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Theorem cdleme26f2ALTN 37044
Description: Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 37042 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.) (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)
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
cdleme26f2ALTN ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 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 cdleme26f2ALTN
StepHypRef Expression
1 simp11 1196 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp23 1201 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑇𝐴 ∧ ¬ 𝑇 𝑊))
3 simp31r 1290 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ¬ 𝑠 (𝑃 𝑄))
4 simp12r 1280 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑇 (𝑃 𝑄))
5 simp12l 1279 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑃𝑄)
63, 4, 53jca 1121 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (¬ 𝑠 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄) ∧ 𝑃𝑄))
7 simp21 1199 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
8 simp22 1200 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
9 simp13 1198 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
10 simp32 1203 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑠𝑇𝑠 (𝑇 𝑉)))
11 simp33 1204 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 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 37027 . . 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 1393 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐺 (𝑂 𝑉))
22 simp23l 1287 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑇𝐴)
23 simp23r 1288 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 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 37037 . . . . 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 1391 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐸𝐵)
28 simp13l 1281 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑠𝐴)
29 simp31 1202 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)))
3024fvexi 6555 . . . . 5 𝐵 ∈ V
3130, 25riotasv 35639 . . . 4 ((𝐸𝐵𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄))) → 𝐸 = 𝑂)
3227, 28, 29, 31syl3anc 1364 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐸 = 𝑂)
3332oveq1d 7034 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠𝑇𝑠 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝐸 𝑉) = (𝑂 𝑉))
3421, 33breqtrrd 4992 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑇 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((¬ 𝑠 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 1080   = wceq 1522  wcel 2080  wne 2983  wral 3104   class class class wbr 4964  cfv 6228  crio 6979  (class class class)co 7019  Basecbs 16312  lecple 16401  joincjn 17383  meetcmee 17384  Atomscatm 35943  HLchlt 36030  LHypclh 36664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322  ax-riotaBAD 35633
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-iun 4829  df-iin 4830  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-1st 7548  df-2nd 7549  df-undef 7793  df-proset 17367  df-poset 17385  df-plt 17397  df-lub 17413  df-glb 17414  df-join 17415  df-meet 17416  df-p0 17478  df-p1 17479  df-lat 17485  df-clat 17547  df-oposet 35856  df-ol 35858  df-oml 35859  df-covers 35946  df-ats 35947  df-atl 35978  df-cvlat 36002  df-hlat 36031  df-llines 36178  df-lplanes 36179  df-lvols 36180  df-lines 36181  df-psubsp 36183  df-pmap 36184  df-padd 36476  df-lhyp 36668
This theorem is referenced by: (None)
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