HomeHome Metamath Proof Explorer
Theorem List (p. 374 of 449)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-28623)
  Hilbert Space Explorer  Hilbert Space Explorer
(28624-30146)
  Users' Mathboxes  Users' Mathboxes
(30147-44804)
 

Theorem List for Metamath Proof Explorer - 37301-37400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdleme16f 37301 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 3rd part of 3rd sentence. 𝐹 and 𝐺 represent f(s) and f(t) respectively. We show, in their notation, (s t) (f(s) f(t))=(f(s) f(t)) w. (Contributed by NM, 11-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → ((𝑆 𝑇) (𝐹 𝐺)) = ((𝐹 𝐺) 𝑊))
 
Theoremcdleme16g 37302 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, Eq. (1). 𝐹 and 𝐺 represent f(s) and f(t) respectively. We show, in their notation, (s t) w=(f(s) f(t)) w. (Contributed by NM, 11-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → ((𝑆 𝑇) 𝑊) = ((𝐹 𝐺) 𝑊))
 
Theoremcdleme16 37303 Part of proof of Lemma E in [Crawley] p. 113, conclusion of 3rd paragraph on p. 114. 𝐹 and 𝐺 represent f(s) and f(t) respectively. We show, in their notation, (s t) w=(f(s) f(t)) w, whether or not u s t. (Contributed by NM, 11-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄))) → ((𝑆 𝑇) 𝑊) = ((𝐹 𝐺) 𝑊))
 
Theoremcdleme17a 37304 Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. 𝐹, 𝐺, and 𝐶 represent f(s), fs(p), and s1 respectively. We show, in their notation, fs(p)=(p q) (q s1). (Contributed by NM, 11-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊)))    &   𝐶 = ((𝑃 𝑆) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝐺 = ((𝑃 𝑄) (𝑄 𝐶)))
 
Theoremcdleme17b 37305 Lemma leading to cdleme17c 37306. (Contributed by NM, 11-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊)))    &   𝐶 = ((𝑃 𝑆) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝐶 (𝑃 𝑄))
 
Theoremcdleme17c 37306 Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. 𝐶 represents s1. We show, in their notation, (p q) (q s1)=q. (Contributed by NM, 11-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊)))    &   𝐶 = ((𝑃 𝑆) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝑃 𝑄) (𝑄 𝐶)) = 𝑄)
 
Theoremcdleme17d1 37307 Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. 𝐹, 𝐺 represent f(s), fs(p) respectively. We show, in their notation, fs(p)=q. (Contributed by NM, 11-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝐺 = 𝑄)
 
Theoremcdleme0nex 37308* Part of proof of Lemma E in [Crawley] p. 114, 4th line of 4th paragraph. Whenever (in their terminology) p q/0 (i.e. the sublattice from 0 to p q) contains precisely three atoms, any atom not under w must equal either p or q. (In case of 3 atoms, one of them must be u - see cdleme0a 37229- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 36361, our (𝑃 𝑟) = (𝑄 𝑟) is a shorter way to express 𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄). Thus, the negated existential condition states there are no atoms different from p or q that are also not under w. (Contributed by NM, 12-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 = 𝑃𝑅 = 𝑄))
 
Theoremcdleme18a 37309 Part of proof of Lemma E in [Crawley] p. 114, 2nd sentence of 4th paragraph. 𝐹, 𝐺 represent f(s), fs(q) respectively. We show ¬ fs(q) w. (Contributed by NM, 12-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝐺 𝑊)
 
Theoremcdleme18b 37310 Part of proof of Lemma E in [Crawley] p. 114, 2nd sentence of 4th paragraph. 𝐹, 𝐺 represent f(s), fs(q) respectively. We show ¬ fs(q) q. (Contributed by NM, 12-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐺𝑄)
 
Theoremcdleme18c 37311* Part of proof of Lemma E in [Crawley] p. 114, 2nd sentence of 4th paragraph. 𝐹, 𝐺 represent f(s), fs(q) respectively. We show ¬ fs(q) = p whenever p q has three atoms under it (implied by the negated existential condition). (Contributed by NM, 10-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐺 = 𝑃)
 
Theoremcdleme22gb 37312 Utility lemma for Lemma E in [Crawley] p. 115. (Contributed by NM, 5-Dec-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))    &   𝐵 = (Base‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → 𝐺𝐵)
 
Theoremcdleme18d 37313* Part of proof of Lemma E in [Crawley] p. 114, 4th sentence of 4th paragraph. 𝐹, 𝐺, 𝐷, 𝐸 represent f(s), fs(r), f(t), ft(r) respectively. We show fs(r)=ft(r) for all possible r (which must equal p or q in the case of exactly 3 atoms in p q/0 i.e. when ¬ ∃𝑟𝐴...). (Contributed by NM, 12-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))    &   𝐷 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐺 = 𝐸)
 
Theoremcdlemesner 37314 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 13-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑆𝑅)
 
Theoremcdlemedb 37315 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. 𝐷 represents s2. (Contributed by NM, 20-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝐵 = (Base‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴)) → 𝐷𝐵)
 
Theoremcdlemeda 37316 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. 𝐷 represents s2. (Contributed by NM, 13-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐷 = ((𝑅 𝑆) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐷𝐴)
 
Theoremcdlemednpq 37317 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. 𝐷 represents s2. (Contributed by NM, 18-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐷 = ((𝑅 𝑆) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝐷 (𝑃 𝑄))
 
TheoremcdlemednuN 37318 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. 𝐷 represents s2. (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐷𝑈)
 
Theoremcdleme20zN 37319 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → ((𝑆 𝑅) 𝑇) = (0.‘𝐾))
 
Theoremcdleme20y 37320 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.) (Proof shortened by OpenAI, 25-Mar-2020.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → ((𝑆 𝑅) (𝑇 𝑅)) = 𝑅)
 
Theoremcdleme19a 37321 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line. 𝐷 represents s2. In their notation, we prove that if r s t, then s2=(s t) w. (Contributed by NM, 13-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)       ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝐷 = ((𝑆 𝑇) 𝑊))
 
Theoremcdleme19b 37322 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line. 𝐷, 𝐹, 𝐺 represent s2, f(s), f(t). In their notation, we prove that if r s t, then s2 f(s) f(t). (Contributed by NM, 13-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐷 (𝐹 𝐺))
 
Theoremcdleme19c 37323 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line. 𝐷, 𝐹 represent s2, f(s). We prove f(s) s2. (Contributed by NM, 13-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐹𝐷)
 
Theoremcdleme19d 37324 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114. 𝐷, 𝐹, 𝐺 represent s2, f(s), f(t). We prove f(s) s2 = f(s) f(t). (Contributed by NM, 14-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝐹 𝐷) = (𝐹 𝐺))
 
Theoremcdleme19e 37325 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, line 2. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We prove f(s) s2=f(t) t2. (Contributed by NM, 14-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝐹 𝐷) = (𝐺 𝑌))
 
Theoremcdleme19f 37326 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, line 3. 𝐷, 𝐹, 𝑁, 𝑌, 𝐺, 𝑂 represent s2, f(s), fs(r), t2, f(t), ft(r). We prove that if r s t, then ft(r) = ft(r). (Contributed by NM, 14-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑂 = ((𝑃 𝑄) (𝐺 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑁 = 𝑂)
 
Theoremcdleme20aN 37327 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 14-Nov-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑉 𝐷) = (((𝑆 𝑅) 𝑇) 𝑊))
 
Theoremcdleme20bN 37328 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We show v s2 = v t2. (Contributed by NM, 15-Nov-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑉 𝐷) = (𝑉 𝑌))
 
Theoremcdleme20c 37329 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 15-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝐷 𝑌) = (((𝑅 𝑆) 𝑇) 𝑊))
 
Theoremcdleme20d 37330 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 17-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ 𝑅 (𝑃 𝑄))) → ((𝐹 𝐺) (𝐷 𝑌)) = 𝑉)
 
Theoremcdleme20e 37331 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, 4th line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We show <f(s),s2,s> and <f(t),t2,t> are centrally perspective. (Contributed by NM, 17-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ 𝑅 (𝑃 𝑄))) → ((𝐹 𝐺) (𝐷 𝑌)) (𝑆 𝑇))
 
Theoremcdleme20f 37332 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, 4th line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We show <f(s),s2,s> and <f(t),t2,t> are axially perspective. (Contributed by NM, 17-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ 𝑅 (𝑃 𝑄))) → ((𝐹 𝐷) (𝐺 𝑌)) (((𝐷 𝑆) (𝑌 𝑇)) ((𝑆 𝐹) (𝑇 𝐺))))
 
Theoremcdleme20g 37333 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 18-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ 𝑅 (𝑃 𝑄))) → (((𝐷 𝑆) (𝑌 𝑇)) ((𝑆 𝐹) (𝑇 𝐺))) = (((𝑆 𝑅) (𝑇 𝑅)) ((𝑆 𝑈) (𝑇 𝑈))))
 
Theoremcdleme20h 37334 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 18-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)) ∧ (¬ 𝑅 (𝑆 𝑇) ∧ ¬ 𝑈 (𝑆 𝑇)))) → (((𝑆 𝑅) (𝑇 𝑅)) ((𝑆 𝑈) (𝑇 𝑈))) = (𝑅 𝑈))
 
Theoremcdleme20i 37335 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We show (f(s) s2) (f(t) t2) p q. (Contributed by NM, 18-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)) ∧ (¬ 𝑅 (𝑆 𝑇) ∧ ¬ 𝑈 (𝑆 𝑇)))) → ((𝐹 𝐷) (𝐺 𝑌)) (𝑃 𝑄))
 
Theoremcdleme20j 37336 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We show s2 t2. (Contributed by NM, 18-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)) ∧ ¬ 𝑅 (𝑆 𝑇))) → 𝐷𝑌)
 
Theoremcdleme20k 37337 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 20-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝐹 𝐷) ≠ (𝑃 𝑄))
 
Theoremcdleme20l1 37338 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝐹 𝐷) ∈ (LLines‘𝐾))
 
Theoremcdleme20l2 37339 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)) ∧ (¬ 𝑅 (𝑆 𝑇) ∧ ¬ 𝑈 (𝑆 𝑇)))) → ((𝐹 𝐷) (𝐺 𝑌)) ∈ 𝐴)
 
Theoremcdleme20l 37340 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)) ∧ (¬ 𝑅 (𝑆 𝑇) ∧ ¬ 𝑈 (𝑆 𝑇)))) → ((𝐹 𝐷) (𝐺 𝑌)) = ((𝑃 𝑄) (𝐹 𝐷)))
 
Theoremcdleme20m 37341 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. 𝐷, 𝐹, 𝑁, 𝑌, 𝐺, 𝑂 represent s2, f(s), fs(r), t2, f(t), ft(r) respectively. We prove that if ¬ r s t and ¬ u s t, then fs(r) = ft(r). (Contributed by NM, 20-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑂 = ((𝑃 𝑄) (𝐺 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)) ∧ (¬ 𝑅 (𝑆 𝑇) ∧ ¬ 𝑈 (𝑆 𝑇)))) → 𝑁 = 𝑂)
 
Theoremcdleme20 37342 Combine cdleme19f 37326 and cdleme20m 37341 to eliminate ¬ 𝑅 (𝑆 𝑇) condition. (Contributed by NM, 28-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑉 = ((𝑆 𝑇) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑂 = ((𝑃 𝑄) (𝐺 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)) ∧ ¬ 𝑈 (𝑆 𝑇))) → 𝑁 = 𝑂)
 
Theoremcdleme21a 37343 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑆𝑧)
 
Theoremcdleme21b 37344 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑧 (𝑃 𝑄))
 
Theoremcdleme21c 37345 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑈 (𝑆 𝑧))
 
Theoremcdleme21at 37346 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ ((𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ 𝑈 (𝑆 𝑇)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑇𝑧)
 
Theoremcdleme21ct 37347 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑈 (𝑆 𝑇))) ∧ ((𝑧𝐴 ∧ ¬ 𝑧 𝑊) ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑈 (𝑇 𝑧))
 
Theoremcdleme21d 37348 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 115, 3rd line. 𝐷, 𝐹, 𝑁, 𝐸, 𝐵, 𝑍 represent s2, f(s), fs(r), z2, f(z), fz(r) respectively. We prove fs(r) = fz(r). (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐵 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝐸 = ((𝑅 𝑧) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑍 = ((𝑃 𝑄) (𝐵 𝐸))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑧𝐴 ∧ ¬ 𝑧 𝑊) ∧ (𝑃 𝑧) = (𝑆 𝑧)))) → 𝑁 = 𝑍)
 
Theoremcdleme21e 37349 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 115, 3rd line. 𝑌, 𝐺, 𝑂, 𝐸, 𝐵, 𝑍 represent s2, f(s), fs(r), z2, f(z), fz(r) respectively. We prove that if u s z, then ft(r) = fz(r). (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐵 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝐸 = ((𝑅 𝑧) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑍 = ((𝑃 𝑄) (𝐵 𝐸))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑂 = ((𝑃 𝑄) (𝐺 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄))) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑈 (𝑆 𝑇)) ∧ ((𝑧𝐴 ∧ ¬ 𝑧 𝑊) ∧ (𝑃 𝑧) = (𝑆 𝑧)))) → 𝑂 = 𝑍)
 
Theoremcdleme21f 37350 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐵 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝐸 = ((𝑅 𝑧) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑍 = ((𝑃 𝑄) (𝐵 𝐸))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑂 = ((𝑃 𝑄) (𝐺 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄))) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑈 (𝑆 𝑇)) ∧ ((𝑧𝐴 ∧ ¬ 𝑧 𝑊) ∧ (𝑃 𝑧) = (𝑆 𝑧)))) → 𝑁 = 𝑂)
 
Theoremcdleme21g 37351 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑂 = ((𝑃 𝑄) (𝐺 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄))) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑈 (𝑆 𝑇)) ∧ ((𝑧𝐴 ∧ ¬ 𝑧 𝑊) ∧ (𝑃 𝑧) = (𝑆 𝑧)))) → 𝑁 = 𝑂)
 
Theoremcdleme21h 37352* Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑂 = ((𝑃 𝑄) (𝐺 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄))) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑈 (𝑆 𝑇)))) → (∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) → 𝑁 = 𝑂))
 
Theoremcdleme21i 37353* Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑂 = ((𝑃 𝑄) (𝐺 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄))) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑈 (𝑆 𝑇)))) → (∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) → 𝑁 = 𝑂))
 
Theoremcdleme21j 37354* Combine cdleme20 37342 and cdleme21i 37353 to eliminate 𝑈 (𝑆 𝑇) condition. (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑂 = ((𝑃 𝑄) (𝐺 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑁 = 𝑂)
 
Theoremcdleme21 37355 Part of proof of Lemma E in [Crawley] p. 113, 3rd line on p. 115. 𝐷, 𝐹, 𝑁, 𝑌, 𝐺, 𝑂 represent s2, f(s), fs(r), t2, f(t), ft(r) respectively. Combine cdleme18d 37313 and cdleme21j 37354 to eliminate existence condition, proving fs(r) = ft(r) with fewer conditions. (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑂 = ((𝑃 𝑄) (𝐺 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) → 𝑁 = 𝑂)
 
Theoremcdleme21k 37356 Eliminate 𝑆𝑇 condition in cdleme21 37355. (Contributed by NM, 26-Dec-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝐷 = ((𝑅 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝑇) 𝑊)    &   𝑁 = ((𝑃 𝑄) (𝐹 𝐷))    &   𝑂 = ((𝑃 𝑄) (𝐺 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) → 𝑁 = 𝑂)
 
Theoremcdleme22aa 37357 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 3rd line on p. 115. Show that t v = p q implies v = u. (Contributed by NM, 2-Dec-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴𝑉 𝑊𝑉 (𝑃 𝑄))) → 𝑉 = 𝑈)
 
Theoremcdleme22a 37358 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 3rd line on p. 115. Show that t v = p q implies v = u. (Contributed by NM, 30-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑇𝐴) ∧ ((𝑉𝐴𝑉 𝑊) ∧ 𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄))) → 𝑉 = 𝑈)
 
Theoremcdleme22b 37359 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 5th line on p. 115. Show that t v =/= p q and s p q implies ¬ t p q. (Contributed by NM, 2-Dec-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑆𝑇)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑇 𝑉) ≠ (𝑃 𝑄) ∧ 𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)))) → ¬ 𝑇 (𝑃 𝑄))
 
Theoremcdleme22cN 37360 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 5th line on p. 115. Show that t v =/= p q and s p q implies ¬ v p q. (Contributed by NM, 3-Dec-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (𝑆 (𝑇 𝑉) ∧ 𝑆 (𝑃 𝑄)) ∧ (𝑇 𝑉) ≠ (𝑃 𝑄))) → ¬ 𝑉 (𝑃 𝑄))
 
Theoremcdleme22d 37361 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 9th line on p. 115. (Contributed by NM, 4-Dec-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑉 = ((𝑆 𝑇) 𝑊))
 
Theoremcdleme22e 37362 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t v = p q, fz(s) fz(t) v. (Contributed by NM, 6-Dec-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑧) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑁 (𝑂 𝑉))
 
Theoremcdleme22eALTN 37363 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t v = p q, fz(s) fz(t) v. (Contributed by NM, 6-Dec-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑦 𝑈) (𝑄 ((𝑃 𝑦) 𝑊)))    &   𝐺 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑦) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝐺 ((𝑇 𝑧) 𝑊)))       (((𝐾 ∈ HL ∧ 𝑊𝐻𝑇𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑃𝑄) ∧ (𝑆𝐴 ∧ (𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ ((𝑦𝐴 ∧ ¬ 𝑦 𝑊) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊)))) → 𝑁 (𝑂 𝑉))
 
Theoremcdleme22f 37364 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝑁 represent f(t), ft(s) respectively. If s t v, then ft(s) f(t) v. (Contributed by NM, 6-Dec-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑇) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑁 (𝐹 𝑉))
 
Theoremcdleme22f2 37365 Part of proof of Lemma E in [Crawley] p. 113. cdleme22f 37364 with s and t swapped (this case is not mentioned by them). If s t v, then f(s) fs(t) v. (Contributed by NM, 7-Dec-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑇 𝑆) 𝑊)))       ((((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))) → 𝐹 (𝑁 𝑉))
 
Theoremcdleme22g 37366 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝐺 represent f(s), f(t) respectively. If s t v and ¬ s p q, then f(s) f(t) v. (Contributed by NM, 6-Dec-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐹 (𝐺 𝑉))
 
Theoremcdleme23a 37367 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 8-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑆 𝑇) (𝑋 𝑊))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑉 𝑊)
 
Theoremcdleme23b 37368 Part of proof of Lemma E in [Crawley] p. 113, 4th paragraph, 6th line on p. 115. (Contributed by NM, 8-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑆 𝑇) (𝑋 𝑊))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑉𝐴)
 
Theoremcdleme23c 37369 Part of proof of Lemma E in [Crawley] p. 113, 4th paragraph, 6th line on p. 115. (Contributed by NM, 8-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑆 𝑇) (𝑋 𝑊))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆 (𝑇 𝑉))
 
Theoremcdleme24 37370* Quantified version of cdleme21k 37356. (Contributed by NM, 26-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊)))    &   𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝐺 ((𝑅 𝑡) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ∀𝑠𝐴𝑡𝐴 (((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄))) → 𝑁 = 𝑂))
 
Theoremcdleme25a 37371* Lemma for cdleme25b 37372. (Contributed by NM, 1-Jan-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ∃𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ 𝑁𝐵))
 
Theoremcdleme25b 37372* Transform cdleme24 37370. TODO get rid of $d's on 𝑈, 𝑁 (Contributed by NM, 1-Jan-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ∃𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑁))
 
Theoremcdleme25c 37373* Transform cdleme25b 37372. (Contributed by NM, 1-Jan-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ∃!𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑁))
 
Theoremcdleme25dN 37374* Transform cdleme25c 37373. (Contributed by NM, 19-Jan-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → ∃!𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ 𝑢 = 𝑁))
 
Theoremcdleme25cl 37375* Show closure of the unique element in cdleme25c 37373. (Contributed by NM, 2-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊)))    &   𝐼 = (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑁))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄))) → 𝐼𝐵)
 
Theoremcdleme25cv 37376* Change bound variables in cdleme25c 37373. (Contributed by NM, 2-Feb-2013.)
𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑠) 𝑊)))    &   𝐺 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝐺 ((𝑅 𝑧) 𝑊)))    &   𝐼 = (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))       𝐼 = 𝐸
 
Theoremcdleme26e 37377* Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t v = p q, fz(s) fz(t) v. TODO: FIX COMMENT. (Contributed by NM, 2-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑧) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊)))    &   𝐼 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐼 (𝐸 𝑉))
 
Theoremcdleme26ee 37378* Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t v = p q, fz(s) fz(t) v. TODO: FIX COMMENT. (Contributed by NM, 2-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑧) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊)))    &   𝐼 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ (𝑇 𝑉) = (𝑃 𝑄))) → 𝐼 (𝐸 𝑉))
 
Theoremcdleme26eALTN 37379* Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t v = p q, fz(s) fz(t) v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑦 𝑈) (𝑄 ((𝑃 𝑦) 𝑊)))    &   𝐺 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑦) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝐺 ((𝑇 𝑧) 𝑊)))    &   𝐼 = (𝑢𝐵𝑦𝐴 ((¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝐼 (𝐸 𝑉))
 
Theoremcdleme26fALTN 37380* Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝑁 represent f(t), ft(s) respectively. If t t v, then ft(s) f(t) v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑡) 𝑊)))    &   𝐼 = (𝑢𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑢 = 𝑁))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐼 (𝐹 𝑉))
 
Theoremcdleme26f 37381* Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝑁 represent f(t), ft(s) respectively. If t t v, then ft(s) f(t) v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑡) 𝑊)))    &   𝐼 = (𝑢𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑢 = 𝑁))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑆 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑆𝑡𝑆 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐼 (𝐹 𝑉))
 
Theoremcdleme26f2ALTN 37382* Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 37380 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.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐺 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝐺 ((𝑇 𝑠) 𝑊)))    &   𝐸 = (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑂))       ((((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))
 
Theoremcdleme26f2 37383* Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 37380 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.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐺 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝐺 ((𝑇 𝑠) 𝑊)))    &   𝐸 = (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = 𝑂))       ((((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))
 
Theoremcdleme27cl 37384* Part of proof of Lemma E in [Crawley] p. 113. Closure of 𝐶. (Contributed by NM, 6-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ 𝑃𝑄)) → 𝐶𝐵)
 
Theoremcdleme27a 37385* Part of proof of Lemma E in [Crawley] p. 113. cdleme26f 37381 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, 3-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)    &   𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))    &   𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))    &   𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉))
 
Theoremcdleme27b 37386* Lemma for cdleme27N 37387. (Contributed by NM, 3-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)    &   𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))    &   𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))    &   𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)       (𝑠 = 𝑡𝐶 = 𝑌)
 
Theoremcdleme27N 37387* Part of proof of Lemma E in [Crawley] p. 113. Eliminate the 𝑠𝑡 antecedent in cdleme27a 37385. (Contributed by NM, 3-Feb-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)    &   𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))    &   𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))    &   𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠 (𝑡 𝑉) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉))
 
Theoremcdleme28a 37388* Lemma for cdleme25b 37372. TODO: FIX COMMENT. (Contributed by NM, 4-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)    &   𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))    &   𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))    &   𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)    &   𝑉 = ((𝑠 𝑡) (𝑋 𝑊))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝐶 (𝑋 𝑊)) (𝑌 (𝑋 𝑊)))
 
Theoremcdleme28b 37389* Lemma for cdleme25b 37372. TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)    &   𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))    &   𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))    &   𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝐶 (𝑋 𝑊)) = (𝑌 (𝑋 𝑊)))
 
Theoremcdleme28c 37390* Part of proof of Lemma E in [Crawley] p. 113. Eliminate the 𝑠𝑡 antecedent in cdleme28b 37389. TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)    &   𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))    &   𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))    &   𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝐶 (𝑋 𝑊)) = (𝑌 (𝑋 𝑊)))
 
Theoremcdleme28 37391* Quantified version of cdleme28c 37390. (Compare cdleme24 37370.) (Contributed by NM, 7-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)    &   𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))    &   𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))    &   𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∀𝑠𝐴𝑡𝐴 (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (¬ 𝑡 𝑊 ∧ (𝑡 (𝑋 𝑊)) = 𝑋)) → (𝐶 (𝑋 𝑊)) = (𝑌 (𝑋 𝑊))))
 
Theoremcdleme29ex 37392* Lemma for cdleme29b 37393. (Compare cdleme25a 37371.) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (𝐶 (𝑋 𝑊)) ∈ 𝐵))
 
Theoremcdleme29b 37393* Transform cdleme28 37391. (Compare cdleme25b 37372.) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑣𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑣 = (𝐶 (𝑋 𝑊))))
 
Theoremcdleme29c 37394* Transform cdleme28b 37389. (Compare cdleme25c 37373.) TODO: FIX COMMENT. (Contributed by NM, 8-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃!𝑣𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑣 = (𝐶 (𝑋 𝑊))))
 
Theoremcdleme29cl 37395* Show closure of the unique element in cdleme28c 37390. (Contributed by NM, 8-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))    &   𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))    &   𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))    &   𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)    &   𝐼 = (𝑣𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑣 = (𝐶 (𝑋 𝑊))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → 𝐼𝐵)
 
Theoremcdleme30a 37396 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐴 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝑌𝐵) ∧ ((𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝑠 (𝑌 𝑊)) = 𝑌)
 
Theoremcdleme31so 37397* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Feb-2013.)
𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐶 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊))))       (𝑋𝐵𝑋 / 𝑥𝑂 = 𝐶)
 
Theoremcdleme31sn 37398* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝐶 = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)       (𝑅𝐴𝑅 / 𝑠𝑁 = 𝐶)
 
Theoremcdleme31sn1 37399* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝐶 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺))       ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝐶)
 
Theoremcdleme31se 37400* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑇) 𝑊)))    &   𝑌 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊)))       (𝑅𝐴𝑅 / 𝑠𝐸 = 𝑌)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
  Copyright terms: Public domain < Previous  Next >