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Theorem dalempnes 35459
Description: Lemma for dath 35544. Frequently-used utility lemma. (Contributed by NM, 13-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)
dalempnes.o 𝑂 = (LPlanes‘𝐾)
dalempnes.y 𝑌 = ((𝑃 𝑄) 𝑅)
Assertion
Ref Expression
dalempnes (𝜑𝑃𝑆)

Proof of Theorem dalempnes
StepHypRef Expression
1 dalema.ph . . . 4 (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)))))
21dalemkelat 35432 . . 3 (𝜑𝐾 ∈ Lat)
3 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 35446 . . 3 (𝜑𝐶 ∈ (Base‘𝐾))
51, 3dalemseb 35450 . . 3 (𝜑𝑆 ∈ (Base‘𝐾))
61, 3dalemteb 35451 . . 3 (𝜑𝑇 ∈ (Base‘𝐾))
7 simp321 1407 . . . 4 ((((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)))) → ¬ 𝐶 (𝑆 𝑇))
81, 7sylbi 207 . . 3 (𝜑 → ¬ 𝐶 (𝑆 𝑇))
9 eqid 2771 . . . 4 (Base‘𝐾) = (Base‘𝐾)
10 dalemc.l . . . 4 = (le‘𝐾)
11 dalemc.j . . . 4 = (join‘𝐾)
129, 10, 11latnlej2l 17280 . . 3 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) ∧ ¬ 𝐶 (𝑆 𝑇)) → ¬ 𝐶 𝑆)
132, 4, 5, 6, 8, 12syl131anc 1489 . 2 (𝜑 → ¬ 𝐶 𝑆)
141dalemclpjs 35442 . . . . 5 (𝜑𝐶 (𝑃 𝑆))
15 oveq1 6800 . . . . . 6 (𝑃 = 𝑆 → (𝑃 𝑆) = (𝑆 𝑆))
1615breq2d 4798 . . . . 5 (𝑃 = 𝑆 → (𝐶 (𝑃 𝑆) ↔ 𝐶 (𝑆 𝑆)))
1714, 16syl5ibcom 235 . . . 4 (𝜑 → (𝑃 = 𝑆𝐶 (𝑆 𝑆)))
181dalemkehl 35431 . . . . . 6 (𝜑𝐾 ∈ HL)
191dalemsea 35437 . . . . . 6 (𝜑𝑆𝐴)
2011, 3hlatjidm 35177 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴) → (𝑆 𝑆) = 𝑆)
2118, 19, 20syl2anc 573 . . . . 5 (𝜑 → (𝑆 𝑆) = 𝑆)
2221breq2d 4798 . . . 4 (𝜑 → (𝐶 (𝑆 𝑆) ↔ 𝐶 𝑆))
2317, 22sylibd 229 . . 3 (𝜑 → (𝑃 = 𝑆𝐶 𝑆))
2423necon3bd 2957 . 2 (𝜑 → (¬ 𝐶 𝑆𝑃𝑆))
2513, 24mpd 15 1 (𝜑𝑃𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  Latclat 17253  Atomscatm 35072  HLchlt 35159  LPlanesclpl 35300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-lat 17254  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160
This theorem is referenced by:  dalempjsen  35461  dalem24  35505
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