Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem23 Structured version   Visualization version   GIF version

Theorem dalem23 33783
Description: Lemma for dath 33823. Show that auxiliary atom 𝐺 is an atom. (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))))
dalem23.m = (meet‘𝐾)
dalem23.o 𝑂 = (LPlanes‘𝐾)
dalem23.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem23.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem23.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
Assertion
Ref Expression
dalem23 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)

Proof of Theorem dalem23
StepHypRef Expression
1 dalem23.g . 2 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
2 dalem.ph . . . . . . . 8 (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)))))
32dalemkehl 33710 . . . . . . 7 (𝜑𝐾 ∈ HL)
43adantr 479 . . . . . 6 ((𝜑𝜓) → 𝐾 ∈ HL)
5 dalem.ps . . . . . . . 8 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
65dalemccea 33770 . . . . . . 7 (𝜓𝑐𝐴)
76adantl 480 . . . . . 6 ((𝜑𝜓) → 𝑐𝐴)
82dalempea 33713 . . . . . . 7 (𝜑𝑃𝐴)
98adantr 479 . . . . . 6 ((𝜑𝜓) → 𝑃𝐴)
105dalemddea 33771 . . . . . . 7 (𝜓𝑑𝐴)
1110adantl 480 . . . . . 6 ((𝜑𝜓) → 𝑑𝐴)
122dalemsea 33716 . . . . . . 7 (𝜑𝑆𝐴)
1312adantr 479 . . . . . 6 ((𝜑𝜓) → 𝑆𝐴)
14 dalem.j . . . . . . 7 = (join‘𝐾)
15 dalem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
1614, 15hlatj4 33461 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑐𝐴𝑃𝐴) ∧ (𝑑𝐴𝑆𝐴)) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑐 𝑑) (𝑃 𝑆)))
174, 7, 9, 11, 13, 16syl122anc 1326 . . . . 5 ((𝜑𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑐 𝑑) (𝑃 𝑆)))
18173adant2 1072 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) = ((𝑐 𝑑) (𝑃 𝑆)))
19 dalem.l . . . . 5 = (le‘𝐾)
20 dalem23.o . . . . 5 𝑂 = (LPlanes‘𝐾)
21 dalem23.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
22 dalem23.z . . . . 5 𝑍 = ((𝑆 𝑇) 𝑈)
232, 19, 14, 15, 5, 20, 21, 22dalem22 33782 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ 𝑂)
2418, 23eqeltrd 2687 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) ∈ 𝑂)
2533ad2ant1 1074 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
262, 19, 14, 15, 20, 21dalemply 33741 . . . . . . . 8 (𝜑𝑃 𝑌)
275dalem-ccly 33772 . . . . . . . 8 (𝜓 → ¬ 𝑐 𝑌)
28 nbrne2 4597 . . . . . . . 8 ((𝑃 𝑌 ∧ ¬ 𝑐 𝑌) → 𝑃𝑐)
2926, 27, 28syl2an 492 . . . . . . 7 ((𝜑𝜓) → 𝑃𝑐)
3029necomd 2836 . . . . . 6 ((𝜑𝜓) → 𝑐𝑃)
31 eqid 2609 . . . . . . 7 (LLines‘𝐾) = (LLines‘𝐾)
3214, 15, 31llni2 33599 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) ∧ 𝑐𝑃) → (𝑐 𝑃) ∈ (LLines‘𝐾))
334, 7, 9, 30, 32syl31anc 1320 . . . . 5 ((𝜑𝜓) → (𝑐 𝑃) ∈ (LLines‘𝐾))
34333adant2 1072 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (LLines‘𝐾))
35103ad2ant3 1076 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
36123ad2ant1 1074 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
372, 19, 14, 15, 22dalemsly 33742 . . . . . . . 8 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
38373adant3 1073 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑆 𝑌)
395dalem-ddly 33773 . . . . . . . 8 (𝜓 → ¬ 𝑑 𝑌)
40393ad2ant3 1076 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
41 nbrne2 4597 . . . . . . 7 ((𝑆 𝑌 ∧ ¬ 𝑑 𝑌) → 𝑆𝑑)
4238, 40, 41syl2anc 690 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝑑)
4342necomd 2836 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝑆)
4414, 15, 31llni2 33599 . . . . 5 (((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) ∧ 𝑑𝑆) → (𝑑 𝑆) ∈ (LLines‘𝐾))
4525, 35, 36, 43, 44syl31anc 1320 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (LLines‘𝐾))
46 dalem23.m . . . . 5 = (meet‘𝐾)
4714, 46, 15, 31, 202llnmj 33647 . . . 4 ((𝐾 ∈ HL ∧ (𝑐 𝑃) ∈ (LLines‘𝐾) ∧ (𝑑 𝑆) ∈ (LLines‘𝐾)) → (((𝑐 𝑃) (𝑑 𝑆)) ∈ 𝐴 ↔ ((𝑐 𝑃) (𝑑 𝑆)) ∈ 𝑂))
4825, 34, 45, 47syl3anc 1317 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝑐 𝑃) (𝑑 𝑆)) ∈ 𝐴 ↔ ((𝑐 𝑃) (𝑑 𝑆)) ∈ 𝑂))
4924, 48mpbird 245 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) ∈ 𝐴)
501, 49syl5eqel 2691 1 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
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 5789  (class class class)co 6526  Basecbs 15643  lecple 15723  joincjn 16715  meetcmee 16716  Atomscatm 33351  HLchlt 33438  LLinesclln 33578  LPlanesclpl 33579
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 4711  ax-pow 4763  ax-pr 4827  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 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-preset 16699  df-poset 16717  df-plt 16729  df-lub 16745  df-glb 16746  df-join 16747  df-meet 16748  df-p0 16810  df-lat 16817  df-clat 16879  df-oposet 33264  df-ol 33266  df-oml 33267  df-covers 33354  df-ats 33355  df-atl 33386  df-cvlat 33410  df-hlat 33439  df-llines 33585  df-lplanes 33586
This theorem is referenced by:  dalem24  33784  dalem27  33786  dalem28  33787  dalem29  33788  dalem38  33797  dalem39  33798  dalem41  33800  dalem42  33801  dalem43  33802  dalem44  33803  dalem45  33804  dalem51  33810  dalem52  33811  dalem54  33813  dalem55  33814  dalem57  33816  dalem58  33817  dalem59  33818
  Copyright terms: Public domain W3C validator