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Theorem 2atjm 37954
Description: The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)
Hypotheses
Ref Expression
2atjm.b 𝐡 = (Baseβ€˜πΎ)
2atjm.l ≀ = (leβ€˜πΎ)
2atjm.j ∨ = (joinβ€˜πΎ)
2atjm.m ∧ = (meetβ€˜πΎ)
2atjm.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
2atjm ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) = 𝑃)

Proof of Theorem 2atjm
StepHypRef Expression
1 hllat 37871 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
213ad2ant1 1134 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ Lat)
3 simp21 1207 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ∈ 𝐴)
4 2atjm.b . . . . . . 7 𝐡 = (Baseβ€˜πΎ)
5 2atjm.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
64, 5atbase 37797 . . . . . 6 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ∈ 𝐡)
8 simp22 1208 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐴)
94, 5atbase 37797 . . . . . 6 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
108, 9syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐡)
11 2atjm.l . . . . . 6 ≀ = (leβ€˜πΎ)
12 2atjm.j . . . . . 6 ∨ = (joinβ€˜πΎ)
134, 11, 12latlej1 18342 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
142, 7, 10, 13syl3anc 1372 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
15 simp3l 1202 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ≀ 𝑋)
16 simp1 1137 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ HL)
174, 12, 5hlatjcl 37875 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
1816, 3, 8, 17syl3anc 1372 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
19 simp23 1209 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑋 ∈ 𝐡)
20 2atjm.m . . . . . 6 ∧ = (meetβ€˜πΎ)
214, 11, 20latlem12 18360 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 ≀ 𝑋) ↔ 𝑃 ≀ ((𝑃 ∨ 𝑄) ∧ 𝑋)))
222, 7, 18, 19, 21syl13anc 1373 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 ≀ 𝑋) ↔ 𝑃 ≀ ((𝑃 ∨ 𝑄) ∧ 𝑋)))
2314, 15, 22mpbi2and 711 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ≀ ((𝑃 ∨ 𝑄) ∧ 𝑋))
24 hlatl 37868 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
25243ad2ant1 1134 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ AtLat)
264, 20latmcom 18357 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) = (𝑋 ∧ (𝑃 ∨ 𝑄)))
272, 18, 19, 26syl3anc 1372 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) = (𝑋 ∧ (𝑃 ∨ 𝑄)))
2819, 3, 83jca 1129 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
29 nbrne2 5126 . . . . . . 7 ((𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝑃 β‰  𝑄)
30293ad2ant3 1136 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 β‰  𝑄)
31 simp3r 1203 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ Β¬ 𝑄 ≀ 𝑋)
324, 12latjcl 18333 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑋 ∨ 𝑄) ∈ 𝐡)
332, 19, 10, 32syl3anc 1372 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑋 ∨ 𝑄) ∈ 𝐡)
344, 11, 12latlej1 18342 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ 𝑋 ≀ (𝑋 ∨ 𝑄))
352, 19, 10, 34syl3anc 1372 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑋 ≀ (𝑋 ∨ 𝑄))
364, 11, 2, 7, 19, 33, 15, 35lattrd 18340 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ≀ (𝑋 ∨ 𝑄))
374, 11, 12, 20, 5cvrat3 37951 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
3837imp 408 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
3916, 28, 30, 31, 36, 38syl23anc 1378 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
4027, 39eqeltrd 2834 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴)
4111, 5atcmp 37819 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴) β†’ (𝑃 ≀ ((𝑃 ∨ 𝑄) ∧ 𝑋) ↔ 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋)))
4225, 3, 40, 41syl3anc 1372 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑃 ≀ ((𝑃 ∨ 𝑄) ∧ 𝑋) ↔ 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋)))
4323, 42mpbid 231 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋))
4443eqcomd 2739 1 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ 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  joincjn 18205  meetcmee 18206  Latclat 18325  Atomscatm 37771  AtLatcal 37772  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:  atbtwn  37955  dalem24  38206  dalem25  38207
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