P. 362 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 36101-36200 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	erngfmul-rN 36101*	Ring multiplication operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
$/\$

Theorem	erngmul-rN 36102	Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	cdlemh1 36103	Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	cdlemh2 36104	Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 16-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	cdlemh 36105	Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemi1 36106	Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemi2 36107	Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemi 36108	Lemma I of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemj1 36109	Part of proof of Lemma J of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemj2 36110	Part of proof of Lemma J of [Crawley] p. 118. Eliminate . (Contributed by NM, 20-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemj3 36111	Part of proof of Lemma J of [Crawley] p. 118. Eliminate . (Contributed by NM, 20-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	tendocan 36112	Cancellation law: if the values of two trace-preserving endormorphisms are equal, so are the endormorphisms. Lemma J of [Crawley] p. 118. (Contributed by NM, 21-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	tendoid0 36113*	A trace-preserving endomorphism is the additive identity iff at least one of its values (at a non-identity translation) is the identity translation. (Contributed by NM, 1-Aug-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendo0mul 36114*	Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 1-Aug-2013.)
$/\$ $/\$

Theorem	tendo0mulr 36115*	Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 13-Feb-2014.)
$/\$ $/\$

Theorem	tendo1ne0 36116*	The identity (unity) is not equal to the zero trace-preserving endomorphism. (Contributed by NM, 8-Aug-2013.)
$/\$

Theorem	tendoconid 36117*	The composition (product) of trace-preserving endormorphisms is nonzero when each argument is nonzero. (Contributed by NM, 8-Aug-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	tendotr 36118*	The trace of the value of a nonzero trace-preserving endomorphism equals the trace of the argument. (Contributed by NM, 11-Aug-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	cdlemk1 36119	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdlemk2 36120	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdlemk3 36121	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk4 36122	Part of proof of Lemma K of [Crawley] p. 118, last line. We use for their h, since is already used. (Contributed by NM, 24-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk5a 36123	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cdlemk5 36124	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 25-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cdlemk6 36125	Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 35172. (Contributed by NM, 25-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdlemk8 36126	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 26-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdlemk9 36127	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	cdlemk9bN 36128	Part of proof of Lemma K of [Crawley] p. 118. TODO: is this needed? If so, shorten with cdlemk9 36127 if that one is also needed. (Contributed by NM, 28-Jun-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	cdlemki 36129*	Part of proof of Lemma K of [Crawley] p. 118. TODO: Eliminate and put into cdlemksel 36133. (Contributed by NM, 25-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkvcl 36130	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk10 36131	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemksv 36132*	Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdlemksel 36133*	Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma(p) function to be a translation. TODO: combine cdlemki 36129? (Contributed by NM, 26-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemksat 36134*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemksv2 36135*	Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function at the fixed parameter. (Contributed by NM, 26-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk7 36136*	Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119. (Contributed by NM, 27-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk11 36137*	Part of proof of Lemma K of [Crawley] p. 118. Eq. 3, line 8, p. 119. (Contributed by NM, 29-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk12 36138*	Part of proof of Lemma K of [Crawley] p. 118. Eq. 4, line 10, p. 119. (Contributed by NM, 30-Jun-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemkoatnle 36139*	Utility lemma. (Contributed by NM, 2-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk13 36140*	Part of proof of Lemma K of [Crawley] p. 118. Line 13 on p. 119. , are k₁, f₁. (Contributed by NM, 1-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemkole 36141*	Utility lemma. (Contributed by NM, 2-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk14 36142*	Part of proof of Lemma K of [Crawley] p. 118. Line 19 on p. 119. , are k₁, f₁. (Contributed by NM, 1-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk15 36143*	Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. , are k₁, f₁. (Contributed by NM, 1-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk16a 36144*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk16 36145*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 1-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk17 36146*	Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. , are k₁, f₁. (Contributed by NM, 1-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk1u 36147*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdlemk5auN 36148*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cdlemk5u 36149*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cdlemk6u 36150*	Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 35172. (Contributed by NM, 4-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdlemkj 36151*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 2-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkuvN 36152*	Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma₁ (p) function . (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdlemkuel 36153*	Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma₁ (p) function to be a translation. TODO: combine cdlemkj 36151? (Contributed by NM, 2-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkuat 36154*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkuv2 36155*	Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 1, where sigma₁ (p) is , f₁ is , and k₁ is . (Contributed by NM, 2-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk18 36156*	Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. , , , are k, sigma₁ (p), k₁, f₁. (Contributed by NM, 2-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk19 36157*	Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. , , , are k, sigma₁ (p), k₁, f₁. (Contributed by NM, 2-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk7u 36158*	Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119 for the sigma₁ case. (Contributed by NM, 3-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk11u 36159*	Part of proof of Lemma K of [Crawley] p. 118. Line 17, p. 119, showing Eq. 3 (line 8, p. 119) for the sigma₁ () case. (Contributed by NM, 4-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk12u 36160*	Part of proof of Lemma K of [Crawley] p. 118. Line 18, p. 119, showing Eq. 4 (line 10, p. 119) for the sigma₁ () case. (Contributed by NM, 4-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk21N 36161*	Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=0 and j=1. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk20 36162*	Part of proof of Lemma K of [Crawley] p. 118. Line 22, p. 119 for the i=2, j=1 case. Note typo on line 22: f should be f_i. Our , , , , , represent their f₁, f₂, k₁, k₂, sigma₁, sigma₂. (Contributed by NM, 5-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkoatnle-2N 36163*	Utility lemma. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk13-2N 36164*	Part of proof of Lemma K of [Crawley] p. 118. Line 13 on p. 119. , are k₂, f₂. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemkole-2N 36165*	Utility lemma. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk14-2N 36166*	Part of proof of Lemma K of [Crawley] p. 118. Line 19 on p. 119. , are k₂, f₂. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk15-2N 36167*	Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. , are k₂, f₂. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk16-2N 36168*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk17-2N 36169*	Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. , are k₂, f₂. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemkj-2N 36170*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkuv-2N 36171*	Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma₂ (p) function, given . (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdlemkuel-2N 36172*	Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma₂ (p) function to be a translation. TODO: combine cdlemkj 36151? (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkuv2-2 36173*	Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 2, where sigma₂ (p) is , f₂ is , and k₂ is . (Contributed by NM, 2-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk18-2N 36174*	Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. , , , are k, sigma₂ (p), k₂, f₂. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk19-2N 36175*	Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. , , , are k, sigma₂ (p), k₂, f₂. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk7u-2N 36176*	Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119 for the sigma₂ case. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk11u-2N 36177*	Part of proof of Lemma K of [Crawley] p. 118. Line 17, p. 119, showing Eq. 3 (line 8, p. 119) for the sigma₂ () case. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk12u-2N 36178*	Part of proof of Lemma K of [Crawley] p. 118. Line 18, p. 119, showing Eq. 4 (line 10, p. 119) for the sigma₂ () case. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk21-2N 36179*	Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=0 and j=2. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk20-2N 36180*	Part of proof of Lemma K of [Crawley] p. 118. Line 22, p. 119 for the i=2, j=1 case. Note typo on line 22: f should be f_i. Our , , , , , represent their f₁, f₂, k₁, k₂, sigma₁, sigma₂. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk22 36181*	Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=1 and j=2. (Contributed by NM, 5-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk30 36182*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 17-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemkuu 36183*	Convert between function and operation forms of . TODO: Use operation form everywhere. (Contributed by NM, 6-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$

Theorem	cdlemk31 36184*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 17-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk32 36185*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 17-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cdlemkuel-3 36186*	Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma₂ (p) function to be a translation. TODO: combine cdlemkj 36151? (Contributed by NM, 11-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkuv2-3N 36187*	Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 1, where sigma₂ (p) is , f₁ is , and k₁ is . (Contributed by NM, 6-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk18-3N 36188*	Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. , , , are k, sigma₂ (p), k₁, f₁. (Contributed by NM, 7-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk22-3 36189*	Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=1 and j=2. (Contributed by NM, 7-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk23-3 36190*	Part of proof of Lemma K of [Crawley] p. 118. Eliminate the requirement from cdlemk22-3 36189. (Contributed by NM, 7-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk24-3 36191*	Part of proof of Lemma K of [Crawley] p. 118. Eliminate the requirement from cdlemk23-3 36190 using . (Contributed by NM, 7-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk25-3 36192*	Part of proof of Lemma K of [Crawley] p. 118. Eliminate the requirement from cdlemk24-3 36191. (Contributed by NM, 7-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk26b-3 36193*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 14-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk26-3 36194*	Part of proof of Lemma K of [Crawley] p. 118. Eliminate the requirements from cdlemk25-3 36192. (Contributed by NM, 10-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk27-3 36195*	Part of proof of Lemma K of [Crawley] p. 118. Eliminate the from the conclusion of cdlemk25-3 36192. (Contributed by NM, 10-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk28-3 36196*	Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 14-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk33N 36197*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. TODO: not needed, is embodied in cdlemk34 36198. (Contributed by NM, 18-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk34 36198*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 18-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cdlemk29-3 36199*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 14-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk35 36200*	Part of proof of Lemma K of [Crawley] p. 118. cdlemk29-3 36199 with shorter hypotheses. (Contributed by NM, 18-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

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