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Theorem cdlemefs44 37681
Description: Value of fs(r) when r is an atom under pq and s is any atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefs45 37684 instead TODO: FIX COMMENT. (Contributed by NM, 31-Mar-2013.)
Hypotheses
Ref Expression
cdlemef44.b 𝐵 = (Base‘𝐾)
cdlemef44.l = (le‘𝐾)
cdlemef44.j = (join‘𝐾)
cdlemef44.m = (meet‘𝐾)
cdlemef44.a 𝐴 = (Atoms‘𝐾)
cdlemef44.h 𝐻 = (LHyp‘𝐾)
cdlemef44.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemef44.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdlemef44.o 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), 𝐼, 𝑠 / 𝑡𝐷) (𝑥 𝑊))))
cdlemef44.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
cdlemefs44.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemefs44.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
Assertion
Ref Expression
cdlemefs44 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐹𝑅) = 𝑅 / 𝑠𝑆 / 𝑡𝐸)
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,𝐴   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧   𝑦,𝐸   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   𝑥,𝐼,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑅,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠   𝑆,𝑠,𝑡,𝑦
Allowed substitution hints:   𝐷(𝑡)   𝑆(𝑥,𝑧)   𝐸(𝑥,𝑧,𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐼(𝑦,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑠)

Proof of Theorem cdlemefs44
StepHypRef Expression
1 cdlemef44.b . . 3 𝐵 = (Base‘𝐾)
2 cdlemef44.l . . 3 = (le‘𝐾)
3 cdlemef44.j . . 3 = (join‘𝐾)
4 cdlemef44.m . . 3 = (meet‘𝐾)
5 cdlemef44.a . . 3 𝐴 = (Atoms‘𝐾)
6 cdlemef44.h . . 3 𝐻 = (LHyp‘𝐾)
7 cdlemef44.u . . 3 𝑈 = ((𝑃 𝑄) 𝑊)
8 cdlemef44.d . . 3 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
9 cdlemefs44.e . . 3 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
10 cdlemefs44.i . . 3 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
11 eqid 2822 . . 3 if(𝑠 (𝑃 𝑄), 𝐼, 𝑠 / 𝑡𝐷) = if(𝑠 (𝑃 𝑄), 𝐼, 𝑠 / 𝑡𝐷)
12 cdlemef44.o . . 3 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), 𝐼, 𝑠 / 𝑡𝐷) (𝑥 𝑊))))
13 cdlemef44.f . . 3 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
14 eqid 2822 . . 3 ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))) = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
15 eqid 2822 . . 3 ((𝑃 𝑄) (((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))) ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))) ((𝑅 𝑆) 𝑊)))
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15cdlemefs31fv1 37679 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐹𝑅) = ((𝑃 𝑄) (((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))) ((𝑅 𝑆) 𝑊))))
17 simp22l 1289 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅𝐴)
18 simp23l 1291 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑆𝐴)
198, 9, 14, 15cdleme31sde 37640 . . 3 ((𝑅𝐴𝑆𝐴) → 𝑅 / 𝑠𝑆 / 𝑡𝐸 = ((𝑃 𝑄) (((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))) ((𝑅 𝑆) 𝑊))))
2017, 18, 19syl2anc 587 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅 / 𝑠𝑆 / 𝑡𝐸 = ((𝑃 𝑄) (((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))) ((𝑅 𝑆) 𝑊))))
2116, 20eqtr4d 2860 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐹𝑅) = 𝑅 / 𝑠𝑆 / 𝑡𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011  wral 3130  csb 3855  ifcif 4439   class class class wbr 5042  cmpt 5122  cfv 6334  crio 7097  (class class class)co 7140  Basecbs 16474  lecple 16563  joincjn 17545  meetcmee 17546  Atomscatm 36518  HLchlt 36605  LHypclh 37239
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-riotaBAD 36208
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-undef 7926  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-p1 17641  df-lat 17647  df-clat 17709  df-oposet 36431  df-ol 36433  df-oml 36434  df-covers 36521  df-ats 36522  df-atl 36553  df-cvlat 36577  df-hlat 36606  df-llines 36753  df-lplanes 36754  df-lvols 36755  df-lines 36756  df-psubsp 36758  df-pmap 36759  df-padd 37051  df-lhyp 37243
This theorem is referenced by:  cdlemefs45  37684
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