P. 86 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8501-8600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ltordlem 8501*	Lemma for eqord1 8502. (Contributed by Mario Carneiro, 14-Jun-2014.)
|- ( x = y -> A = B )   &    |- ( x = C -> A = M )   &    |- ( x = D -> A = N )   &    |- S C_ RR   &    |- ( ( ph /\ x e. S ) -> A e. RR )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> A < B ) )   =>    |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D -> M < N ) )
Theorem	eqord1 8502*	A strictly increasing real function on a subset of RR is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by Jim Kingdon, 20-Dec-2022.)
|- ( x = y -> A = B )   &    |- ( x = C -> A = M )   &    |- ( x = D -> A = N )   &    |- S C_ RR   &    |- ( ( ph /\ x e. S ) -> A e. RR )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> A < B ) )   =>    |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> M = N ) )
Theorem	eqord2 8503*	A strictly decreasing real function on a subset of RR is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.)
|- ( x = y -> A = B )   &    |- ( x = C -> A = M )   &    |- ( x = D -> A = N )   &    |- S C_ RR   &    |- ( ( ph /\ x e. S ) -> A e. RR )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> B < A ) )   =>    |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> M = N ) )
Theorem	leidi 8504	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
|- A e. RR   =>    |- A <_ A
Theorem	gt0ne0i 8505	Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
|- A e. RR   =>    |- ( 0 < A -> A =/= 0 )
Theorem	gt0ne0ii 8506	Positive implies nonzero. (Contributed by NM, 15-May-1999.)
|- A e. RR   &    |- 0 < A   =>    |- A =/= 0
Theorem	addgt0i 8507	Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A + B ) )
Theorem	addge0i 8508	Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> 0 <_ ( A + B ) )
Theorem	addgegt0i 8509	Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 < B ) -> 0 < ( A + B ) )
Theorem	addgt0ii 8510	Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- 0 < A   &    |- 0 < B   =>    |- 0 < ( A + B )
Theorem	add20i 8511	Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
Theorem	ltnegi 8512	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B <-> -u B < -u A )
Theorem	lenegi 8513	Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A <_ B <-> -u B <_ -u A )
Theorem	ltnegcon2i 8514	Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < -u B <-> B < -u A )
Theorem	lesub0i 8515	Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ B <_ ( B - A ) ) <-> A = 0 )
Theorem	ltaddposi 8516	Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( 0 < A <-> B < ( B + A ) )
Theorem	posdifi 8517	Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B <-> 0 < ( B - A ) )
Theorem	ltnegcon1i 8518	Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( -u A < B <-> -u B < A )
Theorem	lenegcon1i 8519	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
|- A e. RR   &    |- B e. RR   =>    |- ( -u A <_ B <-> -u B <_ A )
Theorem	subge0i 8520	Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
|- A e. RR   &    |- B e. RR   =>    |- ( 0 <_ ( A - B ) <-> B <_ A )
Theorem	ltadd1i 8521	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( A < B <-> ( A + C ) < ( B + C ) )
Theorem	leadd1i 8522	Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( A <_ B <-> ( A + C ) <_ ( B + C ) )
Theorem	leadd2i 8523	Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( A <_ B <-> ( C + A ) <_ ( C + B ) )
Theorem	ltsubaddi 8524	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A - B ) < C <-> A < ( C + B ) )
Theorem	lesubaddi 8525	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A - B ) <_ C <-> A <_ ( C + B ) )
Theorem	ltsubadd2i 8526	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A - B ) < C <-> A < ( B + C ) )
Theorem	lesubadd2i 8527	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A - B ) <_ C <-> A <_ ( B + C ) )
Theorem	ltaddsubi 8528	'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A + B ) < C <-> A < ( C - B ) )
Theorem	lt2addi 8529	Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   &    |- D e. RR   =>    |- ( ( A < C /\ B < D ) -> ( A + B ) < ( C + D ) )
Theorem	le2addi 8530	Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   &    |- D e. RR   =>    |- ( ( A <_ C /\ B <_ D ) -> ( A + B ) <_ ( C + D ) )
Theorem	gt0ne0d 8531	Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> 0 < A )   =>    |- ( ph -> A =/= 0 )
Theorem	lt0ne0d 8532	Something less than zero is not zero. Deduction form. See also lt0ap0d 8668 which is similar but for apartness. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A < 0 )   =>    |- ( ph -> A =/= 0 )
Theorem	leidd 8533	'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A <_ A )
Theorem	lt0neg1d 8534	Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A < 0 <-> 0 < -u A ) )
Theorem	lt0neg2d 8535	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( 0 < A <-> -u A < 0 ) )
Theorem	le0neg1d 8536	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A <_ 0 <-> 0 <_ -u A ) )
Theorem	le0neg2d 8537	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( 0 <_ A <-> -u A <_ 0 ) )
Theorem	addgegt0d 8538	Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> 0 < ( A + B ) )
Theorem	addgtge0d 8539	Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 < A )   &    |- ( ph -> 0 <_ B )   =>    |- ( ph -> 0 < ( A + B ) )
Theorem	addgt0d 8540	Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 < A )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> 0 < ( A + B ) )
Theorem	addge0d 8541	Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 0 <_ B )   =>    |- ( ph -> 0 <_ ( A + B ) )
Theorem	ltnegd 8542	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A < B <-> -u B < -u A ) )
Theorem	lenegd 8543	Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A <_ B <-> -u B <_ -u A ) )
Theorem	ltnegcon1d 8544	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> -u A < B )   =>    |- ( ph -> -u B < A )
Theorem	ltnegcon2d 8545	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < -u B )   =>    |- ( ph -> B < -u A )
Theorem	lenegcon1d 8546	Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> -u A <_ B )   =>    |- ( ph -> -u B <_ A )
Theorem	lenegcon2d 8547	Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ -u B )   =>    |- ( ph -> B <_ -u A )
Theorem	ltaddposd 8548	Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 < A <-> B < ( B + A ) ) )
Theorem	ltaddpos2d 8549	Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 < A <-> B < ( A + B ) ) )
Theorem	ltsubposd 8550	Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 < A <-> ( B - A ) < B ) )
Theorem	posdifd 8551	Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
Theorem	addge01d 8552	A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 <_ B <-> A <_ ( A + B ) ) )
Theorem	addge02d 8553	A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 <_ B <-> A <_ ( B + A ) ) )
Theorem	subge0d 8554	Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 <_ ( A - B ) <-> B <_ A ) )
Theorem	suble0d 8555	Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( ( A - B ) <_ 0 <-> A <_ B ) )
Theorem	subge02d 8556	Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 <_ B <-> ( A - B ) <_ A ) )
Theorem	ltadd1d 8557	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A < B <-> ( A + C ) < ( B + C ) ) )
Theorem	leadd1d 8558	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A <_ B <-> ( A + C ) <_ ( B + C ) ) )
Theorem	leadd2d 8559	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A <_ B <-> ( C + A ) <_ ( C + B ) ) )
Theorem	ltsubaddd 8560	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A - B ) < C <-> A < ( C + B ) ) )
Theorem	lesubaddd 8561	'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A - B ) <_ C <-> A <_ ( C + B ) ) )
Theorem	ltsubadd2d 8562	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A - B ) < C <-> A < ( B + C ) ) )
Theorem	lesubadd2d 8563	'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A - B ) <_ C <-> A <_ ( B + C ) ) )
Theorem	ltaddsubd 8564	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A + B ) < C <-> A < ( C - B ) ) )
Theorem	ltaddsub2d 8565	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A + B ) < C <-> B < ( C - A ) ) )
Theorem	leaddsub2d 8566	'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A + B ) <_ C <-> B <_ ( C - A ) ) )
Theorem	subled 8567	Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> ( A - B ) <_ C )   =>    |- ( ph -> ( A - C ) <_ B )
Theorem	lesubd 8568	Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ ( B - C ) )   =>    |- ( ph -> C <_ ( B - A ) )
Theorem	ltsub23d 8569	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> ( A - B ) < C )   =>    |- ( ph -> ( A - C ) < B )
Theorem	ltsub13d 8570	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < ( B - C ) )   =>    |- ( ph -> C < ( B - A ) )
Theorem	lesub1d 8571	Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A <_ B <-> ( A - C ) <_ ( B - C ) ) )
Theorem	lesub2d 8572	Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A <_ B <-> ( C - B ) <_ ( C - A ) ) )
Theorem	ltsub1d 8573	Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A < B <-> ( A - C ) < ( B - C ) ) )
Theorem	ltsub2d 8574	Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A < B <-> ( C - B ) < ( C - A ) ) )
Theorem	ltadd1dd 8575	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> ( A + C ) < ( B + C ) )
Theorem	ltsub1dd 8576	Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> ( A - C ) < ( B - C ) )
Theorem	ltsub2dd 8577	Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> ( C - B ) < ( C - A ) )
Theorem	leadd1dd 8578	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( A + C ) <_ ( B + C ) )
Theorem	leadd2dd 8579	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( C + A ) <_ ( C + B ) )
Theorem	lesub1dd 8580	Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( A - C ) <_ ( B - C ) )
Theorem	lesub2dd 8581	Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( C - B ) <_ ( C - A ) )
Theorem	le2addd 8582	Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A <_ C )   &    |- ( ph -> B <_ D )   =>    |- ( ph -> ( A + B ) <_ ( C + D ) )
Theorem	le2subd 8583	Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A <_ C )   &    |- ( ph -> B <_ D )   =>    |- ( ph -> ( A - D ) <_ ( C - B ) )
Theorem	ltleaddd 8584	Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A < C )   &    |- ( ph -> B <_ D )   =>    |- ( ph -> ( A + B ) < ( C + D ) )
Theorem	leltaddd 8585	Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A <_ C )   &    |- ( ph -> B < D )   =>    |- ( ph -> ( A + B ) < ( C + D ) )
Theorem	lt2addd 8586	Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A < C )   &    |- ( ph -> B < D )   =>    |- ( ph -> ( A + B ) < ( C + D ) )
Theorem	lt2subd 8587	Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A < C )   &    |- ( ph -> B < D )   =>    |- ( ph -> ( A - D ) < ( C - B ) )
Theorem	possumd 8588	Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 < ( A + B ) <-> -u B < A ) )
Theorem	sublt0d 8589	When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( ( A - B ) < 0 <-> A < B ) )
Theorem	ltaddsublt 8590	Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B < C <-> ( ( A + B ) - C ) < A ) )
Theorem	1le1 8591	1 <_ 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
|- 1 <_ 1
Theorem	gt0add 8592	A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.)
|- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < A \/ 0 < B ) )
4.3.5  Real Apartness
Syntax	creap 8593	Class of real apartness relation.
class #ℝ
Definition	df-reap 8594*	Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although #ℝ is an apartness relation on the reals (see df-ap 8601 for more discussion of apartness relations), for our purposes it is just a stepping stone to defining # which is an apartness relation on complex numbers. On the reals, #ℝ and # agree (apreap 8606). (Contributed by Jim Kingdon, 26-Jan-2020.)
|- #ℝ = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ ( x < y \/ y < x ) ) }
Theorem	reapval 8595	Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8607 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B <-> ( A < B \/ B < A ) ) )
Theorem	reapirr 8596	Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8624 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
|- ( A e. RR -> -. A #ℝ A )
Theorem	recexre 8597*	Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
|- ( ( A e. RR /\ A #ℝ 0 ) -> E. x e. RR ( A x. x ) = 1 )
Theorem	reapti 8598	Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8641. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A #ℝ B ) )
Theorem	recexgt0 8599*	Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)
|- ( ( A e. RR /\ 0 < A ) -> E. x e. RR ( 0 < x /\ ( A x. x ) = 1 ) )
4.3.6  Complex Apartness
Syntax	cap 8600	Class of complex apartness relation.
class #