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Theorem dalem41 36914
Description: Lemma for dath 36937. (Contributed by NM, 4-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))))
dalem38.m = (meet‘𝐾)
dalem38.o 𝑂 = (LPlanes‘𝐾)
dalem38.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem38.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem38.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem38.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem38.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
Assertion
Ref Expression
dalem41 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐻)

Proof of Theorem dalem41
StepHypRef Expression
1 dalem.ph . . . . 5 (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)))))
21dalemkehl 36824 . . . 4 (𝜑𝐾 ∈ HL)
323ad2ant1 1130 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
4 dalem.l . . . 4 = (le‘𝐾)
5 dalem.j . . . 4 = (join‘𝐾)
6 dalem.a . . . 4 𝐴 = (Atoms‘𝐾)
7 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
8 dalem38.m . . . 4 = (meet‘𝐾)
9 dalem38.o . . . 4 𝑂 = (LPlanes‘𝐾)
10 dalem38.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
11 dalem38.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
12 dalem38.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
131, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem29 36902 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
14 dalem38.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
151, 4, 5, 6, 7, 8, 9, 10, 11, 14dalem34 36907 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
16 dalem38.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
171, 4, 5, 6, 7, 8, 9, 10, 11, 16dalem23 36897 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
181, 4, 5, 6, 7, 8, 9, 10, 11, 16, 12, 14dalem39 36912 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐻 (𝐼 𝐺))
194, 5, 6atnlej2 36581 . . 3 ((𝐾 ∈ HL ∧ (𝐻𝐴𝐼𝐴𝐺𝐴) ∧ ¬ 𝐻 (𝐼 𝐺)) → 𝐻𝐺)
203, 13, 15, 17, 18, 19syl131anc 1380 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐺)
2120necomd 3068 1 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐻)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3013   class class class wbr 5049  cfv 6338  (class class class)co 7140  Basecbs 16474  lecple 16563  joincjn 17545  meetcmee 17546  Atomscatm 36464  HLchlt 36551  LPlanesclpl 36693
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-riota 7098  df-ov 7143  df-oprab 7144  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-lat 17647  df-clat 17709  df-oposet 36377  df-ol 36379  df-oml 36380  df-covers 36467  df-ats 36468  df-atl 36499  df-cvlat 36523  df-hlat 36552  df-llines 36699  df-lplanes 36700  df-lvols 36701
This theorem is referenced by:  dalem42  36915  dalem54  36927
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