Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme8tN Structured version   Visualization version   GIF version

Theorem cdleme8tN 36329
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. 𝑋 represents t1. In their notation, we prove p t1 = p t. (Contributed by NM, 8-Oct-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme8t.l = (le‘𝐾)
cdleme8t.j = (join‘𝐾)
cdleme8t.m = (meet‘𝐾)
cdleme8t.a 𝐴 = (Atoms‘𝐾)
cdleme8t.h 𝐻 = (LHyp‘𝐾)
cdleme8t.x 𝑋 = ((𝑃 𝑇) 𝑊)
Assertion
Ref Expression
cdleme8tN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑇𝐴) → (𝑃 𝑋) = (𝑃 𝑇))

Proof of Theorem cdleme8tN
StepHypRef Expression
1 cdleme8t.l . 2 = (le‘𝐾)
2 cdleme8t.j . 2 = (join‘𝐾)
3 cdleme8t.m . 2 = (meet‘𝐾)
4 cdleme8t.a . 2 𝐴 = (Atoms‘𝐾)
5 cdleme8t.h . 2 𝐻 = (LHyp‘𝐾)
6 cdleme8t.x . 2 𝑋 = ((𝑃 𝑇) 𝑊)
71, 2, 3, 4, 5, 6cdleme8 36324 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑇𝐴) → (𝑃 𝑋) = (𝑃 𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164   class class class wbr 4875  cfv 6127  (class class class)co 6910  lecple 16319  joincjn 17304  meetcmee 17305  Atomscatm 35337  HLchlt 35424  LHypclh 36058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-p1 17400  df-lat 17406  df-clat 17468  df-oposet 35250  df-ol 35252  df-oml 35253  df-covers 35340  df-ats 35341  df-atl 35372  df-cvlat 35396  df-hlat 35425  df-psubsp 35577  df-pmap 35578  df-padd 35870  df-lhyp 36062
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator