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Theorem List for Metamath Proof Explorer - 31001-31100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremarglem1N 31001 Lemma for Desargues' law. Theorem 13.3 of [Crawley] p. 110, 3rd and 4th lines from bottom. In these lemmas, , , , , , , , , , , and represent Crawley's a0, a1, a2, b0, b1, b2, c, z0, z1, z2, and p respectively. (Contributed by NM, 28-Jun-2012.) (New usage is discouraged.)

Theoremcdlemc1 31002 Part of proof of Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 30675? (Contributed by NM, 29-May-2012.)

Theoremcdlemc2 31003 Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)

Theoremcdlemc3 31004 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)

Theoremcdlemc4 31005 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)

Theoremcdlemc5 31006 Lemma for cdlemc 31008. (Contributed by NM, 26-May-2012.)

Theoremcdlemc6 31007 Lemma for cdlemc 31008. (Contributed by NM, 26-May-2012.)

Theoremcdlemc 31008 Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)

Theoremcdlemd1 31009 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)

Theoremcdlemd2 31010 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)

Theoremcdlemd3 31011 Part of proof of Lemma D in [Crawley] p. 113. The requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012.)

Theoremcdlemd4 31012 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)

Theoremcdlemd5 31013 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)

Theoremcdlemd6 31014 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-May-2012.)

Theoremcdlemd7 31015 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)

Theoremcdlemd8 31016 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)

Theoremcdlemd9 31017 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)

Theoremcdlemd 31018 If two translations agree at any atom not under the fiducial co-atom , then they are equal. Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)

Theoremltrneq3 31019 Two translations agree at any atom not under the fiducial co-atom iff they are equal. (Contributed by NM, 25-Jul-2013.)

Theoremcdleme00a 31020 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme0aa 31021 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme0a 31022 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)

Theoremcdleme0b 31023 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0c 31024 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)

Theoremcdleme0cp 31025 Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 31408- swap consequent equality; make antecedent use df-3an 936. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0cq 31026 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Apr-2013.)

Theoremcdleme0dN 31027 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)

Theoremcdleme0e 31028 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0fN 31029 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)

Theoremcdleme0gN 31030 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)

Theoremcdlemeulpq 31031 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Dec-2012.)

Theoremcdleme01N 31032 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)

Theoremcdleme02N 31033 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Theoremcdleme0ex1N 31034* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Theoremcdleme0ex2N 31035* Part of proof of Lemma E in [Crawley] p. 113. Note that is a shorter way to express . (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Theoremcdleme0moN 31036* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Theoremcdleme1b 31037 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma showing is a lattice element. represents their f(r). (Contributed by NM, 6-Jun-2012.)

Theoremcdleme1 31038 Part of proof of Lemma E in [Crawley] p. 113. represents their f(r). Here we show r f(r) = r u (7th through 5th lines from bottom on p. 113). (Contributed by NM, 4-Jun-2012.)

Theoremcdleme2 31039 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)) w = u in their notation (4th line from bottom on p. 113). (Contributed by NM, 5-Jun-2012.)

Theoremcdleme3b 31040 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 31047 and cdleme3 31048. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3c 31041 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 31047 and cdleme3 31048. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3d 31042 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 31047 and cdleme3 31048. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3e 31043 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 31047 and cdleme3 31048. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3fN 31044 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 31047 and cdleme3 31048. TODO: Delete - duplicates cdleme0e 31028. (Contributed by NM, 6-Jun-2012.) (New usage is discouraged.)

Theoremcdleme3g 31045 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 31047 and cdleme3 31048. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme3h 31046 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 31047 and cdleme3 31048. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3fa 31047 Part of proof of Lemma E in [Crawley] p. 113. See cdleme3 31048. (Contributed by NM, 6-Oct-2012.)

Theoremcdleme3 31048 Part of proof of Lemma E in [Crawley] p. 113. represents f(r). is the fiducial co-atom (hyperplane) w. Here and in cdleme3fa 31047 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 31040 through cdleme3 31048, so there may be a simpler proof that we have overlooked. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme4 31049 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.)

Theoremcdleme4a 31050 Part of proof of Lemma E in [Crawley] p. 114 top. represents fs(r). Auxiliary lemma derived from cdleme5 31051. We show fs(r) p q. (Contributed by NM, 10-Nov-2012.)

Theoremcdleme5 31051 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.)

Theoremcdleme6 31052 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.)

Theoremcdleme7aa 31053 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 31059 and cdleme7 31060. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7a 31054 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 31059 and cdleme7 31060. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7b 31055 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 31059 and cdleme7 31060. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7c 31056 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 31059 and cdleme7 31060. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7d 31057 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 31059 and cdleme7 31060. (Contributed by NM, 8-Jun-2012.)

Theoremcdleme7e 31058 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 31059 and cdleme7 31060. (Contributed by NM, 8-Jun-2012.)

Theoremcdleme7ga 31059 Part of proof of Lemma E in [Crawley] p. 113. See cdleme7 31060. (Contributed by NM, 8-Jun-2012.)

Theoremcdleme7 31060 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 31059 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 31053 through cdleme7 31060, so there may be a simpler proof that we have overlooked. (Contributed by NM, 9-Jun-2012.)

Theoremcdleme8 31061 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.)

Theoremcdleme9a 31062 Part of proof of Lemma E in [Crawley] p. 113. represents s1, which we prove is an atom. (Contributed by NM, 10-Jun-2012.)

Theoremcdleme9b 31063 Utility lemma for Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Oct-2012.)

Theoremcdleme9 31064 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.)

Theoremcdleme10 31065 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.)

Theoremcdleme8tN 31066 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.)

Theoremcdleme9taN 31067 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.)

Theoremcdleme9tN 31068 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.)

Theoremcdleme10tN 31069 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.)

Theoremcdleme16aN 31070 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.)

Theoremcdleme11a 31071 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 31081. (Contributed by NM, 12-Jun-2012.)

Theoremcdleme11c 31072 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 31081. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme11dN 31073 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 31081. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)

Theoremcdleme11e 31074 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 31081. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme11fN 31075 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 31081. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)

Theoremcdleme11g 31076 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 31081. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme11h 31077 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 31081. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme11j 31078 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 31081. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme11k 31079 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 31081. (Contributed by NM, 15-Jun-2012.)

Theoremcdleme11l 31080 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 31081. (Contributed by NM, 15-Jun-2012.)

Theoremcdleme11 31081 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 31071 through cdleme11 31081, so there may be a simpler proof that we have overlooked. (Contributed by NM, 15-Jun-2012.)

Theoremcdleme12 31082 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.)

Theoremcdleme13 31083 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.)

Theoremcdleme14 31084 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 30697 to cdleme13 31083. and represent f(s) and f(t) respectively. (Contributed by NM, 8-Oct-2012.)

Theoremcdleme15a 31085 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.)

Theoremcdleme15b 31086 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.)

Theoremcdleme15c 31087 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.)

Theoremcdleme15d 31088 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.)

Theoremcdleme15 31089 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.)

Theoremcdleme16b 31090 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.)

Theoremcdleme16c 31091 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.)

Theoremcdleme16d 31092 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.)

Theoremcdleme16e 31093 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.)

Theoremcdleme16f 31094 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.)

Theoremcdleme16g 31095 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.)

Theoremcdleme16 31096 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.)

Theoremcdleme17a 31097 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.)

Theoremcdleme17b 31098 Lemma leading to cdleme17c 31099. (Contributed by NM, 11-Oct-2012.)

Theoremcdleme17c 31099 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.)

Theoremcdleme17d1 31100 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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