P. 384 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38301-38400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cdleme16c 38301	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 2nd 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)=s ∨ t ∨ u. (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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∨ (𝐹 ∨ 𝐺)) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	cdleme16d 38302	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)) is an atom. (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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∧ (𝐹 ∨ 𝐺)) ∈ 𝐴)
Theorem	cdleme16e 38303	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))=(s ∨ 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∧ (𝐹 ∨ 𝐺)) = ((𝑆 ∨ 𝑇) ∧ 𝑊))
Theorem	cdleme16f 38304	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∧ (𝐹 ∨ 𝐺)) = ((𝐹 ∨ 𝐺) ∧ 𝑊))
Theorem	cdleme16g 38305	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∧ 𝑊) = ((𝐹 ∨ 𝐺) ∧ 𝑊))
Theorem	cdleme16 38306	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) → ((𝑆 ∨ 𝑇) ∧ 𝑊) = ((𝐹 ∨ 𝐺) ∧ 𝑊))
Theorem	cdleme17a 38307	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝐶)))
Theorem	cdleme17b 38308	Lemma leading to cdleme17c 38309. (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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
Theorem	cdleme17c 38309	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝐶)) = 𝑄)
Theorem	cdleme17d1 38310	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝐺 = 𝑄)
Theorem	cdleme0nex 38311*	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 38232- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 37364, our (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) is a shorter way to express 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (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) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))
Theorem	cdleme18a 38312	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝐺 ≤ 𝑊)
Theorem	cdleme18b 38313	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐺 ≠ 𝑄)
Theorem	cdleme18c 38314*	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝐺 = 𝑃)
Theorem	cdleme22gb 38315	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐺 ∈ 𝐵)
Theorem	cdleme18d 38316*	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 ¬ ∃𝑟 ∈ 𝐴...). (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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝐺 = 𝐸)
Theorem	cdlemesner 38317	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 ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑆 ≠ 𝑅)
Theorem	cdlemedb 38318	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐷 ∈ 𝐵)
Theorem	cdlemeda 38319	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐷 ∈ 𝐴)
Theorem	cdlemednpq 38320	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝐷 ≤ (𝑃 ∨ 𝑄))
Theorem	cdlemednuN 38321	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐷 ≠ 𝑈)
Theorem	cdleme20zN 38322	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 ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑅) ∧ 𝑇) = (0.‘𝐾))
Theorem	cdleme20y 38323	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 ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑅) ∧ (𝑇 ∨ 𝑅)) = 𝑅)
Theorem	cdleme19a 38324	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 ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐷 = ((𝑆 ∨ 𝑇) ∧ 𝑊))
Theorem	cdleme19b 38325	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → 𝐷 ≤ (𝐹 ∨ 𝐺))
Theorem	cdleme19c 38326	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐹 ≠ 𝐷)
Theorem	cdleme19d 38327	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → (𝐹 ∨ 𝐷) = (𝐹 ∨ 𝐺))
Theorem	cdleme19e 38328	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → (𝐹 ∨ 𝐷) = (𝐺 ∨ 𝑌))
Theorem	cdleme19f 38329	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → 𝑁 = 𝑂)
Theorem	cdleme20aN 38330	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑉 ∨ 𝐷) = (((𝑆 ∨ 𝑅) ∨ 𝑇) ∧ 𝑊))
Theorem	cdleme20bN 38331	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑉 ∨ 𝐷) = (𝑉 ∨ 𝑌))
Theorem	cdleme20c 38332	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝐷 ∨ 𝑌) = (((𝑅 ∨ 𝑆) ∨ 𝑇) ∧ 𝑊))
Theorem	cdleme20d 38333	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝐹 ∨ 𝐺) ∧ (𝐷 ∨ 𝑌)) = 𝑉)
Theorem	cdleme20e 38334	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝐹 ∨ 𝐺) ∧ (𝐷 ∨ 𝑌)) ≤ (𝑆 ∨ 𝑇))
Theorem	cdleme20f 38335	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝐹 ∨ 𝐷) ∧ (𝐺 ∨ 𝑌)) ≤ (((𝐷 ∨ 𝑆) ∧ (𝑌 ∨ 𝑇)) ∨ ((𝑆 ∨ 𝐹) ∧ (𝑇 ∨ 𝐺))))
Theorem	cdleme20g 38336	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 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))))
Theorem	cdleme20h 38337	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 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))
Theorem	cdleme20i 38338	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑅 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → ((𝐹 ∨ 𝐷) ∧ (𝐺 ∨ 𝑌)) ≤ (𝑃 ∨ 𝑄))
Theorem	cdleme20j 38339	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐷 ≠ 𝑌)
Theorem	cdleme20k 38340	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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝐹 ∨ 𝐷) ≠ (𝑃 ∨ 𝑄))
Theorem	cdleme20l1 38341	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝐹 ∨ 𝐷) ∈ (LLines‘𝐾))
Theorem	cdleme20l2 38342	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑅 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → ((𝐹 ∨ 𝐷) ∧ (𝐺 ∨ 𝑌)) ∈ 𝐴)
Theorem	cdleme20l 38343	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑅 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → ((𝐹 ∨ 𝐷) ∧ (𝐺 ∨ 𝑌)) = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)))
Theorem	cdleme20m 38344	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑅 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → 𝑁 = 𝑂)
Theorem	cdleme20 38345	Combine cdleme19f 38329 and cdleme20m 38344 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑁 = 𝑂)
Theorem	cdleme21a 38346	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 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑆 ≠ 𝑧)
Theorem	cdleme21b 38347	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 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))
Theorem	cdleme21c 38348	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → ¬ 𝑈 ≤ (𝑆 ∨ 𝑧))
Theorem	cdleme21at 38349	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑇 ≠ 𝑧)
Theorem	cdleme21ct 38350	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇))) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → ¬ 𝑈 ≤ (𝑇 ∨ 𝑧))
Theorem	cdleme21d 38351	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))) → 𝑁 = 𝑍)
Theorem	cdleme21e 38352	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))) → 𝑂 = 𝑍)
Theorem	cdleme21f 38353	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))) → 𝑁 = 𝑂)
Theorem	cdleme21g 38354	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))) → 𝑁 = 𝑂)
Theorem	cdleme21h 38355*	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) → 𝑁 = 𝑂))
Theorem	cdleme21i 38356*	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) → 𝑁 = 𝑂))
Theorem	cdleme21j 38357*	Combine cdleme20 38345 and cdleme21i 38356 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑁 = 𝑂)
Theorem	cdleme21 38358	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 38316 and cdleme21j 38357 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))) → 𝑁 = 𝑂)
Theorem	cdleme21k 38359	Eliminate 𝑆 ≠ 𝑇 condition in cdleme21 38358. (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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))) → 𝑁 = 𝑂)
Theorem	cdleme22aa 38360	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ 𝑉 ≤ (𝑃 ∨ 𝑄))) → 𝑉 = 𝑈)
Theorem	cdleme22a 38361	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → 𝑉 = 𝑈)
Theorem	cdleme22b 38362	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 ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
Theorem	cdleme22cN 38363	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄))) → ¬ 𝑉 ≤ (𝑃 ∨ 𝑄))
Theorem	cdleme22d 38364	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊))
Theorem	cdleme22e 38365	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑁 ≤ (𝑂 ∨ 𝑉))
Theorem	cdleme22eALTN 38366	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 ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑁 ≤ (𝑂 ∨ 𝑉))
Theorem	cdleme22f 38367	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑁 ≤ (𝐹 ∨ 𝑉))
Theorem	cdleme22f2 38368	Part of proof of Lemma E in [Crawley] p. 113. cdleme22f 38367 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐹 ≤ (𝑁 ∨ 𝑉))
Theorem	cdleme22g 38369	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐹 ≤ (𝐺 ∨ 𝑉))
Theorem	cdleme23a 38370	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑉 ≤ 𝑊)
Theorem	cdleme23b 38371	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑉 ∈ 𝐴)
Theorem	cdleme23c 38372	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑆 ≤ (𝑇 ∨ 𝑉))
Theorem	cdleme24 38373*	Quantified version of cdleme21k 38359. (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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))) → 𝑁 = 𝑂))
Theorem	cdleme25a 38374*	Lemma for cdleme25b 38375. (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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑁 ∈ 𝐵))
Theorem	cdleme25b 38375*	Transform cdleme24 38373. 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
Theorem	cdleme25c 38376*	Transform cdleme25b 38375. (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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃!𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
Theorem	cdleme25dN 38377*	Transform cdleme25c 38376. (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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃!𝑢 ∈ 𝐵 ∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑢 = 𝑁))
Theorem	cdleme25cl 38378*	Show closure of the unique element in cdleme25c 38376. (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 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐼 ∈ 𝐵)
Theorem	cdleme25cv 38379*	Change bound variables in cdleme25c 38376. (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 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    &   ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    ⇒   ⊢ 𝐼 = 𝐸
Theorem	cdleme26e 38380*	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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    &   ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄) ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝐼 ≤ (𝐸 ∨ 𝑉))
Theorem	cdleme26ee 38381*	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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    &   ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → 𝐼 ≤ (𝐸 ∨ 𝑉))
Theorem	cdleme26eALTN 38382*	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 = (℩𝑢 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    &   ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝐼 ≤ (𝐸 ∨ 𝑉))
Theorem	cdleme26fALTN 38383*	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 = (℩𝑢 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑆 ≠ 𝑡 ∧ 𝑆 ≤ (𝑡 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐼 ≤ (𝐹 ∨ 𝑉))
Theorem	cdleme26f 38384*	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 = (℩𝑢 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑡 ≤ (𝑃 ∨ 𝑄) ∧ (𝑆 ≠ 𝑡 ∧ 𝑆 ≤ (𝑡 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐼 ≤ (𝐹 ∨ 𝑉))
Theorem	cdleme26f2ALTN 38385*	Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 38383 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 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐺 ≤ (𝐸 ∨ 𝑉))
Theorem	cdleme26f2 38386*	Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 38383 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 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (¬ 𝑠 ≤ (𝑃 ∨ 𝑄) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐺 ≤ (𝐸 ∨ 𝑉))
Theorem	cdleme27cl 38387*	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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    &   ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄)) → 𝐶 ∈ 𝐵)
Theorem	cdleme27a 38388*	Part of proof of Lemma E in [Crawley] p. 113. cdleme26f 38384 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    &   ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ ((𝑠 ≠ 𝑡 ∧ 𝑠 ≤ (𝑡 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐶 ≤ (𝑌 ∨ 𝑉))
Theorem	cdleme27b 38389*	Lemma for cdleme27N 38390. (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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    &   ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺)    ⇒   ⊢ (𝑠 = 𝑡 → 𝐶 = 𝑌)
Theorem	cdleme27N 38390*	Part of proof of Lemma E in [Crawley] p. 113. Eliminate the 𝑠 ≠ 𝑡 antecedent in cdleme27a 38388. (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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    &   ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐶 ≤ (𝑌 ∨ 𝑉))
Theorem	cdleme28a 38391*	Lemma for cdleme25b 38375. 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    &   ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺)    &   ⊢ 𝑉 = ((𝑠 ∨ 𝑡) ∧ (𝑋 ∧ 𝑊))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≠ 𝑡 ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑡 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))) → (𝐶 ∨ (𝑋 ∧ 𝑊)) ≤ (𝑌 ∨ (𝑋 ∧ 𝑊)))
Theorem	cdleme28b 38392*	Lemma for cdleme25b 38375. 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    &   ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≠ 𝑡 ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑡 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))) → (𝐶 ∨ (𝑋 ∧ 𝑊)) = (𝑌 ∨ (𝑋 ∧ 𝑊)))
Theorem	cdleme28c 38393*	Part of proof of Lemma E in [Crawley] p. 113. Eliminate the 𝑠 ≠ 𝑡 antecedent in cdleme28b 38392. 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    &   ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑡 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))) → (𝐶 ∨ (𝑋 ∧ 𝑊)) = (𝑌 ∨ (𝑋 ∧ 𝑊)))
Theorem	cdleme28 38394*	Quantified version of cdleme28c 38393. (Compare cdleme24 38373.) (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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))    &   ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑡 ≤ 𝑊 ∧ (𝑡 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐶 ∨ (𝑋 ∧ 𝑊)) = (𝑌 ∨ (𝑋 ∧ 𝑊))))
Theorem	cdleme29ex 38395*	Lemma for cdleme29b 38396. (Compare cdleme25a 38374.) 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    &   ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (𝐶 ∨ (𝑋 ∧ 𝑊)) ∈ 𝐵))
Theorem	cdleme29b 38396*	Transform cdleme28 38394. (Compare cdleme25b 38375.) 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    &   ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊))))
Theorem	cdleme29c 38397*	Transform cdleme28b 38392. (Compare cdleme25c 38376.) 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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    &   ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃!𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊))))
Theorem	cdleme29cl 38398*	Show closure of the unique element in cdleme28c 38393. (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 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))    &   ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹)    &   ⊢ 𝐼 = (℩𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊))))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝐼 ∈ 𝐵)
Theorem	cdleme30a 38399	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ∨ (𝑌 ∧ 𝑊)) = 𝑌)
Theorem	cdleme31so 38400*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Feb-2013.)
⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))    &   ⊢ 𝐶 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊))))    ⇒   ⊢ (𝑋 ∈ 𝐵 → ⦋𝑋 / 𝑥⦌𝑂 = 𝐶)