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Theorem dalem1 36954
Description: Lemma for dath 37031. 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 36929 . 2 (𝜑𝐶 (𝑃 𝑆))
31dalem-clpjq 36932 . . . . . 6 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
43adantr 484 . . . . 5 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ¬ 𝐶 (𝑃 𝑄))
51dalemkehl 36918 . . . . . . . . . 10 (𝜑𝐾 ∈ HL)
61dalempea 36921 . . . . . . . . . 10 (𝜑𝑃𝐴)
71dalemsea 36924 . . . . . . . . . 10 (𝜑𝑆𝐴)
8 dalemc.l . . . . . . . . . . 11 = (le‘𝐾)
9 dalemc.j . . . . . . . . . . 11 = (join‘𝐾)
10 dalemc.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
118, 9, 10hlatlej1 36670 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑃 (𝑃 𝑆))
125, 6, 7, 11syl3anc 1368 . . . . . . . . 9 (𝜑𝑃 (𝑃 𝑆))
1312adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑃 (𝑃 𝑆))
141dalemqea 36922 . . . . . . . . . . 11 (𝜑𝑄𝐴)
151dalemtea 36925 . . . . . . . . . . 11 (𝜑𝑇𝐴)
168, 9, 10hlatlej1 36670 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑄 (𝑄 𝑇))
175, 14, 15, 16syl3anc 1368 . . . . . . . . . 10 (𝜑𝑄 (𝑄 𝑇))
1817adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑄 (𝑄 𝑇))
19 simpr 488 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑆) = (𝑄 𝑇))
2018, 19breqtrrd 5061 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → 𝑄 (𝑃 𝑆))
211dalemkelat 36919 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
221, 10dalempeb 36934 . . . . . . . . . 10 (𝜑𝑃 ∈ (Base‘𝐾))
231, 10dalemqeb 36935 . . . . . . . . . 10 (𝜑𝑄 ∈ (Base‘𝐾))
24 eqid 2801 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
2524, 9, 10hlatjcl 36662 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
265, 6, 7, 25syl3anc 1368 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
2724, 8, 9latjle12 17668 . . . . . . . . . 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 1369 . . . . . . . . 9 (𝜑 → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
2928adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
3013, 20, 29mpbi2and 711 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑄) (𝑃 𝑆))
311dalemrea 36923 . . . . . . . . . 10 (𝜑𝑅𝐴)
321dalemyeo 36927 . . . . . . . . . 10 (𝜑𝑌𝑂)
33 dalem1.o . . . . . . . . . . 11 𝑂 = (LPlanes‘𝐾)
34 dalem1.y . . . . . . . . . . 11 𝑌 = ((𝑃 𝑄) 𝑅)
359, 10, 33, 34lplnri1 36848 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑌𝑂) → 𝑃𝑄)
365, 6, 14, 31, 32, 35syl131anc 1380 . . . . . . . . 9 (𝜑𝑃𝑄)
378, 9, 10ps-1 36772 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
385, 6, 14, 36, 6, 7, 37syl132anc 1385 . . . . . . . 8 (𝜑 → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
3938adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ((𝑃 𝑄) (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
4030, 39mpbid 235 . . . . . 6 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝑃 𝑄) = (𝑃 𝑆))
4140breq2d 5045 . . . . 5 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → (𝐶 (𝑃 𝑄) ↔ 𝐶 (𝑃 𝑆)))
424, 41mtbid 327 . . . 4 ((𝜑 ∧ (𝑃 𝑆) = (𝑄 𝑇)) → ¬ 𝐶 (𝑃 𝑆))
4342ex 416 . . 3 (𝜑 → ((𝑃 𝑆) = (𝑄 𝑇) → ¬ 𝐶 (𝑃 𝑆)))
4443necon2ad 3005 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16479  lecple 16568  joincjn 17550  Latclat 17651  Atomscatm 36558  HLchlt 36645  LPlanesclpl 36787
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-proset 17534  df-poset 17552  df-plt 17564  df-lub 17580  df-glb 17581  df-join 17582  df-meet 17583  df-p0 17645  df-lat 17652  df-clat 17714  df-oposet 36471  df-ol 36473  df-oml 36474  df-covers 36561  df-ats 36562  df-atl 36593  df-cvlat 36617  df-hlat 36646  df-llines 36793  df-lplanes 36794
This theorem is referenced by:  dalemcea  36955  dalem2  36956
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