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Theorem dalempnes 34937
Description: Lemma for dath 35022. 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 34910 . . 3 (𝜑𝐾 ∈ Lat)
3 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 34924 . . 3 (𝜑𝐶 ∈ (Base‘𝐾))
51, 3dalemseb 34928 . . 3 (𝜑𝑆 ∈ (Base‘𝐾))
61, 3dalemteb 34929 . . 3 (𝜑𝑇 ∈ (Base‘𝐾))
7 simp321 1211 . . . 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 2622 . . . 4 (Base‘𝐾) = (Base‘𝐾)
10 dalemc.l . . . 4 = (le‘𝐾)
11 dalemc.j . . . 4 = (join‘𝐾)
129, 10, 11latnlej2l 17072 . . 3 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) ∧ ¬ 𝐶 (𝑆 𝑇)) → ¬ 𝐶 𝑆)
132, 4, 5, 6, 8, 12syl131anc 1339 . 2 (𝜑 → ¬ 𝐶 𝑆)
141dalemclpjs 34920 . . . . 5 (𝜑𝐶 (𝑃 𝑆))
15 oveq1 6657 . . . . . 6 (𝑃 = 𝑆 → (𝑃 𝑆) = (𝑆 𝑆))
1615breq2d 4665 . . . . 5 (𝑃 = 𝑆 → (𝐶 (𝑃 𝑆) ↔ 𝐶 (𝑆 𝑆)))
1714, 16syl5ibcom 235 . . . 4 (𝜑 → (𝑃 = 𝑆𝐶 (𝑆 𝑆)))
181dalemkehl 34909 . . . . . 6 (𝜑𝐾 ∈ HL)
191dalemsea 34915 . . . . . 6 (𝜑𝑆𝐴)
2011, 3hlatjidm 34655 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴) → (𝑆 𝑆) = 𝑆)
2118, 19, 20syl2anc 693 . . . . 5 (𝜑 → (𝑆 𝑆) = 𝑆)
2221breq2d 4665 . . . 4 (𝜑 → (𝐶 (𝑆 𝑆) ↔ 𝐶 𝑆))
2317, 22sylibd 229 . . 3 (𝜑 → (𝑃 = 𝑆𝐶 𝑆))
2423necon3bd 2808 . 2 (𝜑 → (¬ 𝐶 𝑆𝑃𝑆))
2513, 24mpd 15 1 (𝜑𝑃𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1037   = wceq 1483  wcel 1990  wne 2794   class class class wbr 4653  cfv 5888  (class class class)co 6650  Basecbs 15857  lecple 15948  joincjn 16944  Latclat 17045  Atomscatm 34550  HLchlt 34637  LPlanesclpl 34778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-lat 17046  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638
This theorem is referenced by:  dalempjsen  34939  dalem24  34983
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