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Theorem dalem1 36237
Description: Lemma for dath 36314. 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 36212 . 2 (𝜑𝐶 (𝑃 𝑆))
31dalem-clpjq 36215 . . . . . 6 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
43adantr 473 . . . . 5 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ¬ 𝐶 (𝑃 𝑄))
51dalemkehl 36201 . . . . . . . . . 10 (𝜑𝐾 ∈ HL)
61dalempea 36204 . . . . . . . . . 10 (𝜑𝑃𝐴)
71dalemsea 36207 . . . . . . . . . 10 (𝜑𝑆𝐴)
8 dalemc.l . . . . . . . . . . 11 = (le‘𝐾)
9 dalemc.j . . . . . . . . . . 11 = (join‘𝐾)
10 dalemc.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
118, 9, 10hlatlej1 35953 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑃 (𝑃 𝑆))
125, 6, 7, 11syl3anc 1351 . . . . . . . . 9 (𝜑𝑃 (𝑃 𝑆))
1312adantr 473 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑃 (𝑃 𝑆))
141dalemqea 36205 . . . . . . . . . . 11 (𝜑𝑄𝐴)
151dalemtea 36208 . . . . . . . . . . 11 (𝜑𝑇𝐴)
168, 9, 10hlatlej1 35953 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑄 (𝑄 𝑇))
175, 14, 15, 16syl3anc 1351 . . . . . . . . . 10 (𝜑𝑄 (𝑄 𝑇))
1817adantr 473 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑄 (𝑄 𝑇))
19 simpr 477 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑆) = (𝑄 𝑇))
2018, 19breqtrrd 4957 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑄 (𝑃 𝑆))
211dalemkelat 36202 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
221, 10dalempeb 36217 . . . . . . . . . 10 (𝜑𝑃 ∈ (Base‘𝐾))
231, 10dalemqeb 36218 . . . . . . . . . 10 (𝜑𝑄 ∈ (Base‘𝐾))
24 eqid 2779 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
2524, 9, 10hlatjcl 35945 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
265, 6, 7, 25syl3anc 1351 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
2724, 8, 9latjle12 17530 . . . . . . . . . 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 1352 . . . . . . . . 9 (𝜑 → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
2928adantr 473 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
3013, 20, 29mpbi2and 699 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑄) (𝑃 𝑆))
311dalemrea 36206 . . . . . . . . . 10 (𝜑𝑅𝐴)
321dalemyeo 36210 . . . . . . . . . 10 (𝜑𝑌𝑂)
33 dalem1.o . . . . . . . . . . 11 𝑂 = (LPlanes‘𝐾)
34 dalem1.y . . . . . . . . . . 11 𝑌 = ((𝑃 𝑄) 𝑅)
359, 10, 33, 34lplnri1 36131 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑌𝑂) → 𝑃𝑄)
365, 6, 14, 31, 32, 35syl131anc 1363 . . . . . . . . 9 (𝜑𝑃𝑄)
378, 9, 10ps-1 36055 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
385, 6, 14, 36, 6, 7, 37syl132anc 1368 . . . . . . . 8 (𝜑 → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
3938adantr 473 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
4030, 39mpbid 224 . . . . . 6 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑄) = (𝑃 𝑆))
4140breq2d 4941 . . . . 5 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝐶 (𝑃 𝑄) ↔ 𝐶 (𝑃 𝑆)))
424, 41mtbid 316 . . . 4 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ¬ 𝐶 (𝑃 𝑆))
4342ex 405 . . 3 (𝜑 → ((𝑃 𝑆) = (𝑄 𝑇) → ¬ 𝐶 (𝑃 𝑆)))
4443necon2ad 2983 . 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 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2968   class class class wbr 4929  cfv 6188  (class class class)co 6976  Basecbs 16339  lecple 16428  joincjn 17412  Latclat 17513  Atomscatm 35841  HLchlt 35928  LPlanesclpl 36070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-proset 17396  df-poset 17414  df-plt 17426  df-lub 17442  df-glb 17443  df-join 17444  df-meet 17445  df-p0 17507  df-lat 17514  df-clat 17576  df-oposet 35754  df-ol 35756  df-oml 35757  df-covers 35844  df-ats 35845  df-atl 35876  df-cvlat 35900  df-hlat 35929  df-llines 36076  df-lplanes 36077
This theorem is referenced by:  dalemcea  36238  dalem2  36239
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