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Theorem dalem61 34838
 Description: Lemma for dath 34841. Show that atoms 𝐷, 𝐸, and 𝐹 lie on the same line (axis of perspectivity). Eliminate hypotheses containing dummy atoms 𝑐 and 𝑑. (Contributed by NM, 11-Aug-2012.)
Hypotheses
Ref Expression
dalem.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)))))
dalem.l = (le‘𝐾)
dalem.j = (join‘𝐾)
dalem.a 𝐴 = (Atoms‘𝐾)
dalem.ps (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
dalem61.m = (meet‘𝐾)
dalem61.o 𝑂 = (LPlanes‘𝐾)
dalem61.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem61.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem61.d 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
dalem61.e 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
dalem61.f 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
Assertion
Ref Expression
dalem61 ((𝜑𝑌 = 𝑍𝜓) → 𝐹 (𝐷 𝐸))

Proof of Theorem dalem61
StepHypRef Expression
1 dalem.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)))))
2 dalem.l . . 3 = (le‘𝐾)
3 dalem.j . . 3 = (join‘𝐾)
4 dalem.a . . 3 𝐴 = (Atoms‘𝐾)
5 dalem.ps . . 3 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
6 dalem61.m . . 3 = (meet‘𝐾)
7 dalem61.o . . 3 𝑂 = (LPlanes‘𝐾)
8 dalem61.y . . 3 𝑌 = ((𝑃 𝑄) 𝑅)
9 dalem61.z . . 3 𝑍 = ((𝑆 𝑇) 𝑈)
10 dalem61.f . . 3 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
11 eqid 2620 . . 3 ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑐 𝑃) (𝑑 𝑆))
12 eqid 2620 . . 3 ((𝑐 𝑄) (𝑑 𝑇)) = ((𝑐 𝑄) (𝑑 𝑇))
13 eqid 2620 . . 3 ((𝑐 𝑅) (𝑑 𝑈)) = ((𝑐 𝑅) (𝑑 𝑈))
14 eqid 2620 . . 3 (((((𝑐 𝑃) (𝑑 𝑆)) ((𝑐 𝑄) (𝑑 𝑇))) ((𝑐 𝑅) (𝑑 𝑈))) 𝑌) = (((((𝑐 𝑃) (𝑑 𝑆)) ((𝑐 𝑄) (𝑑 𝑇))) ((𝑐 𝑅) (𝑑 𝑈))) 𝑌)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem59 34836 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐹 (((((𝑐 𝑃) (𝑑 𝑆)) ((𝑐 𝑄) (𝑑 𝑇))) ((𝑐 𝑅) (𝑑 𝑈))) 𝑌))
16 dalem61.d . . 3 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
17 dalem61.e . . 3 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
181, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 11, 12, 13, 14dalem60 34837 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) = (((((𝑐 𝑃) (𝑑 𝑆)) ((𝑐 𝑄) (𝑑 𝑇))) ((𝑐 𝑅) (𝑑 𝑈))) 𝑌))
1915, 18breqtrrd 4672 1 ((𝜑𝑌 = 𝑍𝜓) → 𝐹 (𝐷 𝐸))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988   ≠ wne 2791   class class class wbr 4644  ‘cfv 5876  (class class class)co 6635  Basecbs 15838  lecple 15929  joincjn 16925  meetcmee 16926  Atomscatm 34369  HLchlt 34456  LPlanesclpl 34597 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-preset 16909  df-poset 16927  df-plt 16939  df-lub 16955  df-glb 16956  df-join 16957  df-meet 16958  df-p0 17020  df-lat 17027  df-clat 17089  df-oposet 34282  df-ol 34284  df-oml 34285  df-covers 34372  df-ats 34373  df-atl 34404  df-cvlat 34428  df-hlat 34457  df-llines 34603  df-lplanes 34604  df-lvols 34605 This theorem is referenced by:  dalem62  34839
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