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Theorem dalem1 39642
Description: Lemma for dath 39719. Show the lines 𝑃𝑆 and 𝑄𝑇 are different. (Contributed by NM, 9-Aug-2012.)
Hypotheses
Ref Expression
dalema.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
dalemc.l = (le‘𝐾)
dalemc.j = (join‘𝐾)
dalemc.a 𝐴 = (Atoms‘𝐾)
dalem1.o 𝑂 = (LPlanes‘𝐾)
dalem1.y 𝑌 = ((𝑃 𝑄) 𝑅)
Assertion
Ref Expression
dalem1 (𝜑 → (𝑃 𝑆) ≠ (𝑄 𝑇))

Proof of Theorem dalem1
StepHypRef Expression
1 dalema.ph . . 3 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
21dalemclpjs 39617 . 2 (𝜑𝐶 (𝑃 𝑆))
31dalem-clpjq 39620 . . . . . 6 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
43adantr 480 . . . . 5 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ¬ 𝐶 (𝑃 𝑄))
51dalemkehl 39606 . . . . . . . . . 10 (𝜑𝐾 ∈ HL)
61dalempea 39609 . . . . . . . . . 10 (𝜑𝑃𝐴)
71dalemsea 39612 . . . . . . . . . 10 (𝜑𝑆𝐴)
8 dalemc.l . . . . . . . . . . 11 = (le‘𝐾)
9 dalemc.j . . . . . . . . . . 11 = (join‘𝐾)
10 dalemc.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
118, 9, 10hlatlej1 39357 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑃 (𝑃 𝑆))
125, 6, 7, 11syl3anc 1370 . . . . . . . . 9 (𝜑𝑃 (𝑃 𝑆))
1312adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑃 (𝑃 𝑆))
141dalemqea 39610 . . . . . . . . . . 11 (𝜑𝑄𝐴)
151dalemtea 39613 . . . . . . . . . . 11 (𝜑𝑇𝐴)
168, 9, 10hlatlej1 39357 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑄 (𝑄 𝑇))
175, 14, 15, 16syl3anc 1370 . . . . . . . . . 10 (𝜑𝑄 (𝑄 𝑇))
1817adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑄 (𝑄 𝑇))
19 simpr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑆) = (𝑄 𝑇))
2018, 19breqtrrd 5176 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑄 (𝑃 𝑆))
211dalemkelat 39607 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
221, 10dalempeb 39622 . . . . . . . . . 10 (𝜑𝑃 ∈ (Base‘𝐾))
231, 10dalemqeb 39623 . . . . . . . . . 10 (𝜑𝑄 ∈ (Base‘𝐾))
24 eqid 2735 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
2524, 9, 10hlatjcl 39349 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
265, 6, 7, 25syl3anc 1370 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
2724, 8, 9latjle12 18508 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾))) → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
2821, 22, 23, 26, 27syl13anc 1371 . . . . . . . . 9 (𝜑 → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
2928adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
3013, 20, 29mpbi2and 712 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑄) (𝑃 𝑆))
311dalemrea 39611 . . . . . . . . . 10 (𝜑𝑅𝐴)
321dalemyeo 39615 . . . . . . . . . 10 (𝜑𝑌𝑂)
33 dalem1.o . . . . . . . . . . 11 𝑂 = (LPlanes‘𝐾)
34 dalem1.y . . . . . . . . . . 11 𝑌 = ((𝑃 𝑄) 𝑅)
359, 10, 33, 34lplnri1 39536 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑌𝑂) → 𝑃𝑄)
365, 6, 14, 31, 32, 35syl131anc 1382 . . . . . . . . 9 (𝜑𝑃𝑄)
378, 9, 10ps-1 39460 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
385, 6, 14, 36, 6, 7, 37syl132anc 1387 . . . . . . . 8 (𝜑 → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
3938adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
4030, 39mpbid 232 . . . . . 6 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑄) = (𝑃 𝑆))
4140breq2d 5160 . . . . 5 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝐶 (𝑃 𝑄) ↔ 𝐶 (𝑃 𝑆)))
424, 41mtbid 324 . . . 4 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ¬ 𝐶 (𝑃 𝑆))
4342ex 412 . . 3 (𝜑 → ((𝑃 𝑆) = (𝑄 𝑇) → ¬ 𝐶 (𝑃 𝑆)))
4443necon2ad 2953 . 2 (𝜑 → (𝐶 (𝑃 𝑆) → (𝑃 𝑆) ≠ (𝑄 𝑇)))
452, 44mpd 15 1 (𝜑 → (𝑃 𝑆) ≠ (𝑄 𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Latclat 18489  Atomscatm 39245  HLchlt 39332  LPlanesclpl 39475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482
This theorem is referenced by:  dalemcea  39643  dalem2  39644
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