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	mulgt0d 8301	The product of two 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 x. B ) )
Theorem	letrd 8302	Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   &    |- ( ph -> B <_ C )   =>    |- ( ph -> A <_ C )
Theorem	lelttrd 8303	Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   &    |- ( ph -> B < C )   =>    |- ( ph -> A < C )
Theorem	lttrd 8304	Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> B < C )   =>    |- ( ph -> A < C )
Theorem	0lt1 8305	0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.)
|- 0 < 1
Theorem	ltntri 8306	Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, A < B \/ A = B \/ B < A. Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)
|- ( ( A e. RR /\ B e. RR ) -> -. ( -. A < B /\ -. A = B /\ -. B < A ) )
4.2.5  Initial properties of the complex numbers
Theorem	mul12 8307	Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )
Theorem	mul32 8308	Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )
Theorem	mul31 8309	Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) )
Theorem	mul4 8310	Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )
Theorem	muladd11 8311	A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )
Theorem	1p1times 8312	Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) )
Theorem	peano2cn 8313	A theorem for complex numbers analogous the second Peano postulate peano2 4693. (Contributed by NM, 17-Aug-2005.)
|- ( A e. CC -> ( A + 1 ) e. CC )
Theorem	peano2re 8314	A theorem for reals analogous the second Peano postulate peano2 4693. (Contributed by NM, 5-Jul-2005.)
|- ( A e. RR -> ( A + 1 ) e. RR )
Theorem	addcom 8315	Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
Theorem	addrid 8316	0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
|- ( A e. CC -> ( A + 0 ) = A )
Theorem	addlid 8317	0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( A e. CC -> ( 0 + A ) = A )
Theorem	readdcan 8318	Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) = ( C + B ) <-> A = B ) )
Theorem	00id 8319	0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( 0 + 0 ) = 0
Theorem	addridi 8320	0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
|- A e. CC   =>    |- ( A + 0 ) = A
Theorem	addlidi 8321	0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
|- A e. CC   =>    |- ( 0 + A ) = A
Theorem	addcomi 8322	Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
|- A e. CC   &    |- B e. CC   =>    |- ( A + B ) = ( B + A )
Theorem	addcomli 8323	Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- A e. CC   &    |- B e. CC   &    |- ( A + B ) = C   =>    |- ( B + A ) = C
Theorem	mul12i 8324	Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( A x. ( B x. C ) ) = ( B x. ( A x. C ) )
Theorem	mul32i 8325	Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A x. B ) x. C ) = ( ( A x. C ) x. B )
Theorem	mul4i 8326	Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- D e. CC   =>    |- ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) )
Theorem	addridd 8327	0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A + 0 ) = A )
Theorem	addlidd 8328	0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 0 + A ) = A )
Theorem	addcomd 8329	Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( A + B ) = ( B + A ) )
Theorem	mul12d 8330	Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )
Theorem	mul32d 8331	Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )
Theorem	mul31d 8332	Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) )
Theorem	mul4d 8333	Rearrangement of 4 factors. (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 x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )
Theorem	muladd11r 8334	A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )
Theorem	comraddd 8335	Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
|- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> A = ( B + C ) )   =>    |- ( ph -> A = ( C + B ) )
4.3  Real and complex numbers - basic operations
4.3.1  Addition
Theorem	add12 8336	Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( B + ( A + C ) ) )
Theorem	add32 8337	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 8338	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 8339	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 8340	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 8341	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 8342	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 8343	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 8344	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 8345	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 8346	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 8347	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 8348	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 8349	Extend class notation to include subtraction.
class -
Syntax	cneg 8350	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 8349 ( -) to prevent syntax ambiguity. For example, looking at the syntax definition co 6017, 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 8351*	Define subtraction. Theorem subval 8370 shows its value (and describes how this definition works), Theorem subaddi 8465 relates it to addition, and Theorems subcli 8454 and resubcli 8441 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 8352	Define the negative of a number (unary minus). We use different symbols for unary minus ( -u) and subtraction ( -) to prevent syntax ambiguity. See cneg 8350 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
|- -u A = ( 0 - A )
Theorem	cnegexlem1 8353	Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8356. (Contributed by Eric Schmidt, 22-May-2007.)
|- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )
Theorem	cnegexlem2 8354	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 8356. (Contributed by Eric Schmidt, 22-May-2007.)
|- E. y e. RR ( _i x. y ) e. RR
Theorem	cnegexlem3 8355*	Existence of real number difference. Lemma for cnegex 8356. (Contributed by Eric Schmidt, 22-May-2007.)
|- ( ( b e. RR /\ y e. RR ) -> E. c e. RR ( b + c ) = y )
Theorem	cnegex 8356*	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 8357*	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 8358	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 8359	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 8360	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 8361	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 8362	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 8363	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 8364	Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 8362. (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 8365	Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 8363. (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 8366	Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 8364. Consequence of addcand 8362. (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 8367	Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 8365. Consequence of addcan2d 8363. (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 8368	Alternate proof of 0cn 8170. (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 8369*	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 8370*	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 8371	Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
|- ( A = B -> -u A = -u B )
Theorem	negeqi 8372	Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
|- A = B   =>    |- -u A = -u B
Theorem	negeqd 8373	Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> -u A = -u B )
Theorem	nfnegd 8374	Deduction version of nfneg 8375. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x -u A )
Theorem	nfneg 8375	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 8376	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 8377	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 8378	Closure law for negative. (Contributed by NM, 6-Aug-2003.)
|- ( A e. CC -> -u A e. CC )
Theorem	negicn 8379	-u _i is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
|- -u _i e. CC
Theorem	subf 8380	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 8381	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 8382	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 8383	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 8384	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 8385	Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - A ) = B )
Theorem	pncan3 8386	Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + ( B - A ) ) = B )
Theorem	npcan 8387	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 8388	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 8389	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 8390	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 8391	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 8392	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 8393	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 8394	Subtraction and addition of equals. Almost but not exactly the same as pncan3i 8455 and pncan 8384, 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 8490. (Contributed by David A. Wheeler, 11-Oct-2018.)
|- A e. CC   &    |- B e. CC   =>    |- ( ( A + B ) - B ) = A
Theorem	mvrraddi 8395	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 8396	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 8397	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 8398	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 8399	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 8400	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 ) )