Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemefs27cl Structured version   Visualization version   GIF version

Theorem cdlemefs27cl 38905
Description: Part of proof of Lemma E in [Crawley] p. 113. Closure of 𝑁. TODO FIX COMMENT This is the start of a re-proof of cdleme27cl 38858 etc. with the 𝑠 ≀ (𝑃 ∨ 𝑄) condition (so as to not have the 𝐢 hypothesis). (Contributed by NM, 24-Mar-2013.)
Hypotheses
Ref Expression
cdlemefs26.b 𝐡 = (Baseβ€˜πΎ)
cdlemefs26.l ≀ = (leβ€˜πΎ)
cdlemefs26.j ∨ = (joinβ€˜πΎ)
cdlemefs26.m ∧ = (meetβ€˜πΎ)
cdlemefs26.a 𝐴 = (Atomsβ€˜πΎ)
cdlemefs26.h 𝐻 = (LHypβ€˜πΎ)
cdlemefs27.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdlemefs27.d 𝐷 = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
cdlemefs27.e 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑑) ∧ π‘Š)))
cdlemefs27.i 𝐼 = (℩𝑒 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝐸))
cdlemefs27.n 𝑁 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐼, 𝐢)
Assertion
Ref Expression
cdlemefs27cl ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ 𝑠 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ 𝑁 ∈ 𝐡)
Distinct variable groups:   𝑒,𝑑,𝐴   𝑑,𝐡,𝑒   𝑒,𝐸   𝑑,𝐻   𝑑, ∨ ,𝑒   𝑑,𝐾   𝑑, ≀ ,𝑒   𝑑, ∧ ,𝑒   𝑑,𝑃,𝑒   𝑑,𝑄,𝑒   𝑑,π‘ˆ,𝑒   𝑑,π‘Š,𝑒   𝑑,𝑠,𝑒
Allowed substitution hints:   𝐴(𝑠)   𝐡(𝑠)   𝐢(𝑒,𝑑,𝑠)   𝐷(𝑒,𝑑,𝑠)   𝑃(𝑠)   𝑄(𝑠)   π‘ˆ(𝑠)   𝐸(𝑑,𝑠)   𝐻(𝑒,𝑠)   𝐼(𝑒,𝑑,𝑠)   ∨ (𝑠)   𝐾(𝑒,𝑠)   ≀ (𝑠)   ∧ (𝑠)   𝑁(𝑒,𝑑,𝑠)   π‘Š(𝑠)

Proof of Theorem cdlemefs27cl
StepHypRef Expression
1 cdlemefs27.n . 2 𝑁 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐼, 𝐢)
2 simpr2 1196 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ 𝑠 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ 𝑠 ≀ (𝑃 ∨ 𝑄))
32iftrued 4499 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ 𝑠 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐼, 𝐢) = 𝐼)
4 simpl1 1192 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ 𝑠 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
5 simpl2 1193 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ 𝑠 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
6 simpl3 1194 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ 𝑠 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
7 simpr1 1195 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ 𝑠 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ (𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š))
8 simpr3 1197 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ 𝑠 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ 𝑃 β‰  𝑄)
9 cdlemefs26.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
10 cdlemefs26.l . . . . 5 ≀ = (leβ€˜πΎ)
11 cdlemefs26.j . . . . 5 ∨ = (joinβ€˜πΎ)
12 cdlemefs26.m . . . . 5 ∧ = (meetβ€˜πΎ)
13 cdlemefs26.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
14 cdlemefs26.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
15 cdlemefs27.u . . . . 5 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
16 cdlemefs27.d . . . . 5 𝐷 = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
17 cdlemefs27.e . . . . 5 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑑) ∧ π‘Š)))
18 cdlemefs27.i . . . . 5 𝐼 = (℩𝑒 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝐸))
199, 10, 11, 12, 13, 14, 15, 16, 17, 18cdleme25cl 38849 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑠 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐼 ∈ 𝐡)
204, 5, 6, 7, 8, 2, 19syl312anc 1392 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ 𝑠 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ 𝐼 ∈ 𝐡)
213, 20eqeltrd 2838 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ 𝑠 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐼, 𝐢) ∈ 𝐡)
221, 21eqeltrid 2842 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ 𝑠 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ 𝑁 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  ifcif 4491   class class class wbr 5110  β€˜cfv 6501  β„©crio 7317  (class class class)co 7362  Basecbs 17090  lecple 17147  joincjn 18207  meetcmee 18208  Atomscatm 37754  HLchlt 37841  LHypclh 38476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-proset 18191  df-poset 18209  df-plt 18226  df-lub 18242  df-glb 18243  df-join 18244  df-meet 18245  df-p0 18321  df-p1 18322  df-lat 18328  df-clat 18395  df-oposet 37667  df-ol 37669  df-oml 37670  df-covers 37757  df-ats 37758  df-atl 37789  df-cvlat 37813  df-hlat 37842  df-llines 37990  df-lplanes 37991  df-lvols 37992  df-lines 37993  df-psubsp 37995  df-pmap 37996  df-padd 38288  df-lhyp 38480
This theorem is referenced by:  cdlemefs29bpre0N  38908  cdlemefs29bpre1N  38909  cdlemefs29cpre1N  38910  cdlemefs29clN  38911  cdlemefs32fvaN  38914  cdlemefs32fva1  38915
  Copyright terms: Public domain W3C validator