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Theorem cdleme40w 37610
Description: Part of proof of Lemma E in [Crawley] p. 113. Apply cdleme40v 37609 bound variable change to 𝑆 / 𝑢𝑉. TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013.)
Hypotheses
Ref Expression
cdleme40.b 𝐵 = (Base‘𝐾)
cdleme40.l = (le‘𝐾)
cdleme40.j = (join‘𝐾)
cdleme40.m = (meet‘𝐾)
cdleme40.a 𝐴 = (Atoms‘𝐾)
cdleme40.h 𝐻 = (LHyp‘𝐾)
cdleme40.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme40.e 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme40.g 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
cdleme40.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
cdleme40.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
cdleme40.d 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme40r.y 𝑌 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))
Assertion
Ref Expression
cdleme40w ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑅 / 𝑠𝑁𝑆 / 𝑠𝑁)
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,   𝑢,   𝑢,   𝑢,𝑃   𝑢,𝑄   𝑢,𝑆   𝑢,𝑊   𝑡,𝑠,𝑦,𝐴   𝐵,𝑠,𝑡,𝑦   𝐸,𝑠   𝑡,𝐻,𝑦   ,𝑠,𝑡,𝑦   𝑡,𝐾,𝑦   ,𝑠,𝑡,𝑦   ,𝑠,𝑡,𝑦   𝑃,𝑠,𝑡,𝑦   𝑄,𝑠,𝑡,𝑦   𝑅,𝑠,𝑡,𝑦   𝑡,𝑈,𝑦   𝑊,𝑠,𝑡,𝑦   𝑦,𝑌   𝑡,𝑆,𝑦   𝑦,𝐸   𝑢,𝑁   𝑆,𝑠,𝑢   𝑈,𝑠,𝑢,𝑡,𝑦
Allowed substitution hints:   𝐷(𝑦,𝑢,𝑡,𝑠)   𝑅(𝑢)   𝐸(𝑢,𝑡)   𝐺(𝑦,𝑢,𝑡,𝑠)   𝐻(𝑢,𝑠)   𝐼(𝑦,𝑢,𝑡,𝑠)   𝐾(𝑢,𝑠)   𝑁(𝑦,𝑡,𝑠)   𝑌(𝑢,𝑡,𝑠)

Proof of Theorem cdleme40w
Dummy variables 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdleme40.b . . 3 𝐵 = (Base‘𝐾)
2 cdleme40.l . . 3 = (le‘𝐾)
3 cdleme40.j . . 3 = (join‘𝐾)
4 cdleme40.m . . 3 = (meet‘𝐾)
5 cdleme40.a . . 3 𝐴 = (Atoms‘𝐾)
6 cdleme40.h . . 3 𝐻 = (LHyp‘𝐾)
7 cdleme40.u . . 3 𝑈 = ((𝑃 𝑄) 𝑊)
8 cdleme40.e . . 3 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
9 cdleme40.g . . 3 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
10 cdleme40.i . . 3 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
11 cdleme40.n . . 3 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
12 eqid 2824 . . 3 ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊)))
13 eqid 2824 . . 3 (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊))))) = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊)))))
14 eqid 2824 . . 3 ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) = ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊)))
15 eqid 2824 . . 3 ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑆 𝑣) 𝑊))) = ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑆 𝑣) 𝑊)))
16 eqid 2824 . . 3 ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))) = ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊)))
17 eqid 2824 . . 3 (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))))) = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊)))))
18 eqid 2824 . . 3 if(𝑢 (𝑃 𝑄), (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))))), ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))) = if(𝑢 (𝑃 𝑄), (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))))), ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊))))
19 eqid 2824 . . 3 (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑆 𝑣) 𝑊))))) = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑆 𝑣) 𝑊)))))
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19cdleme40n 37608 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑅 / 𝑠𝑁𝑆 / 𝑢if(𝑢 (𝑃 𝑄), (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))))), ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))))
21 simp23l 1290 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑆𝐴)
22 cdleme40.d . . . 4 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
23 eqid 2824 . . . 4 ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊))) = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 14, 16, 17, 18cdleme40v 37609 . . 3 (𝑆𝐴𝑆 / 𝑠𝑁 = 𝑆 / 𝑢if(𝑢 (𝑃 𝑄), (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))))), ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))))
2521, 24syl 17 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑆 / 𝑠𝑁 = 𝑆 / 𝑢if(𝑢 (𝑃 𝑄), (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))))), ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))))
2620, 25neeqtrrd 3093 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑅 / 𝑠𝑁𝑆 / 𝑠𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  w3a 1083   = wceq 1536  wcel 2113  wne 3019  wral 3141  csb 3886  ifcif 4470   class class class wbr 5069  cfv 6358  crio 7116  (class class class)co 7159  Basecbs 16486  lecple 16575  joincjn 17557  meetcmee 17558  Atomscatm 36403  HLchlt 36490  LHypclh 37124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-riotaBAD 36093
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-undef 7942  df-proset 17541  df-poset 17559  df-plt 17571  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-p0 17652  df-p1 17653  df-lat 17659  df-clat 17721  df-oposet 36316  df-ol 36318  df-oml 36319  df-covers 36406  df-ats 36407  df-atl 36438  df-cvlat 36462  df-hlat 36491  df-llines 36638  df-lplanes 36639  df-lvols 36640  df-lines 36641  df-psubsp 36643  df-pmap 36644  df-padd 36936  df-lhyp 37128
This theorem is referenced by:  cdleme41snaw  37616
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