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Theorem dalem1 34764
Description: Lemma for dath 34841. 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 34739 . 2 (𝜑𝐶 (𝑃 𝑆))
31dalem-clpjq 34742 . . . . . 6 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
43adantr 481 . . . . 5 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ¬ 𝐶 (𝑃 𝑄))
51dalemkehl 34728 . . . . . . . . . 10 (𝜑𝐾 ∈ HL)
61dalempea 34731 . . . . . . . . . 10 (𝜑𝑃𝐴)
71dalemsea 34734 . . . . . . . . . 10 (𝜑𝑆𝐴)
8 dalemc.l . . . . . . . . . . 11 = (le‘𝐾)
9 dalemc.j . . . . . . . . . . 11 = (join‘𝐾)
10 dalemc.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
118, 9, 10hlatlej1 34480 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑃 (𝑃 𝑆))
125, 6, 7, 11syl3anc 1324 . . . . . . . . 9 (𝜑𝑃 (𝑃 𝑆))
1312adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑃 (𝑃 𝑆))
141dalemqea 34732 . . . . . . . . . . 11 (𝜑𝑄𝐴)
151dalemtea 34735 . . . . . . . . . . 11 (𝜑𝑇𝐴)
168, 9, 10hlatlej1 34480 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑄 (𝑄 𝑇))
175, 14, 15, 16syl3anc 1324 . . . . . . . . . 10 (𝜑𝑄 (𝑄 𝑇))
1817adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑄 (𝑄 𝑇))
19 simpr 477 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑆) = (𝑄 𝑇))
2018, 19breqtrrd 4672 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑄 (𝑃 𝑆))
211dalemkelat 34729 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
221, 10dalempeb 34744 . . . . . . . . . 10 (𝜑𝑃 ∈ (Base‘𝐾))
231, 10dalemqeb 34745 . . . . . . . . . 10 (𝜑𝑄 ∈ (Base‘𝐾))
24 eqid 2620 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
2524, 9, 10hlatjcl 34472 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
265, 6, 7, 25syl3anc 1324 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
2724, 8, 9latjle12 17043 . . . . . . . . . 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 1326 . . . . . . . . 9 (𝜑 → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
2928adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
3013, 20, 29mpbi2and 955 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑄) (𝑃 𝑆))
311dalemrea 34733 . . . . . . . . . 10 (𝜑𝑅𝐴)
321dalemyeo 34737 . . . . . . . . . 10 (𝜑𝑌𝑂)
33 dalem1.o . . . . . . . . . . 11 𝑂 = (LPlanes‘𝐾)
34 dalem1.y . . . . . . . . . . 11 𝑌 = ((𝑃 𝑄) 𝑅)
359, 10, 33, 34lplnri1 34658 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑌𝑂) → 𝑃𝑄)
365, 6, 14, 31, 32, 35syl131anc 1337 . . . . . . . . 9 (𝜑𝑃𝑄)
378, 9, 10ps-1 34582 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
385, 6, 14, 36, 6, 7, 37syl132anc 1342 . . . . . . . 8 (𝜑 → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
3938adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
4030, 39mpbid 222 . . . . . 6 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑄) = (𝑃 𝑆))
4140breq2d 4656 . . . . 5 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝐶 (𝑃 𝑄) ↔ 𝐶 (𝑃 𝑆)))
424, 41mtbid 314 . . . 4 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ¬ 𝐶 (𝑃 𝑆))
4342ex 450 . . 3 (𝜑 → ((𝑃 𝑆) = (𝑄 𝑇) → ¬ 𝐶 (𝑃 𝑆)))
4443necon2ad 2806 . 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 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791   class class class wbr 4644  cfv 5876  (class class class)co 6635  Basecbs 15838  lecple 15929  joincjn 16925  Latclat 17026  Atomscatm 34369  HLchlt 34456  LPlanesclpl 34597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-preset 16909  df-poset 16927  df-plt 16939  df-lub 16955  df-glb 16956  df-join 16957  df-meet 16958  df-p0 17020  df-lat 17027  df-clat 17089  df-oposet 34282  df-ol 34284  df-oml 34285  df-covers 34372  df-ats 34373  df-atl 34404  df-cvlat 34428  df-hlat 34457  df-llines 34603  df-lplanes 34604
This theorem is referenced by:  dalemcea  34765  dalem2  34766
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