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Theorem cdleme26f2 36155
 Description: Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 36152 with s and t swapped (this case is not mentioned by them). If s ≤ t ∨ v, then f(s) ≤ fs(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)
cdleme26f2.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme26f2.f 𝐺 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme26f2.n 𝑂 = ((𝑃 𝑄) (𝐺 ((𝑇 𝑠) 𝑊)))
cdleme26f2.e 𝐸 = (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑂))
Assertion
Ref Expression
cdleme26f2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐺 (𝐸 𝑉))
Distinct variable groups:   𝑢,𝑠,𝐴   𝐵,𝑠,𝑢   𝐻,𝑠   ,𝑠,𝑢   𝐾,𝑠   ,𝑠,𝑢   ,𝑠,𝑢   𝑢,𝑂   𝑃,𝑠,𝑢   𝑄,𝑠,𝑢   𝑇,𝑠,𝑢   𝑈,𝑠,𝑢   𝑊,𝑠,𝑢
Allowed substitution hints:   𝐸(𝑢,𝑠)   𝐺(𝑢,𝑠)   𝐻(𝑢)   𝐾(𝑢)   𝑂(𝑠)   𝑉(𝑢,𝑠)

Proof of Theorem cdleme26f2
StepHypRef Expression
1 simp11 1246 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp23 1251 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑇𝐴 ∧ ¬ 𝑇 𝑊))
3 simp31 1252 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ¬ 𝑠 (𝑃 𝑄))
4 simp12r 1372 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑇 (𝑃 𝑄))
5 simp12l 1371 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑃𝑄)
63, 4, 53jca 1123 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (¬ 𝑠 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄) ∧ 𝑃𝑄))
7 simp21 1249 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
8 simp22 1250 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
9 simp13 1248 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
10 simp32 1253 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑠𝑇𝑠 (𝑇 𝑉)))
11 simp33 1254 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑉𝐴𝑉 𝑊))
12 cdleme26.l . . . 4 = (le‘𝐾)
13 cdleme26.j . . . 4 = (join‘𝐾)
14 cdleme26.m . . . 4 = (meet‘𝐾)
15 cdleme26.a . . . 4 𝐴 = (Atoms‘𝐾)
16 cdleme26.h . . . 4 𝐻 = (LHyp‘𝐾)
17 cdleme26f2.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
18 cdleme26f2.f . . . 4 𝐺 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
19 cdleme26f2.n . . . 4 𝑂 = ((𝑃 𝑄) (𝐺 ((𝑇 𝑠) 𝑊)))
2012, 13, 14, 15, 16, 17, 18, 19cdleme22f2 36137 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐺 (𝑂 𝑉))
211, 2, 6, 7, 8, 9, 10, 11, 20syl323anc 1507 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐺 (𝑂 𝑉))
22 simp23l 1379 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑇𝐴)
23 simp23r 1380 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ¬ 𝑇 𝑊)
24 cdleme26.b . . . . . 6 𝐵 = (Base‘𝐾)
25 cdleme26f2.e . . . . . 6 𝐸 = (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑂))
2624, 12, 13, 14, 15, 16, 17, 18, 19, 25cdleme25cl 36147 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑇 (𝑃 𝑄))) → 𝐸𝐵)
271, 7, 8, 22, 23, 5, 4, 26syl322anc 1505 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐸𝐵)
28 simp13l 1373 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑠𝐴)
29 simp13r 1374 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ¬ 𝑠 𝑊)
30 fvex 6362 . . . . . 6 (Base‘𝐾) ∈ V
3124, 30eqeltri 2835 . . . . 5 𝐵 ∈ V
3231, 25riotasv 34748 . . . 4 ((𝐸𝐵𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄))) → 𝐸 = 𝑂)
3327, 28, 29, 3, 32syl112anc 1481 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐸 = 𝑂)
3433oveq1d 6828 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝐸 𝑉) = (𝑂 𝑉))
3521, 34breqtrrd 4832 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐺 (𝐸 𝑉))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  ∀wral 3050  Vcvv 3340   class class class wbr 4804  ‘cfv 6049  ℩crio 6773  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  meetcmee 17146  Atomscatm 35053  HLchlt 35140  LHypclh 35773 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 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-riotaBAD 34742 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 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-undef 7568  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-p1 17241  df-lat 17247  df-clat 17309  df-oposet 34966  df-ol 34968  df-oml 34969  df-covers 35056  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-llines 35287  df-lplanes 35288  df-lvols 35289  df-lines 35290  df-psubsp 35292  df-pmap 35293  df-padd 35585  df-lhyp 35777 This theorem is referenced by:  cdleme27a  36157
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