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Theorem cdleme41sn4aw 40000
Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(r) is for on and off 𝑃 ∨ 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013.)
Hypotheses
Ref Expression
cdleme41.b 𝐡 = (Baseβ€˜πΎ)
cdleme41.l ≀ = (leβ€˜πΎ)
cdleme41.j ∨ = (joinβ€˜πΎ)
cdleme41.m ∧ = (meetβ€˜πΎ)
cdleme41.a 𝐴 = (Atomsβ€˜πΎ)
cdleme41.h 𝐻 = (LHypβ€˜πΎ)
cdleme41.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme41.d 𝐷 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
cdleme41.e 𝐸 = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
cdleme41.g 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑑) ∧ π‘Š)))
cdleme41.i 𝐼 = (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑦 = 𝐺))
cdleme41.n 𝑁 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
Assertion
Ref Expression
cdleme41sn4aw ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑆)) β†’ ⦋𝑅 / π‘ β¦Œπ‘ β‰  ⦋𝑆 / π‘ β¦Œπ‘)
Distinct variable groups:   𝐴,𝑠   ∨ ,𝑠   ≀ ,𝑠   ∧ ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑆,𝑠   π‘ˆ,𝑠   π‘Š,𝑠   𝑦,𝑑,𝐴,𝑠   𝐡,𝑠,𝑑,𝑦   𝑦,𝐷   𝑦,𝐺   𝐸,𝑠,𝑦   𝐻,𝑠,𝑑,𝑦   𝑑, ∨ ,𝑦   𝐾,𝑠,𝑑,𝑦   𝑑, ≀ ,𝑦   𝑑, ∧ ,𝑦   𝑑,𝑃,𝑦   𝑑,𝑄,𝑦   𝑑,𝑅,𝑦   𝑑,𝑆,𝑦   𝑑,π‘ˆ,𝑦   𝑑,π‘Š,𝑦
Allowed substitution hints:   𝐷(𝑑,𝑠)   𝐸(𝑑)   𝐺(𝑑,𝑠)   𝐼(𝑦,𝑑,𝑠)   𝑁(𝑦,𝑑,𝑠)

Proof of Theorem cdleme41sn4aw
StepHypRef Expression
1 simp1 1133 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑆)) β†’ ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)))
2 simp21 1203 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑆)) β†’ 𝑃 β‰  𝑄)
3 simp23 1205 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑆)) β†’ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š))
4 simp22 1204 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑆)) β†’ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))
5 simp32 1207 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑆)) β†’ 𝑆 ≀ (𝑃 ∨ 𝑄))
6 simp31 1206 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑆)) β†’ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))
7 simp33 1208 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑆)) β†’ 𝑅 β‰  𝑆)
87necomd 2986 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑆)) β†’ 𝑆 β‰  𝑅)
9 cdleme41.b . . . 4 𝐡 = (Baseβ€˜πΎ)
10 cdleme41.l . . . 4 ≀ = (leβ€˜πΎ)
11 cdleme41.j . . . 4 ∨ = (joinβ€˜πΎ)
12 cdleme41.m . . . 4 ∧ = (meetβ€˜πΎ)
13 cdleme41.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
14 cdleme41.h . . . 4 𝐻 = (LHypβ€˜πΎ)
15 cdleme41.u . . . 4 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
16 cdleme41.d . . . 4 𝐷 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
17 cdleme41.e . . . 4 𝐸 = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
18 cdleme41.g . . . 4 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑑) ∧ π‘Š)))
19 cdleme41.i . . . 4 𝐼 = (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑦 = 𝐺))
20 cdleme41.n . . . 4 𝑁 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
219, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20cdleme41sn3aw 39999 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) ∧ (𝑆 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 β‰  𝑅)) β†’ ⦋𝑆 / π‘ β¦Œπ‘ β‰  ⦋𝑅 / π‘ β¦Œπ‘)
221, 2, 3, 4, 5, 6, 8, 21syl133anc 1390 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑆)) β†’ ⦋𝑆 / π‘ β¦Œπ‘ β‰  ⦋𝑅 / π‘ β¦Œπ‘)
2322necomd 2986 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑆)) β†’ ⦋𝑅 / π‘ β¦Œπ‘ β‰  ⦋𝑆 / π‘ β¦Œπ‘)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  β¦‹csb 3886  ifcif 4525   class class class wbr 5144  β€˜cfv 6543  β„©crio 7368  (class class class)co 7413  Basecbs 17174  lecple 17234  joincjn 18297  meetcmee 18298  Atomscatm 38787  HLchlt 38874  LHypclh 39509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-riotaBAD 38477
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-iin 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-undef 8272  df-proset 18281  df-poset 18299  df-plt 18316  df-lub 18332  df-glb 18333  df-join 18334  df-meet 18335  df-p0 18411  df-p1 18412  df-lat 18418  df-clat 18485  df-oposet 38700  df-ol 38702  df-oml 38703  df-covers 38790  df-ats 38791  df-atl 38822  df-cvlat 38846  df-hlat 38875  df-llines 39023  df-lplanes 39024  df-lvols 39025  df-lines 39026  df-psubsp 39028  df-pmap 39029  df-padd 39321  df-lhyp 39513
This theorem is referenced by:  cdleme41snaw  40001
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