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Theorem cdleme50trn2 40250
Description: Part of proof that 𝐹 is a translation. Remove 𝑆 hypotheses no longer needed from cdleme50trn2a 40249. TODO: fix comment. (Contributed by NM, 10-Apr-2013.)
Hypotheses
Ref Expression
cdlemef50.b 𝐵 = (Base‘𝐾)
cdlemef50.l = (le‘𝐾)
cdlemef50.j = (join‘𝐾)
cdlemef50.m = (meet‘𝐾)
cdlemef50.a 𝐴 = (Atoms‘𝐾)
cdlemef50.h 𝐻 = (LHyp‘𝐾)
cdlemef50.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemef50.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdlemefs50.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemef50.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))
Assertion
Ref Expression
cdleme50trn2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → ((𝑅 (𝐹𝑅)) 𝑊) = 𝑈)
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,   ,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐴,𝑠,𝑡,𝑥,𝑦,𝑧   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,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
Allowed substitution hints:   𝐷(𝑡)   𝐸(𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)

Proof of Theorem cdleme50trn2
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 simp11 1200 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp12 1201 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simp13 1202 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simp2l 1196 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → 𝑃𝑄)
5 cdlemef50.l . . . 4 = (le‘𝐾)
6 cdlemef50.j . . . 4 = (join‘𝐾)
7 cdlemef50.a . . . 4 𝐴 = (Atoms‘𝐾)
8 cdlemef50.h . . . 4 𝐻 = (LHyp‘𝐾)
95, 6, 7, 8cdlemb2 39740 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑒𝐴𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄)))
101, 2, 3, 4, 9syl121anc 1372 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → ∃𝑒𝐴𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄)))
11 simp1 1133 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄))))) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
12 simp2l 1196 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄))))) → 𝑃𝑄)
13 simp2r 1197 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄))))) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
14 simp3rl 1243 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄))))) → 𝑒𝐴)
15 simprrl 779 . . . . . . . . . 10 ((𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄)))) → ¬ 𝑒 𝑊)
16153ad2ant3 1132 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄))))) → ¬ 𝑒 𝑊)
1714, 16jca 510 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄))))) → (𝑒𝐴 ∧ ¬ 𝑒 𝑊))
18 simp3l 1198 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄))))) → 𝑅 (𝑃 𝑄))
19 simprrr 780 . . . . . . . . 9 ((𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄)))) → ¬ 𝑒 (𝑃 𝑄))
20193ad2ant3 1132 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄))))) → ¬ 𝑒 (𝑃 𝑄))
21 cdlemef50.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
22 cdlemef50.m . . . . . . . . 9 = (meet‘𝐾)
23 cdlemef50.u . . . . . . . . 9 𝑈 = ((𝑃 𝑄) 𝑊)
24 cdlemef50.d . . . . . . . . 9 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
25 cdlemefs50.e . . . . . . . . 9 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
26 cdlemef50.f . . . . . . . . 9 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))
2721, 5, 6, 22, 7, 8, 23, 24, 25, 26cdleme50trn2a 40249 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑒𝐴 ∧ ¬ 𝑒 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑒 (𝑃 𝑄))) → ((𝑅 (𝐹𝑅)) 𝑊) = 𝑈)
2811, 12, 13, 17, 18, 20, 27syl132anc 1385 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄))))) → ((𝑅 (𝐹𝑅)) 𝑊) = 𝑈)
29283exp 1116 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝑅 (𝑃 𝑄) ∧ (𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄)))) → ((𝑅 (𝐹𝑅)) 𝑊) = 𝑈)))
3029exp4a 430 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 (𝑃 𝑄) → ((𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄))) → ((𝑅 (𝐹𝑅)) 𝑊) = 𝑈))))
31303imp 1108 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → ((𝑒𝐴 ∧ (¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄))) → ((𝑅 (𝐹𝑅)) 𝑊) = 𝑈))
3231expd 414 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑒𝐴 → ((¬ 𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄)) → ((𝑅 (𝐹𝑅)) 𝑊) = 𝑈)))
3332rexlimdv 3143 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (∃𝑒𝐴𝑒 𝑊 ∧ ¬ 𝑒 (𝑃 𝑄)) → ((𝑅 (𝐹𝑅)) 𝑊) = 𝑈))
3410, 33mpd 15 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → ((𝑅 (𝐹𝑅)) 𝑊) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  csb 3892  ifcif 4533   class class class wbr 5153  cmpt 5236  cfv 6554  crio 7379  (class class class)co 7424  Basecbs 17213  lecple 17273  joincjn 18336  meetcmee 18337  Atomscatm 38961  HLchlt 39048  LHypclh 39683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-riotaBAD 38651
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-undef 8288  df-proset 18320  df-poset 18338  df-plt 18355  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-p0 18450  df-p1 18451  df-lat 18457  df-clat 18524  df-oposet 38874  df-ol 38876  df-oml 38877  df-covers 38964  df-ats 38965  df-atl 38996  df-cvlat 39020  df-hlat 39049  df-llines 39197  df-lplanes 39198  df-lvols 39199  df-lines 39200  df-psubsp 39202  df-pmap 39203  df-padd 39495  df-lhyp 39687
This theorem is referenced by:  cdleme50trn12  40251
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