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Theorem dalem1 33761
Description: Lemma for dath 33838. 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 33736 . 2 (𝜑𝐶 (𝑃 𝑆))
31dalem-clpjq 33739 . . . . . 6 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
43adantr 479 . . . . 5 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ¬ 𝐶 (𝑃 𝑄))
51dalemkehl 33725 . . . . . . . . . 10 (𝜑𝐾 ∈ HL)
61dalempea 33728 . . . . . . . . . 10 (𝜑𝑃𝐴)
71dalemsea 33731 . . . . . . . . . 10 (𝜑𝑆𝐴)
8 dalemc.l . . . . . . . . . . 11 = (le‘𝐾)
9 dalemc.j . . . . . . . . . . 11 = (join‘𝐾)
10 dalemc.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
118, 9, 10hlatlej1 33477 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑃 (𝑃 𝑆))
125, 6, 7, 11syl3anc 1317 . . . . . . . . 9 (𝜑𝑃 (𝑃 𝑆))
1312adantr 479 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑃 (𝑃 𝑆))
141dalemqea 33729 . . . . . . . . . . 11 (𝜑𝑄𝐴)
151dalemtea 33732 . . . . . . . . . . 11 (𝜑𝑇𝐴)
168, 9, 10hlatlej1 33477 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑄 (𝑄 𝑇))
175, 14, 15, 16syl3anc 1317 . . . . . . . . . 10 (𝜑𝑄 (𝑄 𝑇))
1817adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑄 (𝑄 𝑇))
19 simpr 475 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑆) = (𝑄 𝑇))
2018, 19breqtrrd 4600 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑄 (𝑃 𝑆))
211dalemkelat 33726 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
221, 10dalempeb 33741 . . . . . . . . . 10 (𝜑𝑃 ∈ (Base‘𝐾))
231, 10dalemqeb 33742 . . . . . . . . . 10 (𝜑𝑄 ∈ (Base‘𝐾))
24 eqid 2604 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
2524, 9, 10hlatjcl 33469 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
265, 6, 7, 25syl3anc 1317 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
2724, 8, 9latjle12 16826 . . . . . . . . . 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 1319 . . . . . . . . 9 (𝜑 → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
2928adantr 479 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
3013, 20, 29mpbi2and 957 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑄) (𝑃 𝑆))
311dalemrea 33730 . . . . . . . . . 10 (𝜑𝑅𝐴)
321dalemyeo 33734 . . . . . . . . . 10 (𝜑𝑌𝑂)
33 dalem1.o . . . . . . . . . . 11 𝑂 = (LPlanes‘𝐾)
34 dalem1.y . . . . . . . . . . 11 𝑌 = ((𝑃 𝑄) 𝑅)
359, 10, 33, 34lplnri1 33655 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑌𝑂) → 𝑃𝑄)
365, 6, 14, 31, 32, 35syl131anc 1330 . . . . . . . . 9 (𝜑𝑃𝑄)
378, 9, 10ps-1 33579 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
385, 6, 14, 36, 6, 7, 37syl132anc 1335 . . . . . . . 8 (𝜑 → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
3938adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
4030, 39mpbid 220 . . . . . 6 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑄) = (𝑃 𝑆))
4140breq2d 4584 . . . . 5 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝐶 (𝑃 𝑄) ↔ 𝐶 (𝑃 𝑆)))
424, 41mtbid 312 . . . 4 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ¬ 𝐶 (𝑃 𝑆))
4342ex 448 . . 3 (𝜑 → ((𝑃 𝑆) = (𝑄 𝑇) → ¬ 𝐶 (𝑃 𝑆)))
4443necon2ad 2791 . 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2774   class class class wbr 4572  cfv 5785  (class class class)co 6522  Basecbs 15636  lecple 15716  joincjn 16708  Latclat 16809  Atomscatm 33366  HLchlt 33453  LPlanesclpl 33594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-preset 16692  df-poset 16710  df-plt 16722  df-lub 16738  df-glb 16739  df-join 16740  df-meet 16741  df-p0 16803  df-lat 16810  df-clat 16872  df-oposet 33279  df-ol 33281  df-oml 33282  df-covers 33369  df-ats 33370  df-atl 33401  df-cvlat 33425  df-hlat 33454  df-llines 33600  df-lplanes 33601
This theorem is referenced by:  dalemcea  33762  dalem2  33763
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