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Theorem cdleme26fALTN 34668
 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.) (New usage is discouraged.)
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
cdleme26fALTN ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 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 cdleme26fALTN
StepHypRef Expression
1 simp11 1084 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp21 1087 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simp22 1088 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simp23l 1175 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑆𝐴)
5 simp23r 1176 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ¬ 𝑆 𝑊)
6 simp12l 1167 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑃𝑄)
7 simp12r 1168 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 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 34663 . . . 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 1346 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐼𝐵)
20 simp13l 1169 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑡𝐴)
21 simp31 1090 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)))
22 fvex 6113 . . . . 5 (Base‘𝐾) ∈ V
238, 22eqeltri 2684 . . . 4 𝐵 ∈ V
2423, 17riotasv 33263 . . 3 ((𝐼𝐵𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝐼 = 𝑁)
2519, 20, 21, 24syl3anc 1318 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐼 = 𝑁)
26 simp23 1089 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
27 simp33 1092 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑉𝐴𝑉 𝑊))
28 simp32 1091 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑆𝑡𝑆 (𝑡 𝑉)))
299, 10, 11, 12, 13, 14, 15, 16cdleme22f 34652 . . 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 1343 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑁 (𝐹 𝑉))
3125, 30eqbrtrd 4605 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   class class class wbr 4583  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  HLchlt 33655  LHypclh 34288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-undef 7286  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292 This theorem is referenced by: (None)
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