P. 84 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8301-8400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	add32 8301	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
Theorem	add32r 8302	Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( ( A + C ) + B ) )
Theorem	add4 8303	Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) )
Theorem	add42 8304	Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )
Theorem	add12i 8305	Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( A + ( B + C ) ) = ( B + ( A + C ) )
Theorem	add32i 8306	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A + B ) + C ) = ( ( A + C ) + B )
Theorem	add4i 8307	Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- D e. CC   =>    |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) )
Theorem	add42i 8308	Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- D e. CC   =>    |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) )
Theorem	add12d 8309	Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A + ( B + C ) ) = ( B + ( A + C ) ) )
Theorem	add32d 8310	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
Theorem	add4d 8311	Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) )
Theorem	add42d 8312	Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )
4.3.2  Subtraction
Syntax	cmin 8313	Extend class notation to include subtraction.
class -
Syntax	cneg 8314	Extend class notation to include unary minus. The symbol -u is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus ( -u) and subtraction cmin 8313 ( -) to prevent syntax ambiguity. For example, looking at the syntax definition co 6000, if we used the same symbol then " ( - A - B ) " could mean either " - A " minus " B", or it could represent the (meaningless) operation of classes " - " and " - B " connected with "operation" " A". On the other hand, " ( -u A - B ) " is unambiguous.
class -u A
Definition	df-sub 8315*	Define subtraction. Theorem subval 8334 shows its value (and describes how this definition works), Theorem subaddi 8429 relates it to addition, and Theorems subcli 8418 and resubcli 8405 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
|- - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
Definition	df-neg 8316	Define the negative of a number (unary minus). We use different symbols for unary minus ( -u) and subtraction ( -) to prevent syntax ambiguity. See cneg 8314 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
|- -u A = ( 0 - A )
Theorem	cnegexlem1 8317	Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8320. (Contributed by Eric Schmidt, 22-May-2007.)
|- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )
Theorem	cnegexlem2 8318	Existence of a real number which produces a real number when multiplied by _i. (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 8320. (Contributed by Eric Schmidt, 22-May-2007.)
|- E. y e. RR ( _i x. y ) e. RR
Theorem	cnegexlem3 8319*	Existence of real number difference. Lemma for cnegex 8320. (Contributed by Eric Schmidt, 22-May-2007.)
|- ( ( b e. RR /\ y e. RR ) -> E. c e. RR ( b + c ) = y )
Theorem	cnegex 8320*	Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)
|- ( A e. CC -> E. x e. CC ( A + x ) = 0 )
Theorem	cnegex2 8321*	Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( A e. CC -> E. x e. CC ( x + A ) = 0 )
Theorem	addcan 8322	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )
Theorem	addcan2 8323	Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) = ( B + C ) <-> A = B ) )
Theorem	addcani 8324	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A + B ) = ( A + C ) <-> B = C )
Theorem	addcan2i 8325	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A + C ) = ( B + C ) <-> A = B )
Theorem	addcand 8326	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) )
Theorem	addcan2d 8327	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )
Theorem	addcanad 8328	Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 8326. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> ( A + B ) = ( A + C ) )   =>    |- ( ph -> B = C )
Theorem	addcan2ad 8329	Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 8327. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> ( A + C ) = ( B + C ) )   =>    |- ( ph -> A = B )
Theorem	addneintrd 8330	Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 8328. Consequence of addcand 8326. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> ( A + B ) =/= ( A + C ) )
Theorem	addneintr2d 8331	Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 8329. Consequence of addcan2d 8327. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> A =/= B )   =>    |- ( ph -> ( A + C ) =/= ( B + C ) )
Theorem	0cnALT 8332	Alternate proof of 0cn 8134. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
|- 0 e. CC
Theorem	negeu 8333*	Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> E! x e. CC ( A + x ) = B )
Theorem	subval 8334*	Value of subtraction, which is the (unique) element x such that B + x = A. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
|- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )
Theorem	negeq 8335	Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
|- ( A = B -> -u A = -u B )
Theorem	negeqi 8336	Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
|- A = B   =>    |- -u A = -u B
Theorem	negeqd 8337	Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> -u A = -u B )
Theorem	nfnegd 8338	Deduction version of nfneg 8339. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x -u A )
Theorem	nfneg 8339	Bound-variable hypothesis builder for the negative of a complex number. (Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   =>    |- F/_ x -u A
Theorem	csbnegg 8340	Move class substitution in and out of the negative of a number. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( A e. V -> [_ A / x ]_ -u B = -u [_ A / x ]_ B )
Theorem	subcl 8341	Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
|- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
Theorem	negcl 8342	Closure law for negative. (Contributed by NM, 6-Aug-2003.)
|- ( A e. CC -> -u A e. CC )
Theorem	negicn 8343	-u _i is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
|- -u _i e. CC
Theorem	subf 8344	Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
|- - : ( CC X. CC ) --> CC
Theorem	subadd 8345	Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( B + C ) = A ) )
Theorem	subadd2 8346	Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( C + B ) = A ) )
Theorem	subsub23 8347	Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( A - C ) = B ) )
Theorem	pncan 8348	Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A )
Theorem	pncan2 8349	Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - A ) = B )
Theorem	pncan3 8350	Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + ( B - A ) ) = B )
Theorem	npcan 8351	Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + B ) = A )
Theorem	addsubass 8352	Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - C ) = ( A + ( B - C ) ) )
Theorem	addsub 8353	Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - C ) = ( ( A - C ) + B ) )
Theorem	subadd23 8354	Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + C ) = ( A + ( C - B ) ) )
Theorem	addsub12 8355	Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B - C ) ) = ( B + ( A - C ) ) )
Theorem	2addsub 8356	Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A + B ) + C ) - D ) = ( ( ( A + C ) - D ) + B ) )
Theorem	addsubeq4 8357	Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) )
Theorem	pncan3oi 8358	Subtraction and addition of equals. Almost but not exactly the same as pncan3i 8419 and pncan 8348, this order happens often when applying "operations to both sides" so create a theorem specifically for it. A deduction version of this is available as pncand 8454. (Contributed by David A. Wheeler, 11-Oct-2018.)
|- A e. CC   &    |- B e. CC   =>    |- ( ( A + B ) - B ) = A
Theorem	mvrraddi 8359	Move RHS right addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
|- B e. CC   &    |- C e. CC   &    |- A = ( B + C )   =>    |- ( A - C ) = B
Theorem	mvlladdi 8360	Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
|- A e. CC   &    |- B e. CC   &    |- ( A + B ) = C   =>    |- B = ( C - A )
Theorem	subid 8361	Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> ( A - A ) = 0 )
Theorem	subid1 8362	Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> ( A - 0 ) = A )
Theorem	npncan 8363	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( B - C ) ) = ( A - C ) )
Theorem	nppcan 8364	Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) + C ) + B ) = ( A + C ) )
Theorem	nnpcan 8365	Cancellation law for subtraction: ((a-b)-c)+b = a-c holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) - C ) + B ) = ( A - C ) )
Theorem	nppcan3 8366	Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( C + B ) ) = ( A + C ) )
Theorem	subcan2 8367	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) = ( B - C ) <-> A = B ) )
Theorem	subeq0 8368	If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = 0 <-> A = B ) )
Theorem	npncan2 8369	Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + ( B - A ) ) = 0 )
Theorem	subsub2 8370	Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( A + ( C - B ) ) )
Theorem	nncan 8371	Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( A e. CC /\ B e. CC ) -> ( A - ( A - B ) ) = B )
Theorem	subsub 8372	Law for double subtraction. (Contributed by NM, 13-May-2004.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( ( A - B ) + C ) )
Theorem	nppcan2 8373	Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - ( B + C ) ) + C ) = ( A - B ) )
Theorem	subsub3 8374	Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( ( A + C ) - B ) )
Theorem	subsub4 8375	Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) - C ) = ( A - ( B + C ) ) )
Theorem	sub32 8376	Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) - C ) = ( ( A - C ) - B ) )
Theorem	nnncan 8377	Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - ( B - C ) ) - C ) = ( A - B ) )
Theorem	nnncan1 8378	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) - ( A - C ) ) = ( C - B ) )
Theorem	nnncan2 8379	Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) - ( B - C ) ) = ( A - B ) )
Theorem	npncan3 8380	Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( C - A ) ) = ( C - B ) )
Theorem	pnpcan 8381	Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A + C ) ) = ( B - C ) )
Theorem	pnpcan2 8382	Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( B + C ) ) = ( A - B ) )
Theorem	pnncan 8383	Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A - C ) ) = ( B + C ) )
Theorem	ppncan 8384	Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + ( C - B ) ) = ( A + C ) )
Theorem	addsub4 8385	Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) - ( C + D ) ) = ( ( A - C ) + ( B - D ) ) )
Theorem	subadd4 8386	Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) - ( C - D ) ) = ( ( A + D ) - ( B + C ) ) )
Theorem	sub4 8387	Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) - ( C - D ) ) = ( ( A - C ) - ( B - D ) ) )
Theorem	neg0 8388	Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
|- -u 0 = 0
Theorem	negid 8389	Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
|- ( A e. CC -> ( A + -u A ) = 0 )
Theorem	negsub 8390	Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
Theorem	subneg 8391	Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( A - -u B ) = ( A + B ) )
Theorem	negneg 8392	A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> -u -u A = A )
Theorem	neg11 8393	Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( -u A = -u B <-> A = B ) )
Theorem	negcon1 8394	Negative contraposition law. (Contributed by NM, 9-May-2004.)
|- ( ( A e. CC /\ B e. CC ) -> ( -u A = B <-> -u B = A ) )
Theorem	negcon2 8395	Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
|- ( ( A e. CC /\ B e. CC ) -> ( A = -u B <-> B = -u A ) )
Theorem	negeq0 8396	A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> ( A = 0 <-> -u A = 0 ) )
Theorem	subcan 8397	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = ( A - C ) <-> B = C ) )
Theorem	negsubdi 8398	Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( -u A + B ) )
Theorem	negdi 8399	Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A + -u B ) )
Theorem	negdi2 8400	Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
|- ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A - B ) )