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Theorem cdleme40n 34774
Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 𝑄 line. TODO: FIX COMMENT. TODO get rid of '.<' class? (Contributed by NM, 18-Mar-2013.)
Hypotheses
Ref Expression
cdleme40.b 𝐵 = (Base‘𝐾)
cdleme40.l = (le‘𝐾)
cdleme40.j = (join‘𝐾)
cdleme40.m = (meet‘𝐾)
cdleme40.a 𝐴 = (Atoms‘𝐾)
cdleme40.h 𝐻 = (LHyp‘𝐾)
cdleme40.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme40.e 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme40.g 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
cdleme40.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
cdleme40.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
cdleme40a1.y 𝑌 = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊)))
cdleme40a1.c 𝐶 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌))
cdleme40.t 𝑇 = ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊)))
cdleme40.f 𝐹 = ((𝑃 𝑄) (𝑇 ((𝑆 𝑣) 𝑊)))
cdleme40a1.x 𝑋 = ((𝑃 𝑄) (𝑇 ((𝑢 𝑣) 𝑊)))
cdleme40.o 𝑂 = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))
cdleme40.v 𝑉 = if(𝑢 (𝑃 𝑄), 𝑂, < )
cdleme40a1.z 𝑍 = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝐹))
Assertion
Ref Expression
cdleme40n ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑅 / 𝑠𝑁𝑆 / 𝑢𝑉)
Distinct variable groups:   𝑣,𝑢,𝑧,𝐴   𝑢,𝐵,𝑣,𝑧   𝑧,𝐹   𝑣,𝐻,𝑧   𝑢, ,𝑣,𝑧   𝑣,𝐾,𝑧   𝑢, ,𝑣,𝑧   𝑢, ,𝑣,𝑧   𝑢,𝑃,𝑣,𝑧   𝑢,𝑄,𝑣,𝑧   𝑣,𝑅,𝑧   𝑢,𝑆,𝑧   𝑢,𝑇   𝑣,𝑈,𝑧   𝑢,𝑊,𝑣,𝑧   𝑣,𝑠,𝑡,𝑦,𝐴   𝐵,𝑠,𝑡,𝑦   𝐸,𝑠   𝑡,𝐹   𝑡,𝐻,𝑦   ,𝑠,𝑡,𝑦   𝑡,𝐾,𝑦   ,𝑠,𝑡,𝑦   ,𝑠,𝑡,𝑦   𝑃,𝑠,𝑡,𝑦   𝑄,𝑠,𝑡,𝑦   𝑅,𝑠,𝑡,𝑦   𝑡,𝑈,𝑦   𝑊,𝑠,𝑡,𝑦   𝑦,𝑌   𝑡,𝑆,𝑣,𝑦   𝑇,𝑠,𝑡,𝑦   𝑣,𝐷   𝑣,𝐼   𝑣,𝑁
Allowed substitution hints:   𝐶(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝐷(𝑦,𝑧,𝑢,𝑡,𝑠)   𝑅(𝑢)   𝑆(𝑠)   < (𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑇(𝑧,𝑣)   𝑈(𝑢,𝑠)   𝐸(𝑦,𝑧,𝑣,𝑢,𝑡)   𝐹(𝑦,𝑣,𝑢,𝑠)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝐻(𝑢,𝑠)   𝐼(𝑦,𝑧,𝑢,𝑡,𝑠)   𝐾(𝑢,𝑠)   𝑁(𝑦,𝑧,𝑢,𝑡,𝑠)   𝑂(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑉(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑋(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑌(𝑧,𝑣,𝑢,𝑡,𝑠)   𝑍(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)

Proof of Theorem cdleme40n
StepHypRef Expression
1 cdleme40.b . . . 4 𝐵 = (Base‘𝐾)
2 fvex 6113 . . . 4 (Base‘𝐾) ∈ V
31, 2eqeltri 2684 . . 3 𝐵 ∈ V
4 nfv 1830 . . . 4 𝑣(((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆))
5 nfcv 2751 . . . . . 6 𝑣𝑅 / 𝑠𝑁
6 cdleme40a1.z . . . . . . 7 𝑍 = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝐹))
7 nfra1 2925 . . . . . . . 8 𝑣𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝐹)
8 nfcv 2751 . . . . . . . 8 𝑣𝐵
97, 8nfriota 6520 . . . . . . 7 𝑣(𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝐹))
106, 9nfcxfr 2749 . . . . . 6 𝑣𝑍
115, 10nfne 2882 . . . . 5 𝑣𝑅 / 𝑠𝑁𝑍
1211a1i 11 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → Ⅎ𝑣𝑅 / 𝑠𝑁𝑍)
136a1i 11 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑍 = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝐹)))
14 neeq2 2845 . . . . 5 (𝐹 = 𝑍 → (𝑅 / 𝑠𝑁𝐹𝑅 / 𝑠𝑁𝑍))
1514adantl 481 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ 𝐹 = 𝑍) → (𝑅 / 𝑠𝑁𝐹𝑅 / 𝑠𝑁𝑍))
16 simpl11 1129 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
17 simpl12 1130 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
18 simpl13 1131 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
19 simpl21 1132 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → 𝑃𝑄)
20 simpl22 1133 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
21 simpl23 1134 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
22 simpl3 1059 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆))
23 simprl 790 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → 𝑣𝐴)
24 simprrl 800 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → ¬ 𝑣 𝑊)
25 simprrr 801 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → ¬ 𝑣 (𝑃 𝑄))
2623, 24, 253jca 1235 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → (𝑣𝐴 ∧ ¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))
27 cdleme40.l . . . . . . 7 = (le‘𝐾)
28 cdleme40.j . . . . . . 7 = (join‘𝐾)
29 cdleme40.m . . . . . . 7 = (meet‘𝐾)
30 cdleme40.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
31 cdleme40.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
32 cdleme40.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
33 cdleme40.e . . . . . . 7 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
34 cdleme40.g . . . . . . 7 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
35 cdleme40.i . . . . . . 7 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
36 cdleme40.n . . . . . . 7 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
37 cdleme40a1.y . . . . . . 7 𝑌 = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊)))
38 cdleme40a1.c . . . . . . 7 𝐶 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌))
39 cdleme40.t . . . . . . 7 𝑇 = ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊)))
40 cdleme40.f . . . . . . 7 𝐹 = ((𝑃 𝑄) (𝑇 ((𝑆 𝑣) 𝑊)))
411, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40cdleme40m 34773 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ (𝑣𝐴 ∧ ¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → 𝑅 / 𝑠𝑁𝐹)
4216, 17, 18, 19, 20, 21, 22, 26, 41syl332anc 1349 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ (𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → 𝑅 / 𝑠𝑁𝐹)
4342ex 449 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → ((𝑣𝐴 ∧ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄))) → 𝑅 / 𝑠𝑁𝐹))
44 simp1 1054 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
45 simp23l 1175 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑆𝐴)
46 simp23r 1176 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → ¬ 𝑆 𝑊)
47 simp21 1087 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑃𝑄)
48 simp32 1091 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑆 (𝑃 𝑄))
491, 27, 28, 29, 30, 31, 32, 39, 40, 6cdleme25cl 34663 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑃𝑄𝑆 (𝑃 𝑄))) → 𝑍𝐵)
5044, 45, 46, 47, 48, 49syl122anc 1327 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑍𝐵)
51 simp11 1084 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
52 simp12 1085 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
53 simp13 1086 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
5427, 28, 30, 31cdlemb2 34345 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑣𝐴𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))
5551, 52, 53, 47, 54syl121anc 1323 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → ∃𝑣𝐴𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))
564, 12, 13, 15, 43, 50, 55riotasv3d 33264 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) ∧ 𝐵 ∈ V) → 𝑅 / 𝑠𝑁𝑍)
573, 56mpan2 703 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑅 / 𝑠𝑁𝑍)
58 cdleme40a1.x . . . 4 𝑋 = ((𝑃 𝑄) (𝑇 ((𝑢 𝑣) 𝑊)))
59 cdleme40.o . . . 4 𝑂 = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))
60 cdleme40.v . . . 4 𝑉 = if(𝑢 (𝑃 𝑄), 𝑂, < )
6158, 59, 60, 40, 6cdleme31sn1c 34694 . . 3 ((𝑆𝐴𝑆 (𝑃 𝑄)) → 𝑆 / 𝑢𝑉 = 𝑍)
6245, 48, 61syl2anc 691 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑆 / 𝑢𝑉 = 𝑍)
6357, 62neeqtrrd 2856 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑅 / 𝑠𝑁𝑆 / 𝑢𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wnf 1699  wcel 1977  wne 2780  wral 2896  wrex 2897  Vcvv 3173  csb 3499  ifcif 4036   class class class wbr 4583  cfv 5804  crio 6510  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  HLchlt 33655  LHypclh 34288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-undef 7286  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292
This theorem is referenced by:  cdleme40w  34776
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