πΎ)𝑋 ∧ 𝑄(leβ€˜πΎ)𝑋) ↔ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋))
3635biimpd 228 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃(leβ€˜πΎ)𝑋 ∧ 𝑄(leβ€˜πΎ)𝑋) β†’ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋))
3733, 36nsyld 156 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 < (𝑃 ∨ 𝑄) β†’ Β¬ (𝑃(leβ€˜πΎ)𝑋 ∧ 𝑄(leβ€˜πΎ)𝑋)))
38 ianor 978 . . . . . 6 (Β¬ (𝑃(leβ€˜πΎ)𝑋 ∧ 𝑄(leβ€˜πΎ)𝑋) ↔ (Β¬ 𝑃(leβ€˜πΎ)𝑋 ∨ Β¬ 𝑄(leβ€˜πΎ)𝑋))
3937, 38imbitrdi 250 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 < (𝑃 ∨ 𝑄) β†’ (Β¬ 𝑃(leβ€˜πΎ)𝑋 ∨ Β¬ 𝑄(leβ€˜πΎ)𝑋)))
4039imp 406 . . . 4 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 < (𝑃 ∨ 𝑄)) β†’ (Β¬ 𝑃(leβ€˜πΎ)𝑋 ∨ Β¬ 𝑄(leβ€˜πΎ)𝑋))
4140adantrl 713 . . 3 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) β†’ (Β¬ 𝑃(leβ€˜πΎ)𝑋 ∨ Β¬ 𝑄(leβ€˜πΎ)𝑋))
426, 25, 41mpjaod 857 . 2 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) β†’ 𝑋 ∈ 𝐴)
4342ex 412 1 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) β†’ 𝑋 ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932   class class class wbr 5139  β€˜cfv 6534  (class class class)co 7402  Basecbs 17149  lecple 17209  Posetcpo 18268  ltcplt 18269  joincjn 18272  0.cp0 18384  Latclat 18392  Atomscatm 38637  HLchlt 38724
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-proset 18256  df-poset 18274  df-plt 18291  df-lub 18307  df-glb 18308  df-join 18309  df-meet 18310  df-p0 18386  df-lat 18393  df-clat 18460  df-oposet 38550  df-ol 38552  df-oml 38553  df-covers 38640  df-ats 38641  df-atl 38672  df-cvlat 38696  df-hlat 38725
This theorem is referenced by:  cvrat2  38804
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