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Theorem cvrat2 38811
Description: A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 32145 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 2726 . . . . . . . . 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 38810 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = (0.β€˜πΎ) ↔ 𝑃 = 𝑄))
763expa 1115 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = (0.β€˜πΎ) ↔ 𝑃 = 𝑄))
87necon3bid 2979 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 β‰  (0.β€˜πΎ) ↔ 𝑃 β‰  𝑄))
9 simpl 482 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
10 simpr1 1191 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
11 hllat 38744 . . . . . . . . . . 11 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1211adantr 480 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
13 simpr2 1192 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
141, 5atbase 38670 . . . . . . . . . . 11 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
1513, 14syl 17 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐡)
16 simpr3 1193 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
171, 5atbase 38670 . . . . . . . . . . 11 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1816, 17syl 17 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
191, 2latjcl 18402 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
2012, 15, 18, 19syl3anc 1368 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
21 eqid 2726 . . . . . . . . . . 11 (ltβ€˜πΎ) = (ltβ€˜πΎ)
221, 21, 4cvrlt 38651 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄))
2322ex 412 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄)))
249, 10, 20, 23syl3anc 1368 . . . . . . . 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 38804 . . . . . . . . 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 416 . . . . . . . 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 406 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 β‰  (0.β€˜πΎ) β†’ 𝑋 ∈ 𝐴))
298, 28sylbird 260 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑃 β‰  𝑄 β†’ 𝑋 ∈ 𝐴))
3029ex 412 . . . 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 410 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ 𝑋 ∈ 𝐴))
33323impia 1114 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  ltcplt 18271  joincjn 18274  0.cp0 18386  Latclat 18394   β‹– ccvr 38643  Atomscatm 38644  HLchlt 38731
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-lat 18395  df-clat 18462  df-oposet 38557  df-ol 38559  df-oml 38560  df-covers 38647  df-ats 38648  df-atl 38679  df-cvlat 38703  df-hlat 38732
This theorem is referenced by:  cvrat3  38824  atcvrlln  38902  lncvrelatN  39163
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