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Theorem cvrat 38895
Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 32209 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 38894 . . 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 38835 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
87adantr 480 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
9 simpr2 1193 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
101, 5atbase 38761 . . . . . . . . 9 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
119, 10syl 17 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐡)
12 simpr3 1194 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
131, 5atbase 38761 . . . . . . . . 9 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1412, 13syl 17 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
151, 3latjcom 18439 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
168, 11, 14, 15syl3anc 1369 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
1716breq2d 5160 . . . . . 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 482 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
20 simpr1 1192 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
211, 2, 3, 4, 5cvratlem 38894 . . . . . . 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 1370 . . . . 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 38838 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
2726adantr 480 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Poset)
281, 3latjcl 18431 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
298, 11, 14, 28syl3anc 1369 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
30 eqid 2728 . . . . . . . . . 10 (leβ€˜πΎ) = (leβ€˜πΎ)
311, 30, 2pltnle 18330 . . . . . . . . 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 1369 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 < (𝑃 ∨ 𝑄) β†’ Β¬ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋))
341, 30, 3latjle12 18442 . . . . . . . . 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 1370 . . . . . . . 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 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 715 . . 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 859 . 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 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2937   class class class wbr 5148  β€˜cfv 6548  (class class class)co 7420  Basecbs 17180  lecple 17240  Posetcpo 18299  ltcplt 18300  joincjn 18303  0.cp0 18415  Latclat 18423  Atomscatm 38735  HLchlt 38822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-proset 18287  df-poset 18305  df-plt 18322  df-lub 18338  df-glb 18339  df-join 18340  df-meet 18341  df-p0 18417  df-lat 18424  df-clat 18491  df-oposet 38648  df-ol 38650  df-oml 38651  df-covers 38738  df-ats 38739  df-atl 38770  df-cvlat 38794  df-hlat 38823
This theorem is referenced by:  cvrat2  38902
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