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Theorem dalem41 35514
Description: Lemma for dath 35537. (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 35424 . . . 4 (𝜑𝐾 ∈ HL)
323ad2ant1 1127 . . 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 35502 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
14 dalem38.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
151, 4, 5, 6, 7, 8, 9, 10, 11, 14dalem34 35507 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
16 dalem38.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
171, 4, 5, 6, 7, 8, 9, 10, 11, 16dalem23 35497 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
181, 4, 5, 6, 7, 8, 9, 10, 11, 16, 12, 14dalem39 35512 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐻 (𝐼 𝐺))
194, 5, 6atnlej2 35181 . . 3 ((𝐾 ∈ HL ∧ (𝐻𝐴𝐼𝐴𝐺𝐴) ∧ ¬ 𝐻 (𝐼 𝐺)) → 𝐻𝐺)
203, 13, 15, 17, 18, 19syl131anc 1489 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐺)
2120necomd 2998 1 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐻)
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 6029  (class class class)co 6791  Basecbs 16057  lecple 16149  joincjn 17145  meetcmee 17146  Atomscatm 35065  HLchlt 35152  LPlanesclpl 35293
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 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-lat 17247  df-clat 17309  df-oposet 34978  df-ol 34980  df-oml 34981  df-covers 35068  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153  df-llines 35299  df-lplanes 35300  df-lvols 35301
This theorem is referenced by:  dalem42  35515  dalem54  35527
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