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Theorem cdlemefs27cl 39279
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 39232 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 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 ≀ (𝑃 ∨ 𝑄))
32iftrued 4536 . . 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 1191 . . . 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 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†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
6 simpl3 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†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
7 simpr1 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†’ (𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š))
8 simpr3 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†’ 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 39223 . . . 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 1391 . . 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 2833 . 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 2837 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  ifcif 4528   class class class wbr 5148  β€˜cfv 6543  β„©crio 7363  (class class class)co 7408  Basecbs 17143  lecple 17203  joincjn 18263  meetcmee 18264  Atomscatm 38128  HLchlt 38215  LHypclh 38850
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-llines 38364  df-lplanes 38365  df-lvols 38366  df-lines 38367  df-psubsp 38369  df-pmap 38370  df-padd 38662  df-lhyp 38854
This theorem is referenced by:  cdlemefs29bpre0N  39282  cdlemefs29bpre1N  39283  cdlemefs29cpre1N  39284  cdlemefs29clN  39285  cdlemefs32fvaN  39288  cdlemefs32fva1  39289
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