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Theorem dalem43 34520
Description: Lemma for dath 34541. Planes 𝐺𝐻𝐼 and 𝑌 are different. (Contributed by NM, 8-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
dalem43 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ≠ 𝑌)

Proof of Theorem dalem43
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)))))
21dalemkelat 34429 . . . 4 (𝜑𝐾 ∈ Lat)
323ad2ant1 1080 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
41dalemkehl 34428 . . . . 5 (𝜑𝐾 ∈ HL)
543ad2ant1 1080 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
6 dalem.l . . . . 5 = (le‘𝐾)
7 dalem.j . . . . 5 = (join‘𝐾)
8 dalem.a . . . . 5 𝐴 = (Atoms‘𝐾)
9 dalem.ps . . . . 5 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
10 dalem38.m . . . . 5 = (meet‘𝐾)
11 dalem38.o . . . . 5 𝑂 = (LPlanes‘𝐾)
12 dalem38.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
13 dalem38.z . . . . 5 𝑍 = ((𝑆 𝑇) 𝑈)
14 dalem38.g . . . . 5 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
151, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem23 34501 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
16 dalem38.h . . . . 5 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
171, 6, 7, 8, 9, 10, 11, 12, 13, 16dalem29 34506 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
18 eqid 2621 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
1918, 7, 8hlatjcl 34172 . . . 4 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
205, 15, 17, 19syl3anc 1323 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
21 dalem38.i . . . . 5 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
221, 6, 7, 8, 9, 10, 11, 12, 13, 21dalem34 34511 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2318, 8atbase 34095 . . . 4 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
2422, 23syl 17 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
2518, 6, 7latlej2 17001 . . 3 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → 𝐼 ((𝐺 𝐻) 𝐼))
263, 20, 24, 25syl3anc 1323 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ((𝐺 𝐻) 𝐼))
271, 6, 7, 8, 9, 10, 11, 12, 13, 21dalem35 34512 . 2 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐼 𝑌)
28 nbrne1 4642 . 2 ((𝐼 ((𝐺 𝐻) 𝐼) ∧ ¬ 𝐼 𝑌) → ((𝐺 𝐻) 𝐼) ≠ 𝑌)
2926, 27, 28syl2anc 692 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ≠ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  lecple 15888  joincjn 16884  meetcmee 16885  Latclat 16985  Atomscatm 34069  HLchlt 34156  LPlanesclpl 34297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-preset 16868  df-poset 16886  df-plt 16898  df-lub 16914  df-glb 16915  df-join 16916  df-meet 16917  df-p0 16979  df-lat 16986  df-clat 17048  df-oposet 33982  df-ol 33984  df-oml 33985  df-covers 34072  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157  df-llines 34303  df-lplanes 34304
This theorem is referenced by:  dalem44  34521  dalem51  34528
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