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Theorem atbtwn 37909
Description: Property of a 3rd atom 𝑅 on a line 𝑃 𝑄 intersecting element 𝑋 at 𝑃. (Contributed by NM, 30-Jul-2012.)
Hypotheses
Ref Expression
atbtwn.b 𝐵 = (Base‘𝐾)
atbtwn.l = (le‘𝐾)
atbtwn.j = (join‘𝐾)
atbtwn.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atbtwn (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅𝑃 ↔ ¬ 𝑅 𝑋))

Proof of Theorem atbtwn
StepHypRef Expression
1 simpl33 1256 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝑅 (𝑃 𝑄))
2 simpr 485 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝑅 𝑋)
3 simpl11 1248 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝐾 ∈ HL)
43hllatd 37826 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝐾 ∈ Lat)
5 simpl2l 1226 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝑅𝐴)
6 atbtwn.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
7 atbtwn.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
86, 7atbase 37751 . . . . . . . . 9 (𝑅𝐴𝑅𝐵)
95, 8syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝑅𝐵)
10 simpl1 1191 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → (𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴))
11 atbtwn.j . . . . . . . . . 10 = (join‘𝐾)
126, 11, 7hlatjcl 37829 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
1310, 12syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → (𝑃 𝑄) ∈ 𝐵)
14 simpl2r 1227 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝑋𝐵)
15 atbtwn.l . . . . . . . . 9 = (le‘𝐾)
16 eqid 2736 . . . . . . . . 9 (meet‘𝐾) = (meet‘𝐾)
176, 15, 16latlem12 18355 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑅𝐵 ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵)) → ((𝑅 (𝑃 𝑄) ∧ 𝑅 𝑋) ↔ 𝑅 ((𝑃 𝑄)(meet‘𝐾)𝑋)))
184, 9, 13, 14, 17syl13anc 1372 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → ((𝑅 (𝑃 𝑄) ∧ 𝑅 𝑋) ↔ 𝑅 ((𝑃 𝑄)(meet‘𝐾)𝑋)))
191, 2, 18mpbi2and 710 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝑅 ((𝑃 𝑄)(meet‘𝐾)𝑋))
20 simpl12 1249 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝑃𝐴)
21 simpl13 1250 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝑄𝐴)
22 simpl31 1254 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝑃 𝑋)
23 simpl32 1255 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → ¬ 𝑄 𝑋)
246, 15, 11, 16, 72atjm 37908 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋) = 𝑃)
253, 20, 21, 14, 22, 23, 24syl132anc 1388 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → ((𝑃 𝑄)(meet‘𝐾)𝑋) = 𝑃)
2619, 25breqtrd 5131 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝑅 𝑃)
27 hlatl 37822 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
283, 27syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝐾 ∈ AtLat)
2915, 7atcmp 37773 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑅𝐴𝑃𝐴) → (𝑅 𝑃𝑅 = 𝑃))
3028, 5, 20, 29syl3anc 1371 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → (𝑅 𝑃𝑅 = 𝑃))
3126, 30mpbid 231 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅 𝑋) → 𝑅 = 𝑃)
3231ex 413 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 𝑋𝑅 = 𝑃))
3332necon3ad 2956 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅𝑃 → ¬ 𝑅 𝑋))
34 simp31 1209 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → 𝑃 𝑋)
35 nbrne2 5125 . . . . 5 ((𝑃 𝑋 ∧ ¬ 𝑅 𝑋) → 𝑃𝑅)
3635necomd 2999 . . . 4 ((𝑃 𝑋 ∧ ¬ 𝑅 𝑋) → 𝑅𝑃)
3736ex 413 . . 3 (𝑃 𝑋 → (¬ 𝑅 𝑋𝑅𝑃))
3834, 37syl 17 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (¬ 𝑅 𝑋𝑅𝑃))
3933, 38impbid 211 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅𝑃 ↔ ¬ 𝑅 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Latclat 18320  Atomscatm 37725  AtLatcal 37726  HLchlt 37812
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813
This theorem is referenced by:  atbtwnexOLDN  37910  atbtwnex  37911
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