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Theorem cdleme40w 36544
 Description: Part of proof of Lemma E in [Crawley] p. 113. Apply cdleme40v 36543 bound variable change to ⦋𝑆 / 𝑢⦌𝑉. 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 2825 . . 3 ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊)))
13 eqid 2825 . . 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 2825 . . 3 ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))) = ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊)))
15 eqid 2825 . . 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 2825 . . 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 2825 . . 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 2825 . . 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 2825 . . 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 36542 . 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 1397 . . 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 2825 . . . 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 36543 . . 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 3073 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 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∀wral 3117  ⦋csb 3757  ifcif 4308   class class class wbr 4875  ‘cfv 6127  ℩crio 6870  (class class class)co 6910  Basecbs 16229  lecple 16319  joincjn 17304  meetcmee 17305  Atomscatm 35337  HLchlt 35424  LHypclh 36058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-riotaBAD 35027 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-undef 7669  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-p1 17400  df-lat 17406  df-clat 17468  df-oposet 35250  df-ol 35252  df-oml 35253  df-covers 35340  df-ats 35341  df-atl 35372  df-cvlat 35396  df-hlat 35425  df-llines 35572  df-lplanes 35573  df-lvols 35574  df-lines 35575  df-psubsp 35577  df-pmap 35578  df-padd 35870  df-lhyp 36062 This theorem is referenced by:  cdleme41snaw  36550
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