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Theorem cdleme26f2 37368
Description: Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 37365 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 1197 . . 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 1202 . . 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 1203 . . . 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 1281 . . . 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 1280 . . . 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 1122 . . 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 1200 . . 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 1201 . . 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 1199 . . 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 1204 . . 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 1205 . . 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 37350 . . 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 1394 . 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 1288 . . . . 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 1289 . . . . 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 37360 . . . . 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 1392 . . . 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 1282 . . . 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 1283 . . . 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)
3024fvexi 6680 . . . . 5 𝐵 ∈ V
3130, 25riotasv 35962 . . . 4 ((𝐸𝐵𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄))) → 𝐸 = 𝑂)
3227, 28, 29, 3, 31syl112anc 1368 . . 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)
3332oveq1d 7166 . 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))
3421, 33breqtrrd 5090 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 396  w3a 1081   = wceq 1530  wcel 2107  wne 3020  wral 3142   class class class wbr 5062  cfv 6351  crio 7108  (class class class)co 7151  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Atomscatm 36266  HLchlt 36353  LHypclh 36987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-riotaBAD 35956
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-undef 7933  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36179  df-ol 36181  df-oml 36182  df-covers 36269  df-ats 36270  df-atl 36301  df-cvlat 36325  df-hlat 36354  df-llines 36501  df-lplanes 36502  df-lvols 36503  df-lines 36504  df-psubsp 36506  df-pmap 36507  df-padd 36799  df-lhyp 36991
This theorem is referenced by:  cdleme27a  37370
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