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Theorem cdleme26fALTN 36171
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 1246 . . . 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 1249 . . . 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 1250 . . . 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 1379 . . . 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 1380 . . . 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 1371 . . . 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 1372 . . . 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 36166 . . . 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 1505 . . 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 1373 . . 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 1252 . . 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 6364 . . . . 5 (Base‘𝐾) ∈ V
238, 22eqeltri 2836 . . . 4 𝐵 ∈ V
2423, 17riotasv 34767 . . 3 ((𝐼𝐵𝑡𝐴 ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝐼 = 𝑁)
2519, 20, 21, 24syl3anc 1477 . 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 1251 . . 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 1254 . . 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 1253 . . 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 36155 . . 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 1502 . 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 4827 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 1072   = wceq 1632  wcel 2140  wne 2933  wral 3051  Vcvv 3341   class class class wbr 4805  cfv 6050  crio 6775  (class class class)co 6815  Basecbs 16080  lecple 16171  joincjn 17166  meetcmee 17167  Atomscatm 35072  HLchlt 35159  LHypclh 35792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-riotaBAD 34761
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-iin 4676  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-undef 7570  df-preset 17150  df-poset 17168  df-plt 17180  df-lub 17196  df-glb 17197  df-join 17198  df-meet 17199  df-p0 17261  df-p1 17262  df-lat 17268  df-clat 17330  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307  df-lvols 35308  df-lines 35309  df-psubsp 35311  df-pmap 35312  df-padd 35604  df-lhyp 35796
This theorem is referenced by: (None)
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