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Theorem cdleme26f 38356
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝑁 represent f(t), ft(s) respectively. If t t v, then ft(s) f(t) v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.)
Hypotheses
Ref Expression
cdleme26.b 𝐵 = (Base‘𝐾)
cdleme26.l = (le‘𝐾)
cdleme26.j = (join‘𝐾)
cdleme26.m = (meet‘𝐾)
cdleme26.a 𝐴 = (Atoms‘𝐾)
cdleme26.h 𝐻 = (LHyp‘𝐾)
cdleme26f.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme26f.f 𝐹 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme26f.n 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑡) 𝑊)))
cdleme26f.i 𝐼 = (𝑢𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑢 = 𝑁))
Assertion
Ref Expression
cdleme26f ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐼 (𝐹 𝑉))
Distinct variable groups:   𝑢,𝑡,𝐴   𝑡,𝐵,𝑢   𝑡,𝐻   𝑡, ,𝑢   𝑡,𝐾   𝑡, ,𝑢   𝑡, ,𝑢   𝑢,𝑁   𝑡,𝑃,𝑢   𝑡,𝑄,𝑢   𝑡,𝑆,𝑢   𝑡,𝑈,𝑢   𝑡,𝑊,𝑢
Allowed substitution hints:   𝐹(𝑢,𝑡)   𝐻(𝑢)   𝐼(𝑢,𝑡)   𝐾(𝑢)   𝑁(𝑡)   𝑉(𝑢,𝑡)

Proof of Theorem cdleme26f
StepHypRef Expression
1 simp11 1201 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp21 1204 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simp22 1205 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simp23l 1292 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑆𝐴)
5 simp23r 1293 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ¬ 𝑆 𝑊)
6 simp12l 1284 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑃𝑄)
7 simp12r 1285 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑆 (𝑃 𝑄))
8 cdleme26.b . . . . 5 𝐵 = (Base‘𝐾)
9 cdleme26.l . . . . 5 = (le‘𝐾)
10 cdleme26.j . . . . 5 = (join‘𝐾)
11 cdleme26.m . . . . 5 = (meet‘𝐾)
12 cdleme26.a . . . . 5 𝐴 = (Atoms‘𝐾)
13 cdleme26.h . . . . 5 𝐻 = (LHyp‘𝐾)
14 cdleme26f.u . . . . 5 𝑈 = ((𝑃 𝑄) 𝑊)
15 cdleme26f.f . . . . 5 𝐹 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
16 cdleme26f.n . . . . 5 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑡) 𝑊)))
17 cdleme26f.i . . . . 5 𝐼 = (𝑢𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑢 = 𝑁))
188, 9, 10, 11, 12, 13, 14, 15, 16, 17cdleme25cl 38350 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑃𝑄𝑆 (𝑃 𝑄))) → 𝐼𝐵)
191, 2, 3, 4, 5, 6, 7, 18syl322anc 1396 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐼𝐵)
20 simp13l 1286 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑡𝐴)
21 simp13r 1287 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ¬ 𝑡 𝑊)
22 simp31 1207 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ¬ 𝑡 (𝑃 𝑄))
238fvexi 6782 . . . 4 𝐵 ∈ V
2423, 17riotasv 36952 . . 3 ((𝐼𝐵𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝐼 = 𝑁)
2519, 20, 21, 22, 24syl112anc 1372 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐼 = 𝑁)
26 simp23 1206 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
27 simp33 1209 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑉𝐴𝑉 𝑊))
28 simp32 1208 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑆𝑡𝑆 (𝑡 𝑉)))
299, 10, 11, 12, 13, 14, 15, 16cdleme22f 38339 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑡𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉))) → 𝑁 (𝐹 𝑉))
301, 2, 3, 26, 20, 27, 28, 29syl331anc 1393 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑁 (𝐹 𝑉))
3125, 30eqbrtrd 5100 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐼 (𝐹 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1085   = wceq 1541  wcel 2109  wne 2944  wral 3065   class class class wbr 5078  cfv 6430  crio 7224  (class class class)co 7268  Basecbs 16893  lecple 16950  joincjn 18010  meetcmee 18011  Atomscatm 37256  HLchlt 37343  LHypclh 37977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-riotaBAD 36946
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-iin 4932  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-undef 8073  df-proset 17994  df-poset 18012  df-plt 18029  df-lub 18045  df-glb 18046  df-join 18047  df-meet 18048  df-p0 18124  df-p1 18125  df-lat 18131  df-clat 18198  df-oposet 37169  df-ol 37171  df-oml 37172  df-covers 37259  df-ats 37260  df-atl 37291  df-cvlat 37315  df-hlat 37344  df-llines 37491  df-lplanes 37492  df-lvols 37493  df-lines 37494  df-psubsp 37496  df-pmap 37497  df-padd 37789  df-lhyp 37981
This theorem is referenced by:  cdleme27a  38360
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