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Theorem cvrat2 38295
Description: A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 31635 analog.) (Contributed by NM, 30-Nov-2011.)
Hypotheses
Ref Expression
cvrat2.b 𝐡 = (Baseβ€˜πΎ)
cvrat2.j ∨ = (joinβ€˜πΎ)
cvrat2.c 𝐢 = ( β‹– β€˜πΎ)
cvrat2.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
cvrat2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢(𝑃 ∨ 𝑄))) β†’ 𝑋 ∈ 𝐴)
Proof of Theorem cvrat2
StepHypRef Expression
1 cvrat2.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΎ)
2 cvrat2.j . . . . . . . . 9 ∨ = (joinβ€˜πΎ)
3 eqid 2732 . . . . . . . . 9 (0.β€˜πΎ) = (0.β€˜πΎ)
4 cvrat2.c . . . . . . . . 9 𝐢 = ( β‹– β€˜πΎ)
5 cvrat2.a . . . . . . . . 9 𝐴 = (Atomsβ€˜πΎ)
61, 2, 3, 4, 5atcvrj0 38294 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = (0.β€˜πΎ) ↔ 𝑃 = 𝑄))
763expa 1118 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = (0.β€˜πΎ) ↔ 𝑃 = 𝑄))
87necon3bid 2985 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 β‰  (0.β€˜πΎ) ↔ 𝑃 β‰  𝑄))
9 simpl 483 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
10 simpr1 1194 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
11 hllat 38228 . . . . . . . . . . 11 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1211adantr 481 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
13 simpr2 1195 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
141, 5atbase 38154 . . . . . . . . . . 11 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
1513, 14syl 17 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐡)
16 simpr3 1196 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
171, 5atbase 38154 . . . . . . . . . . 11 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1816, 17syl 17 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
191, 2latjcl 18391 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
2012, 15, 18, 19syl3anc 1371 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
21 eqid 2732 . . . . . . . . . . 11 (ltβ€˜πΎ) = (ltβ€˜πΎ)
221, 21, 4cvrlt 38135 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄))
2322ex 413 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄)))
249, 10, 20, 23syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄)))
251, 21, 2, 3, 5cvrat 38288 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  (0.β€˜πΎ) ∧ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄)) β†’ 𝑋 ∈ 𝐴))
2625expcomd 417 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄) β†’ (𝑋 β‰  (0.β€˜πΎ) β†’ 𝑋 ∈ 𝐴)))
2724, 26syld 47 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ (𝑋 β‰  (0.β€˜πΎ) β†’ 𝑋 ∈ 𝐴)))
2827imp 407 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 β‰  (0.β€˜πΎ) β†’ 𝑋 ∈ 𝐴))
298, 28sylbird 259 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑃 β‰  𝑄 β†’ 𝑋 ∈ 𝐴))
3029ex 413 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ (𝑃 β‰  𝑄 β†’ 𝑋 ∈ 𝐴)))
3130com23 86 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 β‰  𝑄 β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 𝑋 ∈ 𝐴)))
3231impd 411 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ 𝑋 ∈ 𝐴))
33323impia 1117 1 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢(𝑃 ∨ 𝑄))) β†’ 𝑋 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  ltcplt 18260  joincjn 18263  0.cp0 18375  Latclat 18383   β‹– ccvr 38127  Atomscatm 38128  HLchlt 38215
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-lat 18384  df-clat 18451  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216
This theorem is referenced by:  cvrat3  38308  atcvrlln  38386  lncvrelatN  38647
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