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Theorem cdleme42b 38932
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 6-Mar-2013.)
Hypotheses
Ref Expression
cdleme41.b 𝐵 = (Base‘𝐾)
cdleme41.l = (le‘𝐾)
cdleme41.j = (join‘𝐾)
cdleme41.m = (meet‘𝐾)
cdleme41.a 𝐴 = (Atoms‘𝐾)
cdleme41.h 𝐻 = (LHyp‘𝐾)
cdleme41.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme41.d 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme41.e 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme41.g 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
cdleme41.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
cdleme41.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
cdleme41.o 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
cdleme41.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
Assertion
Ref Expression
cdleme42b ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = (𝑅 / 𝑠𝑁 (𝑋 𝑊)))
Distinct variable groups:   𝐴,𝑠   ,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑈,𝑠   𝑊,𝑠   𝑦,𝑡,𝐴,𝑠   𝐵,𝑠,𝑡,𝑦   𝑦,𝐷   𝑦,𝐺   𝐸,𝑠,𝑦   𝐻,𝑠,𝑡,𝑦   𝑡, ,𝑦   𝐾,𝑠,𝑡,𝑦   𝑡, ,𝑦   𝑡, ,𝑦   𝑡,𝑃,𝑦   𝑡,𝑄,𝑦   𝑡,𝑅,𝑦   𝑡,𝑈,𝑦   𝑡,𝑊,𝑦   𝑥,𝑧,𝐴   𝑥,𝐵,𝑧   𝑧,𝐸,𝑠   𝑧,𝐻   𝑥, ,𝑧   𝑧,𝐾   𝑥, ,𝑧   𝑥, ,𝑧   𝑥,𝑁,𝑧   𝑥,𝑃,𝑧   𝑥,𝑄,𝑧   𝑥,𝑅,𝑧   𝑥,𝑈,𝑧   𝑥,𝑊,𝑧,𝑠,𝑡,𝑦   𝑋,𝑠,𝑡,𝑥,𝑧
Allowed substitution hints:   𝐷(𝑥,𝑧,𝑡,𝑠)   𝐸(𝑥,𝑡)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐺(𝑥,𝑧,𝑡,𝑠)   𝐻(𝑥)   𝐼(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐾(𝑥)   𝑁(𝑦,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑋(𝑦)

Proof of Theorem cdleme42b
StepHypRef Expression
1 cdleme41.b . . 3 𝐵 = (Base‘𝐾)
21fvexi 6856 . 2 𝐵 ∈ V
3 nfv 1917 . . 3 𝑠(((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋))
4 nfcsb1v 3880 . . . . 5 𝑠𝑅 / 𝑠𝑁
5 nfcv 2907 . . . . 5 𝑠
6 nfcv 2907 . . . . 5 𝑠(𝑋 𝑊)
74, 5, 6nfov 7386 . . . 4 𝑠(𝑅 / 𝑠𝑁 (𝑋 𝑊))
87a1i 11 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → 𝑠(𝑅 / 𝑠𝑁 (𝑋 𝑊)))
9 nfvd 1918 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → Ⅎ𝑠𝑅 𝑊 ∧ (𝑅 (𝑋 𝑊)) = 𝑋))
10 cdleme41.o . . . . 5 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
11 cdleme41.f . . . . 5 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
12 eqid 2736 . . . . 5 (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))) = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊))))
1310, 11, 12cdleme31fv1 38845 . . . 4 ((𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
14133ad2ant2 1134 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
15 breq1 5108 . . . . . 6 (𝑠 = 𝑅 → (𝑠 𝑊𝑅 𝑊))
1615notbid 317 . . . . 5 (𝑠 = 𝑅 → (¬ 𝑠 𝑊 ↔ ¬ 𝑅 𝑊))
17 oveq1 7363 . . . . . 6 (𝑠 = 𝑅 → (𝑠 (𝑋 𝑊)) = (𝑅 (𝑋 𝑊)))
1817eqeq1d 2738 . . . . 5 (𝑠 = 𝑅 → ((𝑠 (𝑋 𝑊)) = 𝑋 ↔ (𝑅 (𝑋 𝑊)) = 𝑋))
1916, 18anbi12d 631 . . . 4 (𝑠 = 𝑅 → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ↔ (¬ 𝑅 𝑊 ∧ (𝑅 (𝑋 𝑊)) = 𝑋)))
2019adantl 482 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) ∧ 𝑠 = 𝑅) → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ↔ (¬ 𝑅 𝑊 ∧ (𝑅 (𝑋 𝑊)) = 𝑋)))
21 csbeq1a 3869 . . . . 5 (𝑠 = 𝑅𝑁 = 𝑅 / 𝑠𝑁)
2221oveq1d 7371 . . . 4 (𝑠 = 𝑅 → (𝑁 (𝑋 𝑊)) = (𝑅 / 𝑠𝑁 (𝑋 𝑊)))
2322adantl 482 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) ∧ 𝑠 = 𝑅) → (𝑁 (𝑋 𝑊)) = (𝑅 / 𝑠𝑁 (𝑋 𝑊)))
24 simp1 1136 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
25 simp2l 1199 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → 𝑋𝐵)
26 cdleme41.l . . . . 5 = (le‘𝐾)
27 cdleme41.j . . . . 5 = (join‘𝐾)
28 cdleme41.m . . . . 5 = (meet‘𝐾)
29 cdleme41.a . . . . 5 𝐴 = (Atoms‘𝐾)
30 cdleme41.h . . . . 5 𝐻 = (LHyp‘𝐾)
31 cdleme41.u . . . . 5 𝑈 = ((𝑃 𝑄) 𝑊)
32 cdleme41.d . . . . 5 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
33 cdleme41.e . . . . 5 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
34 cdleme41.g . . . . 5 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
35 cdleme41.i . . . . 5 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
36 cdleme41.n . . . . 5 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
371, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 10, 11cdleme32fvcl 38894 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
3824, 25, 37syl2anc 584 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) ∈ 𝐵)
39 simp3ll 1244 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → 𝑅𝐴)
40 simp3lr 1245 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → ¬ 𝑅 𝑊)
41 simp3r 1202 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → (𝑅 (𝑋 𝑊)) = 𝑋)
4240, 41jca 512 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → (¬ 𝑅 𝑊 ∧ (𝑅 (𝑋 𝑊)) = 𝑋))
433, 8, 9, 14, 20, 23, 38, 39, 42riotasv2d 37410 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) ∧ 𝐵 ∈ V) → (𝐹𝑋) = (𝑅 / 𝑠𝑁 (𝑋 𝑊)))
442, 43mpan2 689 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = (𝑅 / 𝑠𝑁 (𝑋 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wnfc 2887  wne 2943  wral 3064  Vcvv 3445  csb 3855  ifcif 4486   class class class wbr 5105  cmpt 5188  cfv 6496  crio 7311  (class class class)co 7356  Basecbs 17082  lecple 17139  joincjn 18199  meetcmee 18200  Atomscatm 37716  HLchlt 37803  LHypclh 38438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671  ax-riotaBAD 37406
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7312  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7920  df-2nd 7921  df-undef 8203  df-proset 18183  df-poset 18201  df-plt 18218  df-lub 18234  df-glb 18235  df-join 18236  df-meet 18237  df-p0 18313  df-p1 18314  df-lat 18320  df-clat 18387  df-oposet 37629  df-ol 37631  df-oml 37632  df-covers 37719  df-ats 37720  df-atl 37751  df-cvlat 37775  df-hlat 37804  df-llines 37952  df-lplanes 37953  df-lvols 37954  df-lines 37955  df-psubsp 37957  df-pmap 37958  df-padd 38250  df-lhyp 38442
This theorem is referenced by:  cdleme42e  38933  cdleme48fv  38953
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