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Theorem dalem34 33806
Description: Lemma for dath 33836. Analogue of dalem23 33796 for 𝐼. (Contributed by NM, 2-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
dalem34 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)

Proof of Theorem dalem34
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 33757 . . 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 1074 . 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 33758 . . 3 ((𝜑𝑌 = 𝑍) → ((𝑄 𝑅) 𝑃) = ((𝑇 𝑈) 𝑆))
1093adant3 1073 . 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 33791 . . 3 ((𝜑𝜓) → ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))))
13123adant2 1072 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))))
14 biid 249 . . 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 249 . . 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 2609 . . 3 ((𝑄 𝑅) 𝑃) = ((𝑄 𝑅) 𝑃)
19 eqid 2609 . . 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, 20dalem29 33801 . 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)))) → 𝐼𝐴)
228, 10, 13, 21syl3anc 1317 1 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  meetcmee 16714  Atomscatm 33364  HLchlt 33451  LPlanesclpl 33592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-lat 16815  df-clat 16877  df-oposet 33277  df-ol 33279  df-oml 33280  df-covers 33367  df-ats 33368  df-atl 33399  df-cvlat 33423  df-hlat 33452  df-llines 33598  df-lplanes 33599
This theorem is referenced by:  dalem38  33810  dalem39  33811  dalem41  33813  dalem42  33814  dalem43  33815  dalem44  33816  dalem45  33817  dalem51  33823  dalem52  33824  dalem54  33826  dalem55  33827  dalem58  33830  dalem59  33831
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