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Theorem cvrat 37931
Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 31370 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 37930 . . 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 37871 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
87adantr 482 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
9 simpr2 1196 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
101, 5atbase 37797 . . . . . . . . 9 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
119, 10syl 17 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐡)
12 simpr3 1197 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
131, 5atbase 37797 . . . . . . . . 9 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1412, 13syl 17 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
151, 3latjcom 18341 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
168, 11, 14, 15syl3anc 1372 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
1716breq2d 5118 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 < (𝑃 ∨ 𝑄) ↔ 𝑋 < (𝑄 ∨ 𝑃)))
1817anbi2d 630 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) ↔ (𝑋 β‰  0 ∧ 𝑋 < (𝑄 ∨ 𝑃))))
19 simpl 484 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
20 simpr1 1195 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
211, 2, 3, 4, 5cvratlem 37930 . . . . . . 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 414 . . . . . 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 1373 . . . . 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 408 . . 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 37874 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
2726adantr 482 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Poset)
281, 3latjcl 18333 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
298, 11, 14, 28syl3anc 1372 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
30 eqid 2733 . . . . . . . . . 10 (leβ€˜πΎ) = (leβ€˜πΎ)
311, 30, 2pltnle 18232 . . . . . . . . 9 (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) ∧ 𝑋 < (𝑃 ∨ 𝑄)) β†’ Β¬ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋)
3231ex 414 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋 < (𝑃 ∨ 𝑄) β†’ Β¬ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋))
3327, 20, 29, 32syl3anc 1372 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 < (𝑃 ∨ 𝑄) β†’ Β¬ (𝑃 ∨ 𝑄)(leβ€˜πΎ)𝑋))
341, 30, 3latjle12 18344 . . . . . . . . 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 1373 . . . . . . . 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 981 . . . . . 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, 38syl6ib 251 . . . . 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 408 . . . 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 414 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  Posetcpo 18201  ltcplt 18202  joincjn 18205  0.cp0 18317  Latclat 18325  Atomscatm 37771  HLchlt 37858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-lat 18326  df-clat 18393  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859
This theorem is referenced by:  cvrat2  37938
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