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Theorem dalemqnet 35430
Description: Lemma for dath 35514. 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
dalemqnet (𝜑𝑄𝑇)

Proof of Theorem dalemqnet
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 35402 . . 3 (𝜑𝐾 ∈ Lat)
3 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 35416 . . 3 (𝜑𝐶 ∈ (Base‘𝐾))
51, 3dalemteb 35421 . . 3 (𝜑𝑇 ∈ (Base‘𝐾))
61, 3dalemueb 35422 . . 3 (𝜑𝑈 ∈ (Base‘𝐾))
7 simp322 1416 . . . 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 (𝑇 𝑈))
81, 7sylbi 208 . . 3 (𝜑 → ¬ 𝐶 (𝑇 𝑈))
9 eqid 2804 . . . 4 (Base‘𝐾) = (Base‘𝐾)
10 dalemc.l . . . 4 = (le‘𝐾)
11 dalemc.j . . . 4 = (join‘𝐾)
129, 10, 11latnlej2l 17275 . . 3 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) ∧ ¬ 𝐶 (𝑇 𝑈)) → ¬ 𝐶 𝑇)
132, 4, 5, 6, 8, 12syl131anc 1495 . 2 (𝜑 → ¬ 𝐶 𝑇)
141dalemclqjt 35413 . . . . 5 (𝜑𝐶 (𝑄 𝑇))
15 oveq1 6879 . . . . . 6 (𝑄 = 𝑇 → (𝑄 𝑇) = (𝑇 𝑇))
1615breq2d 4854 . . . . 5 (𝑄 = 𝑇 → (𝐶 (𝑄 𝑇) ↔ 𝐶 (𝑇 𝑇)))
1714, 16syl5ibcom 236 . . . 4 (𝜑 → (𝑄 = 𝑇𝐶 (𝑇 𝑇)))
181dalemkehl 35401 . . . . . 6 (𝜑𝐾 ∈ HL)
191dalemtea 35408 . . . . . 6 (𝜑𝑇𝐴)
2011, 3hlatjidm 35147 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑇𝐴) → (𝑇 𝑇) = 𝑇)
2118, 19, 20syl2anc 575 . . . . 5 (𝜑 → (𝑇 𝑇) = 𝑇)
2221breq2d 4854 . . . 4 (𝜑 → (𝐶 (𝑇 𝑇) ↔ 𝐶 𝑇))
2317, 22sylibd 230 . . 3 (𝜑 → (𝑄 = 𝑇𝐶 𝑇))
2423necon3bd 2990 . 2 (𝜑 → (¬ 𝐶 𝑇𝑄𝑇))
2513, 24mpd 15 1 (𝜑𝑄𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2976   class class class wbr 4842  cfv 6099  (class class class)co 6872  Basecbs 16066  lecple 16158  joincjn 17147  Latclat 17248  Atomscatm 35041  HLchlt 35128  LPlanesclpl 35270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5094  ax-un 7177
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ne 2977  df-ral 3099  df-rex 3100  df-reu 3101  df-rab 3103  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4115  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5217  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-rn 5320  df-res 5321  df-ima 5322  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6833  df-ov 6875  df-oprab 6876  df-proset 17131  df-poset 17149  df-lub 17177  df-glb 17178  df-join 17179  df-meet 17180  df-lat 17249  df-ats 35045  df-atl 35076  df-cvlat 35100  df-hlat 35129
This theorem is referenced by:  dalemcea  35438  dalem2  35439  dalemdnee  35444
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