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Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltrnatneq 29501 If any atom (under ) is not equal to its translation, so is any other atom. TODO: isn't needed to prove this. Will removing it shorten (and not lengthen) proofs using it? (Contributed by NM, 6-May-2013.)

Theoremltrnatlw 29502 If the value of an atom equals the atom in a non-identity translation, the atom is under the fiducial hyperplane. (Contributed by NM, 15-May-2013.)

Theoremtrlle 29503 The trace of a lattice translation is less than the fiducial co-atom .. (Contributed by NM, 25-May-2012.)

Theoremtrlne 29504 The trace of a lattice translation is not equal to any atom not under the fiducial co-atom . Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)

Theoremtrlnle 29505 The atom not under the fiducial co-atom is not less than the trace of a lattice translation. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 26-May-2012.)

Theoremtrlval3 29506 The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (Contributed by NM, 3-May-2013.)

Theoremtrlval4 29507 The value of the trace of a lattice translation in terms of 2 atoms. (Contributed by NM, 3-May-2013.)

Theoremtrlval5 29508 The value of the trace of a lattice translation in terms of itself. (Contributed by NM, 19-Jul-2013.)

Theoremarglem1N 29509 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 29510 Part of proof of Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 29183? (Contributed by NM, 29-May-2012.)

Theoremcdlemc2 29511 Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)

Theoremcdlemc3 29512 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)

Theoremcdlemc4 29513 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)

Theoremcdlemc5 29514 Lemma for cdlemc 29516. (Contributed by NM, 26-May-2012.)

Theoremcdlemc6 29515 Lemma for cdlemc 29516. (Contributed by NM, 26-May-2012.)

Theoremcdlemc 29516 Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)

Theoremcdlemd1 29517 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)

Theoremcdlemd2 29518 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)

Theoremcdlemd3 29519 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 29520 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)

Theoremcdlemd5 29521 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)

Theoremcdlemd6 29522 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-May-2012.)

Theoremcdlemd7 29523 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)

Theoremcdlemd8 29524 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)

Theoremcdlemd9 29525 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)

Theoremcdlemd 29526 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 29527 Two translations agree at any atom not under the fiducial co-atom iff they are equal. (Contributed by NM, 25-Jul-2013.)

Theoremcdleme00a 29528 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme0aa 29529 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme0a 29530 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)

Theoremcdleme0b 29531 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0c 29532 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)

Theoremcdleme0cp 29533 Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 29916- swap consequent equality; make antecedent use df-3an 941. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0cq 29534 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Apr-2013.)

Theoremcdleme0dN 29535 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)

Theoremcdleme0e 29536 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0fN 29537 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)

Theoremcdleme0gN 29538 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)

Theoremcdlemeulpq 29539 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Dec-2012.)

Theoremcdleme01N 29540 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)

Theoremcdleme02N 29541 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Theoremcdleme0ex1N 29542* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Theoremcdleme0ex2N 29543* 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 29544* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Theoremcdleme1b 29545 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 29546 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 29547 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 29548 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29555 and cdleme3 29556. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3c 29549 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29555 and cdleme3 29556. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3d 29550 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29555 and cdleme3 29556. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3e 29551 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29555 and cdleme3 29556. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3fN 29552 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29555 and cdleme3 29556. TODO: Delete - duplicates cdleme0e 29536. (Contributed by NM, 6-Jun-2012.) (New usage is discouraged.)

Theoremcdleme3g 29553 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29555 and cdleme3 29556. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme3h 29554 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29555 and cdleme3 29556. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3fa 29555 Part of proof of Lemma E in [Crawley] p. 113. See cdleme3 29556. (Contributed by NM, 6-Oct-2012.)

Theoremcdleme3 29556 Part of proof of Lemma E in [Crawley] p. 113. represents f(r). is the fiducial co-atom (hyperplane) w. Here and in cdleme3fa 29555 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 29548 through cdleme3 29556, so there may be a simpler proof that we have overlooked. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme4 29557 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 29558 Part of proof of Lemma E in [Crawley] p. 114 top. represents fs(r). Auxiliary lemma derived from cdleme5 29559. We show fs(r) p q. (Contributed by NM, 10-Nov-2012.)

Theoremcdleme5 29559 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 29560 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 29561 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 29567 and cdleme7 29568. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7a 29562 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 29567 and cdleme7 29568. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7b 29563 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 29567 and cdleme7 29568. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7c 29564 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 29567 and cdleme7 29568. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7d 29565 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 29567 and cdleme7 29568. (Contributed by NM, 8-Jun-2012.)

Theoremcdleme7e 29566 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 29567 and cdleme7 29568. (Contributed by NM, 8-Jun-2012.)

Theoremcdleme7ga 29567 Part of proof of Lemma E in [Crawley] p. 113. See cdleme7 29568. (Contributed by NM, 8-Jun-2012.)

Theoremcdleme7 29568 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 29567 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 29561 through cdleme7 29568, so there may be a simpler proof that we have overlooked. (Contributed by NM, 9-Jun-2012.)

Theoremcdleme8 29569 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 29570 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 29571 Utility lemma for Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Oct-2012.)

Theoremcdleme9 29572 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 29573 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 29574 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 29575 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 29576 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 29577 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 29578 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 29579 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29589. (Contributed by NM, 12-Jun-2012.)

Theoremcdleme11c 29580 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29589. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme11dN 29581 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29589. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)

Theoremcdleme11e 29582 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29589. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme11fN 29583 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29589. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)

Theoremcdleme11g 29584 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29589. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme11h 29585 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29589. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme11j 29586 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29589. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme11k 29587 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29589. (Contributed by NM, 15-Jun-2012.)

Theoremcdleme11l 29588 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29589. (Contributed by NM, 15-Jun-2012.)

Theoremcdleme11 29589 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 29579 through cdleme11 29589, so there may be a simpler proof that we have overlooked. (Contributed by NM, 15-Jun-2012.)

Theoremcdleme12 29590 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 29591 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 29592 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 29205 to cdleme13 29591. and represent f(s) and f(t) respectively. (Contributed by NM, 8-Oct-2012.)

Theoremcdleme15a 29593 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 29594 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 29595 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 29596 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 29597 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 29598 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 29599 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 29600 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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