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Theorem dalem37 36963
Description: Lemma for dath 36990. Analogue of dalem28 36954 for 𝐼. (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))))
dalem34.m = (meet‘𝐾)
dalem34.o 𝑂 = (LPlanes‘𝐾)
dalem34.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem34.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem34.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
Assertion
Ref Expression
dalem37 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝐼 𝑐))

Proof of Theorem dalem37
StepHypRef Expression
1 dalem.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)))))
2 dalem.l . . . 4 = (le‘𝐾)
3 dalem.j . . . 4 = (join‘𝐾)
4 dalem.a . . . 4 𝐴 = (Atoms‘𝐾)
5 dalem34.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
6 dalem34.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
71, 2, 3, 4, 5, 6dalemrot 36911 . . 3 (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))))
873ad2ant1 1130 . 2 ((𝜑𝑌 = 𝑍𝜓) → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))))
91, 2, 3, 4, 5, 6dalemrotyz 36912 . . 3 ((𝜑𝑌 = 𝑍) → ((𝑄 𝑅) 𝑃) = ((𝑇 𝑈) 𝑆))
1093adant3 1129 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑄 𝑅) 𝑃) = ((𝑇 𝑈) 𝑆))
11 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
121, 2, 3, 4, 11, 5dalemrotps 36945 . . 3 ((𝜑𝜓) → ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))))
13123adant2 1128 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))))
14 biid 264 . . 3 ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))) ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))))
15 biid 264 . . 3 (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))) ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))))
16 dalem34.m . . 3 = (meet‘𝐾)
17 dalem34.o . . 3 𝑂 = (LPlanes‘𝐾)
18 eqid 2822 . . 3 ((𝑄 𝑅) 𝑃) = ((𝑄 𝑅) 𝑃)
19 eqid 2822 . . 3 ((𝑇 𝑈) 𝑆) = ((𝑇 𝑈) 𝑆)
20 dalem34.i . . 3 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
2114, 2, 3, 4, 15, 16, 17, 18, 19, 20dalem33 36959 . 2 (((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))) ∧ ((𝑄 𝑅) 𝑃) = ((𝑇 𝑈) 𝑆) ∧ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑)))) → 𝑅 (𝐼 𝑐))
228, 10, 13, 21syl3anc 1368 1 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝐼 𝑐))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011   class class class wbr 5042  cfv 6334  (class class class)co 7140  Basecbs 16474  lecple 16563  joincjn 17545  meetcmee 17546  Atomscatm 36517  HLchlt 36604  LPlanesclpl 36746
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  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 36430  df-ol 36432  df-oml 36433  df-covers 36520  df-ats 36521  df-atl 36552  df-cvlat 36576  df-hlat 36605  df-llines 36752  df-lplanes 36753
This theorem is referenced by:  dalem38  36964  dalem44  36970
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