P. 83 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)

Type	Label	Description
Statement
4.2.2  Infinity and the extended real number system
Syntax	cpnf 8201	Plus infinity.
class +oo
Syntax	cmnf 8202	Minus infinity.
class -oo
Syntax	cxr 8203	The set of extended reals (includes plus and minus infinity).
class RR*
Syntax	clt 8204	'Less than' predicate (extended to include the extended reals).
class <
Syntax	cle 8205	Extend wff notation to include the 'less than or equal to' relation.
class <_
Definition	df-pnf 8206	Define plus infinity. Note that the definition is arbitrary, requiring only that +oo be a set not in RR and different from -oo (df-mnf 8207). We use ~P U. CC to make it independent of the construction of CC, and Cantor's Theorem will show that it is different from any member of CC and therefore RR. See pnfnre 8211 and mnfnre 8212, and we'll also be able to prove +oo =/= -oo. A simpler possibility is to define +oo as CC and -oo as { CC }, but that approach requires the Axiom of Regularity to show that +oo and -oo are different from each other and from all members of RR. (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
|- +oo = ~P U. CC
Definition	df-mnf 8207	Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that -oo be a set not in RR and different from +oo (see mnfnre 8212). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
|- -oo = ~P +oo
Definition	df-xr 8208	Define the set of extended reals that includes plus and minus infinity. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 13-Oct-2005.)
|- RR* = ( RR u. { +oo , -oo } )
Definition	df-ltxr 8209*	Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers, <RR is primitive and not necessarily a relation on RR. (Contributed by NM, 13-Oct-2005.)
|- < = ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) )
Definition	df-le 8210	Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)
|- <_ = ( ( RR* X. RR* ) \ `' < )
Theorem	pnfnre 8211	Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
|- +oo e/ RR
Theorem	mnfnre 8212	Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
|- -oo e/ RR
Theorem	ressxr 8213	The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
|- RR C_ RR*
Theorem	rexpssxrxp 8214	The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( RR X. RR ) C_ ( RR* X. RR* )
Theorem	rexr 8215	A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR -> A e. RR* )
Theorem	0xr 8216	Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
|- 0 e. RR*
Theorem	renepnf 8217	No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( A e. RR -> A =/= +oo )
Theorem	renemnf 8218	No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( A e. RR -> A =/= -oo )
Theorem	rexrd 8219	A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A e. RR* )
Theorem	renepnfd 8220	No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A =/= +oo )
Theorem	renemnfd 8221	No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A =/= -oo )
Theorem	pnfxr 8222	Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)
|- +oo e. RR*
Theorem	pnfex 8223	Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- +oo e. _V
Theorem	pnfnemnf 8224	Plus and minus infinity are different elements of RR*. (Contributed by NM, 14-Oct-2005.)
|- +oo =/= -oo
Theorem	mnfnepnf 8225	Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -oo =/= +oo
Theorem	mnfxr 8226	Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- -oo e. RR*
Theorem	rexri 8227	A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
|- A e. RR   =>    |- A e. RR*
Theorem	1xr 8228	1 is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
|- 1 e. RR*
Theorem	renfdisj 8229	The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( RR i^i { +oo , -oo } ) = (/)
Theorem	ltrelxr 8230	'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- < C_ ( RR* X. RR* )
Theorem	ltrel 8231	'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
|- Rel <
Theorem	lerelxr 8232	'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- <_ C_ ( RR* X. RR* )
Theorem	lerel 8233	'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- Rel <_
Theorem	xrlenlt 8234	'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
Theorem	ltxrlt 8235	The standard less-than <RR and the extended real less-than < are identical when restricted to the non-extended reals RR. (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
4.2.3  Restate the ordering postulates with extended real "less than"
Theorem	axltirr 8236	Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 8134 with ordering on the extended reals. New proofs should use ltnr 8246 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
|- ( A e. RR -> -. A < A )
Theorem	axltwlin 8237	Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 8135 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( A < C \/ C < B ) ) )
Theorem	axlttrn 8238	Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 8136 with ordering on the extended reals. New proofs should use lttr 8243 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
Theorem	axltadd 8239	Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 8138 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( C + A ) < ( C + B ) ) )
Theorem	axapti 8240	Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 8137 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
|- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )
Theorem	axmulgt0 8241	The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 8139 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) )
Theorem	axsuploc 8242*	An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 8143 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.)
|- ( ( ( A C_ RR /\ E. x x e. A ) /\ ( E. x e. RR A. y e. A y < x /\ A. x e. RR A. y e. RR ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) ) ) -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
4.2.4  Ordering on reals
Theorem	lttr 8243	Alias for axlttrn 8238, for naming consistency with lttri 8274. New proofs should generally use this instead of ax-pre-lttrn 8136. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
Theorem	mulgt0 8244	The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.)
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A x. B ) )
Theorem	lenlt 8245	'Less than or equal to' expressed in terms of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
Theorem	ltnr 8246	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> -. A < A )
Theorem	ltso 8247	'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
|- < Or RR
Theorem	gtso 8248	'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
|- `' < Or RR
Theorem	lttri3 8249	Tightness of real apartness. (Contributed by NM, 5-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	letri3 8250	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	ltleletr 8251	Transitive law, weaker form of ( A < B /\ B <_ C ) -> A < C. (Contributed by AV, 14-Oct-2018.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B <_ C ) -> A <_ C ) )
Theorem	letr 8252	Transitive law. (Contributed by NM, 12-Nov-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ B /\ B <_ C ) -> A <_ C ) )
Theorem	leid 8253	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> A <_ A )
Theorem	ltne 8254	'Less than' implies not equal. See also ltap 8803 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)
|- ( ( A e. RR /\ A < B ) -> B =/= A )
Theorem	ltnsym 8255	'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
Theorem	eqlelt 8256	Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ -. A < B ) ) )
Theorem	ltle 8257	'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A <_ B ) )
Theorem	lelttr 8258	Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ B /\ B < C ) -> A < C ) )
Theorem	ltletr 8259	Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B <_ C ) -> A < C ) )
Theorem	ltnsym2 8260	'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( A e. RR /\ B e. RR ) -> -. ( A < B /\ B < A ) )
Theorem	eqle 8261	Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
|- ( ( A e. RR /\ A = B ) -> A <_ B )
Theorem	ltnri 8262	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
|- A e. RR   =>    |- -. A < A
Theorem	eqlei 8263	Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.)
|- A e. RR   =>    |- ( A = B -> A <_ B )
Theorem	eqlei2 8264	Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
|- A e. RR   =>    |- ( B = A -> B <_ A )
Theorem	gtneii 8265	'Less than' implies not equal. See also gtapii 8804 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
|- A e. RR   &    |- A < B   =>    |- B =/= A
Theorem	ltneii 8266	'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
|- A e. RR   &    |- A < B   =>    |- A =/= B
Theorem	lttri3i 8267	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A = B <-> ( -. A < B /\ -. B < A ) )
Theorem	letri3i 8268	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A = B <-> ( A <_ B /\ B <_ A ) )
Theorem	ltnsymi 8269	'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> -. B < A )
Theorem	lenlti 8270	'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A <_ B <-> -. B < A )
Theorem	ltlei 8271	'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> A <_ B )
Theorem	ltleii 8272	'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- A < B   =>    |- A <_ B
Theorem	ltnei 8273	'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> B =/= A )
Theorem	lttri 8274	'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A < B /\ B < C ) -> A < C )
Theorem	lelttri 8275	'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A <_ B /\ B < C ) -> A < C )
Theorem	ltletri 8276	'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A < B /\ B <_ C ) -> A < C )
Theorem	letri 8277	'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A <_ B /\ B <_ C ) -> A <_ C )
Theorem	le2tri3i 8278	Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A <_ B /\ B <_ C /\ C <_ A ) <-> ( A = B /\ B = C /\ C = A ) )
Theorem	mulgt0i 8279	The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) )
Theorem	mulgt0ii 8280	The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- 0 < A   &    |- 0 < B   =>    |- 0 < ( A x. B )
Theorem	ltnrd 8281	'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> -. A < A )
Theorem	gtned 8282	'Less than' implies not equal. See also gtapd 8807 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> B =/= A )
Theorem	ltned 8283	'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A =/= B )
Theorem	lttri3d 8284	Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	letri3d 8285	Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	eqleltd 8286	Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A = B <-> ( A <_ B /\ -. A < B ) ) )
Theorem	lenltd 8287	'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A <_ B <-> -. B < A ) )
Theorem	ltled 8288	'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A <_ B )
Theorem	ltnsymd 8289	'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> -. B < A )
Theorem	nltled 8290	'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> -. B < A )   =>    |- ( ph -> A <_ B )
Theorem	lensymd 8291	'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> -. B < A )
Theorem	mulgt0d 8292	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 8293	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 8294	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 8295	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 8296	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 8297	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 8298	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 8299	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 8300	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 ) )