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Theorem cvrat 38281
Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 31626 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 38280 . . 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 38221 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
87adantr 481 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
9 simpr2 1195 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
101, 5atbase 38147 . . . . . . . . 9 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
119, 10syl 17 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐡)
12 simpr3 1196 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
131, 5atbase 38147 . . . . . . . . 9 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1412, 13syl 17 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
151, 3latjcom 18396 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
168, 11, 14, 15syl3anc 1371 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
1716breq2d 5159 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 < (𝑃 ∨ 𝑄) ↔ 𝑋 < (𝑄 ∨ 𝑃)))
1817anbi2d 629 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) ↔ (𝑋 β‰  0 ∧ 𝑋 < (𝑄 ∨ 𝑃))))
19 simpl 483 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
20 simpr1 1194 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
211, 2, 3, 4, 5cvratlem 38280 . . . . . . 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 413 . . . . . 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 1372 . . . . 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 407 . . 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 38224 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
2726adantr 481 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Poset)
281, 3latjcl 18388 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
298, 11, 14, 28syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
30 eqid 2732 . . . . . . . . . 10 (leβ€˜πΎ) = (leβ€˜πΎ)
311, 30, 2pltnle 18287 . . . . . . . . 9 (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) ∧ 𝑋 < (𝑃 ∨ 𝑄)) β†’ Β¬ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋)
3231ex 413 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋 < (𝑃 ∨ 𝑄) β†’ Β¬ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋))
3327, 20, 29, 32syl3anc 1371 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 < (𝑃 ∨ 𝑄) β†’ Β¬ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋))
341, 30, 3latjle12 18399 . . . . . . . . 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 1372 . . . . . . . 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 980 . . . . . 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 407 . . . 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 714 . . 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 858 . 2 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) β†’ 𝑋 ∈ 𝐴)
4342ex 413 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 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  Posetcpo 18256  ltcplt 18257  joincjn 18260  0.cp0 18372  Latclat 18380  Atomscatm 38121  HLchlt 38208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209
This theorem is referenced by:  cvrat2  38288
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