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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | cdleme2 34901 |
Part of proof of Lemma E in [Crawley] p. 113.
 represents f(r).
 is the
fiducial co-atom (hyperplane) w. Here we show that (r
 f(r)) ![/\](wedge.gif "/\"){kind=small} w = u in their
notation (4th line from bottom on
p. 113). (Contributed by NM, 5-Jun-2012.)
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 ![./\](_.wedge.gif "./\"){kind=small}
       
  
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| Theorem | cdleme3b 34902 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 34909 and cdleme3 34910. (Contributed by NM, 6-Jun-2012.) |
 | ||
| Theorem | cdleme3c 34903 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 34909 and cdleme3 34910. (Contributed by NM, 6-Jun-2012.) |
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| Theorem | cdleme3d 34904 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 34909 and cdleme3 34910. (Contributed by NM, 6-Jun-2012.) |
 | ||
| Theorem | cdleme3e 34905 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 34909 and cdleme3 34910. (Contributed by NM, 6-Jun-2012.) |
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| Theorem | cdleme3fN 34906 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 34909 and cdleme3 34910. TODO: Delete - duplicates cdleme0e 34890. (Contributed by NM, 6-Jun-2012.) (New usage is discouraged.) |
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| Theorem | cdleme3g 34907 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 34909 and cdleme3 34910. (Contributed by NM, 7-Jun-2012.) |
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| Theorem | cdleme3h 34908 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 34909 and cdleme3 34910. (Contributed by NM, 6-Jun-2012.) |
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| Theorem | cdleme3fa 34909 | Part of proof of Lemma E in [Crawley] p. 113. See cdleme3 34910. (Contributed by NM, 6-Oct-2012.) |
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| Theorem | cdleme3 34910 |
Part of proof of Lemma E in [Crawley] p. 113.
represents f(r).
is the
fiducial co-atom (hyperplane) w. Here and in cdleme3fa 34909
above, we show that f(r) W (4th line from bottom on p. 113),
meaning it is an atom and not under w, which in our notation is
expressed as
. Their proof provides no
details
of our lemmas cdleme3b 34902 through cdleme3 34910, so there may be a simpler
proof that we have overlooked. (Contributed by NM, 7-Jun-2012.)
|
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| Theorem | cdleme4 34911 |
Part of proof of Lemma E in [Crawley] p. 113.
and  represent
f(s) and fs(r). Here show p q = r u at the top of
p. 114. (Contributed by NM, 7-Jun-2012.)
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| Theorem | cdleme4a 34912 |
Part of proof of Lemma E in [Crawley] p. 114
top. represents
fs(r). Auxiliary lemma derived from
cdleme5 34913. We show fs(r) 
p q.
(Contributed by NM, 10-Nov-2012.)
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| Theorem | cdleme5 34913 |
Part of proof of Lemma E in [Crawley] p. 113.
represents
fs(r).
We show r
fs(r)) = p q at the top of p. 114.
(Contributed by NM, 7-Jun-2012.)
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| Theorem | cdleme6 34914 |
Part of proof of Lemma E in [Crawley] p. 113.
This expresses (r
fs(r)) w = u at the top of p. 114. (Contributed by
NM,
7-Jun-2012.)
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| Theorem | cdleme7aa 34915 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 34921 and cdleme7 34922. (Contributed by NM, 7-Jun-2012.) |
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| Theorem | cdleme7a 34916 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 34921 and cdleme7 34922. (Contributed by NM, 7-Jun-2012.) |
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| Theorem | cdleme7b 34917 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 34921 and cdleme7 34922. (Contributed by NM, 7-Jun-2012.) |
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| Theorem | cdleme7c 34918 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 34921 and cdleme7 34922. (Contributed by NM, 7-Jun-2012.) |
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| Theorem | cdleme7d 34919 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 34921 and cdleme7 34922. (Contributed by NM, 8-Jun-2012.) |
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| Theorem | cdleme7e 34920 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 34921 and cdleme7 34922. (Contributed by NM, 8-Jun-2012.) |
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| Theorem | cdleme7ga 34921 | Part of proof of Lemma E in [Crawley] p. 113. See cdleme7 34922. (Contributed by NM, 8-Jun-2012.) |
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| Theorem | cdleme7 34922 |
Part of proof of Lemma E in [Crawley] p. 113.
and represent
fs(r) and f(s) respectively. is the fiducial co-atom
(hyperplane)
that they call w. Here and in cdleme7ga 34921 above, we show that fs(r)
W (top of p.
114), meaning it is an atom and not under w, which
in our notation is expressed as
. (Note that
we do not have a symbol for their W.) Their proof provides no details of
our cdleme7aa 34915 through cdleme7 34922, so there may be a simpler proof that
we have overlooked. (Contributed by NM, 9-Jun-2012.)
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| Theorem | cdleme8 34923 |
Part of proof of Lemma E in [Crawley] p. 113,
2nd paragraph on p. 114.
 represents
s1. In their notation, we prove p s1 = p
s.
(Contributed by NM, 9-Jun-2012.)
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| Theorem | cdleme9a 34924 |
Part of proof of Lemma E in [Crawley] p. 113.
represents
s1,
which we prove is an atom. (Contributed by NM, 10-Jun-2012.)
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| Theorem | cdleme9b 34925 | Utility lemma for Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Oct-2012.) |
 
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| Theorem | cdleme9 34926 |
Part of proof of Lemma E in [Crawley] p. 113,
2nd paragraph on p. 114.
and represent s1 and f(s) respectively. In their notation,
we prove f(s) s1 = q s1.
(Contributed by NM,
10-Jun-2012.)
|
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| Theorem | cdleme10 34927 |
Part of proof of Lemma E in [Crawley] p. 113,
2nd paragraph on p. 114.
 represents
s2. In their notation, we prove s s2 = s
r.
(Contributed by NM, 9-Jun-2012.)
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| Theorem | cdleme8tN 34928 |
Part of proof of Lemma E in [Crawley] p. 113,
2nd paragraph on p. 114.
 represents
t1. In their notation, we prove p t1 = p
t.
(Contributed by NM, 8-Oct-2012.)
(New usage is discouraged.)
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| Theorem | cdleme9taN 34929 |
Part of proof of Lemma E in [Crawley] p. 113.
represents
t1,
which we prove is an atom. (Contributed by NM, 8-Oct-2012.)
(New usage is discouraged.)
|
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| Theorem | cdleme9tN 34930 |
Part of proof of Lemma E in [Crawley] p. 113,
2nd paragraph on p. 114.
and represent t1 and f(t) respectively. In their notation,
we prove f(t) t1 = q t1.
(Contributed by NM,
8-Oct-2012.) (New usage is discouraged.)
|
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| Theorem | cdleme10tN 34931 |
Part of proof of Lemma E in [Crawley] p. 113,
2nd paragraph on p. 114.
 represents
t2. In their notation, we prove t t2 = t
r.
(Contributed by NM, 8-Oct-2012.)
(New usage is discouraged.)
|
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| Theorem | cdleme16aN 34932 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
showing, in their notation, s u t u.
(Contributed
by NM, 9-Oct-2012.) (New usage is discouraged.)
|
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| Theorem | cdleme11a 34933 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 34943. (Contributed by NM, 12-Jun-2012.) |
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| Theorem | cdleme11c 34934 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 34943. (Contributed by NM, 13-Jun-2012.) |
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| Theorem | cdleme11dN 34935 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 34943. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.) |
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| Theorem | cdleme11e 34936 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 34943. (Contributed by NM, 13-Jun-2012.) |
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| Theorem | cdleme11fN 34937 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 34943. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.) |
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| Theorem | cdleme11g 34938 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 34943. (Contributed by NM, 14-Jun-2012.) |
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| Theorem | cdleme11h 34939 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 34943. (Contributed by NM, 14-Jun-2012.) |
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| Theorem | cdleme11j 34940 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 34943. (Contributed by NM, 14-Jun-2012.) |
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| Theorem | cdleme11k 34941 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 34943. (Contributed by NM, 15-Jun-2012.) |
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| Theorem | cdleme11l 34942 | Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 34943. (Contributed by NM, 15-Jun-2012.) |
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| Theorem | cdleme11 34943 |
Part of proof of Lemma E in [Crawley] p. 113,
1st sentence of 3rd
paragraph on p. 114. and
represent f(s) and f(t)
respectively. Their proof provides no details of our cdleme11a 34933
through cdleme11 34943, so there may be a simpler proof that we
have
overlooked. (Contributed by NM, 15-Jun-2012.)
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| Theorem | cdleme12 34944 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
first part of 3rd sentence. and
represent f(s) and f(t)
respectively. (Contributed by NM, 16-Jun-2012.)
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| Theorem | cdleme13 34945 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
"<s,t,p> and <f(s),f(t),q> are centrally
perspective."
and
represent f(s) and f(t) respectively. (Contributed by NM,
7-Oct-2012.)
|
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| Theorem | cdleme14 34946 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
"<s,t,p> and <f(s),f(t),q> ... are axially
perspective." We apply
dalaw 34559 to cdleme13 34945. and
represent f(s) and f(t)
respectively. (Contributed by NM, 8-Oct-2012.)
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| Theorem | cdleme15a 34947 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
showing, in their notation, ((s p) (f(s)
q))
((t p) (f(t) q))=((p s1) (q
s1)) ((p t1) (q t1)). We
represent f(s), f(t), s1, and
t1 with , ,
, and
respectively. The order of our operations is slightly
different. (Contributed by NM, 9-Oct-2012.)
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| Theorem | cdleme15b 34948 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
showing, in their notation, (p s1) (q s1)=s1.
We represent s1 with . (Contributed by NM,
10-Oct-2012.)
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| Theorem | cdleme15c 34949 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
showing, in their notation, ((p s1) (q s1))
((p t1) (q
t1))=s1
t1.
and
represent s1 and t1 respectively. The order of our
operations is slightly different. (Contributed by NM,
10-Oct-2012.)
|
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| Theorem | cdleme15d 34950 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
showing, in their notation, s1 t1 w. and
represent s1 and t1 respectively. The order of our operations is
slightly different. (Contributed by NM, 10-Oct-2012.)
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| Theorem | cdleme15 34951 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
showing, in their notation, (s t) (f(s)
f(t))
w. We
use , for f(s), f(t)
respectively.
(Contributed by NM, 10-Oct-2012.)
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| Theorem | cdleme16b 34952 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
first part of 3rd sentence. and
represent f(s) and f(t)
respectively. It is unclear how this follows from s u t
u, as the
authors state, and we used a different proof. (Note:
the antecedent is not used.) (Contributed
by
NM, 11-Oct-2012.)
|
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| Theorem | cdleme16c 34953 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
2nd part of 3rd sentence. and
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.)
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| Theorem | cdleme16d 34954 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
3rd part of 3rd sentence. and
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.)
|
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| Theorem | cdleme16e 34955 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
3rd part of 3rd sentence. and
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.)
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| Theorem | cdleme16f 34956 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
3rd part of 3rd sentence. and
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.)
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| Theorem | cdleme16g 34957 |
Part of proof of Lemma E in [Crawley] p. 113,
3rd paragraph on p. 114,
Eq. (1). and
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.)
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| Theorem | cdleme16 34958 |
Part of proof of Lemma E in [Crawley] p. 113,
conclusion of 3rd
paragraph on p. 114. and
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.)
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| Theorem | cdleme17a 34959 |
Part of proof of Lemma E in [Crawley] p. 114,
first part of 4th
paragraph. ,
, and 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.)
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| Theorem | cdleme17b 34960 | Lemma leading to cdleme17c 34961. (Contributed by NM, 11-Oct-2012.) |
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| Theorem | cdleme17c 34961 |
Part of proof of Lemma E in [Crawley] p. 114,
first part of 4th
paragraph.
represents s1. We show, in their notation,
(p
q) (q s1)=q. (Contributed by NM,
11-Oct-2012.)
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| Theorem | cdleme17d1 34962 |
Part of proof of Lemma E in [Crawley] p. 114,
first part of 4th
paragraph. ,
represent f(s),
fs(p) respectively. We show,
in their notation, fs(p)=q.
(Contributed by NM, 11-Oct-2012.)
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| Theorem | cdleme0nex 34963* |
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 34884- which is under w, so the only 2
left
not under w are p and q themselves.) Note that by cvlsupr2 34017, our
 is a shorter way to express
. 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.)
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| Theorem | cdleme18a 34964 |
Part of proof of Lemma E in [Crawley] p. 114,
2nd sentence of 4th
paragraph. ,
represent f(s),
fs(q) respectively. We show
fs(q) w. (Contributed by NM, 12-Oct-2012.)
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| Theorem | cdleme18b 34965 |
Part of proof of Lemma E in [Crawley] p. 114,
2nd sentence of 4th
paragraph. ,
represent f(s),
fs(q) respectively. We show
fs(q) q. (Contributed by NM, 12-Oct-2012.)
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| Theorem | cdleme18c 34966* |
Part of proof of Lemma E in [Crawley] p. 114,
2nd sentence of 4th
paragraph. ,
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.)
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| Theorem | cdleme22gb 34967 | Utility lemma for Lemma E in [Crawley] p. 115. (Contributed by NM, 5-Dec-2012.) |
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| Theorem | cdleme18d 34968* |
Part of proof of Lemma E in [Crawley] p. 114,
4th sentence of 4th
paragraph. ,
, ,  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 ...). (Contributed by NM, 12-Nov-2012.)
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| Theorem | cdlemesner 34969 | Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 13-Nov-2012.) |
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| Theorem | cdlemedb 34970 |
Part of proof of Lemma E in [Crawley] p. 113.
Utility lemma.
represents s2. (Contributed by NM,
20-Nov-2012.)
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| Theorem | cdlemeda 34971 |
Part of proof of Lemma E in [Crawley] p. 113.
Utility lemma.
represents s2. (Contributed by NM,
13-Nov-2012.)
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| Theorem | cdlemednpq 34972 |
Part of proof of Lemma E in [Crawley] p. 113.
Utility lemma.
represents s2. (Contributed by NM,
18-Nov-2012.)
|
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| Theorem | cdlemednuN 34973 |
Part of proof of Lemma E in [Crawley] p. 113.
Utility lemma.
represents s2. (Contributed by NM,
18-Nov-2012.)
(New usage is discouraged.)
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| Theorem | cdleme20zN 34974 | Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.) (New usage is discouraged.) |
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| Theorem | cdleme20y 34975 | Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.) |
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| Theorem | cdleme19a 34976 |
Part of proof of Lemma E in [Crawley] p. 113,
5th paragraph on p. 114,
1st line.
represents s2. In their notation, we prove
that if r
s t, then s2=(s t) w.
(Contributed by NM,
13-Nov-2012.)
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| Theorem | cdleme19b 34977 |
Part of proof of Lemma E in [Crawley] p. 113,
5th paragraph on p. 114,
1st line. ,
, 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.)
|
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| Theorem | cdleme19c 34978 |
Part of proof of Lemma E in [Crawley] p. 113,
5th paragraph on p. 114,
1st line. ,
represent s2, f(s). We prove f(s)
s2. (Contributed by NM, 13-Nov-2012.)
|
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| Theorem | cdleme19d 34979 |
Part of proof of Lemma E in [Crawley] p. 113,
5th paragraph on p. 114.
, , represent s2, f(s), f(t). We prove f(s)
s2 = f(s) f(t). (Contributed by NM, 14-Nov-2012.)
|
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| Theorem | cdleme19e 34980 |
Part of proof of Lemma E in [Crawley] p. 113,
5th paragraph on p. 114,
line 2. , , , represent s2, f(s),
t2, f(t).
We prove f(s) s2=f(t) t2.
(Contributed by NM,
14-Nov-2012.)
|
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| Theorem | cdleme19f 34981 |
Part of proof of Lemma E in [Crawley] p. 113,
5th paragraph on p. 114,
line 3. ,
,  , , ,
 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.)
|
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| Theorem | cdleme20aN 34982 |
Part of proof of Lemma E in [Crawley] p. 113,
last paragraph on p. 114.
, , , represent s2, f(s),
t2, f(t).
(Contributed by NM, 14-Nov-2012.) (New usage is discouraged.)
|
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| Theorem | cdleme20bN 34983 |
Part of proof of Lemma E in [Crawley] p. 113,
last paragraph on p. 114,
second line. ,
, , represent s2, f(s),
t2,
f(t). We show v s2 = v t2.
(Contributed by NM,
15-Nov-2012.) (New usage is discouraged.)
|
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| Theorem | cdleme20c 34984 |
Part of proof of Lemma E in [Crawley] p. 113,
last paragraph on p. 114,
second line. ,
, , represent s2, f(s),
t2,
f(t). (Contributed by NM, 15-Nov-2012.)
|
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| Theorem | cdleme20d 34985 |
Part of proof of Lemma E in [Crawley] p. 113,
last paragraph on p. 114,
second line. ,
, , represent s2, f(s),
t2,
f(t). (Contributed by NM, 17-Nov-2012.)
|
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| Theorem | cdleme20e 34986 |
Part of proof of Lemma E in [Crawley] p. 113,
last paragraph on p. 114,
4th line. ,
, , 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.)
|
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| Theorem | cdleme20f 34987 |
Part of proof of Lemma E in [Crawley] p. 113,
last paragraph on p. 114,
4th line. ,
, , 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.)
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| Theorem | cdleme20g 34988 |
Part of proof of Lemma E in [Crawley] p. 113,
last paragraph on p. 114,
antepenultimate line. , , , represent s2, f(s),
t2, f(t). (Contributed by NM,
18-Nov-2012.)
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