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Theorem List for Metamath Proof Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmax0sub 10401 Decompose a real number into positive and negative parts. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( A  e.  RR  ->  ( if ( 0 
 <_  A ,  A , 
 0 )  -  if ( 0  <_  -u A ,  -u A ,  0 ) )  =  A )
 
Theoremifle 10402 An if statement transforms an implication into an inequality of terms. (Contributed by Mario Carneiro, 31-Aug-2014.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B ) )
 
Theoremz2ge 10403* There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  E. k  e.  ZZ  ( M  <_  k  /\  N  <_  k ) )
 
Theoremqbtwnre 10404* The rational numbers are dense in 
RR: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.) (Proof shortened by Mario Carneiro, 13-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) 
 ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
 
Theoremqbtwnxr 10405* The rational numbers are dense in  RR*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  E. x  e.  QQ  ( A  <  x 
 /\  x  <  B ) )
 
Theoremqsqueeze 10406* If a nonnegative real is less than any positive rational, it is zero. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  QQ  ( 0  <  x  ->  A  <  x ) )  ->  A  =  0 )
 
Theoremqextltlem 10407* Lemma for qextlt 10408 and qextle . (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( -.  ( x  <  A 
 <->  x  <  B ) 
 /\  -.  ( x  <_  A  <->  x  <_  B ) ) ) )
 
Theoremqextlt 10408* An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  A. x  e.  QQ  ( x  <  A  <->  x  <  B ) ) )
 
Theoremqextle 10409* An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  A. x  e.  QQ  ( x  <_  A  <->  x  <_  B ) ) )
 
Theoremxralrple 10410* Show that  A is less than  B by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  <_  B  <->  A. x  e.  RR+  A  <_  ( B  +  x ) ) )
 
Theoremalrple 10411* Show that  A is less than  B by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  A. x  e.  RR+  A  <_  ( B  +  x ) ) )
 
Theoremxnegeq 10412 Equality of two extended numbers with  - e in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  =  B  -> 
 - e A  =  - e B )
 
Theoremxnegex 10413 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  - e A  e.  _V
 
Theoremxnegpnf 10414 Minus  +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  - e  +oo  =  -oo
 
Theoremxnegmnf 10415 Minus  -oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
 |-  - e  -oo  =  +oo
 
Theoremrexneg 10416 Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR  -> 
 - e A  =  -u A )
 
Theoremxneg0 10417 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  - e 0  =  0
 
Theoremxnegcl 10418 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  - e A  e.  RR* )
 
Theoremxnegneg 10419 Extended real version of negneg 8977. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  - e  - e A  =  A )
 
Theoremxneg11 10420 Extended real version of neg11 8978. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  - e A  =  - e B  <->  A  =  B ) )
 
Theoremxltnegi 10421 Forward direction of xltneg 10422. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  - e B  <  - e A )
 
Theoremxltneg 10422 Extended real version of ltneg 9154. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  - e B  <  - e A ) )
 
Theoremxleneg 10423 Extended real version of leneg 9157. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  - e B  <_  - e A ) )
 
Theoremxlt0neg1 10424 Extended real version of lt0neg1 9160. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  - e A ) )
 
Theoremxlt0neg2 10425 Extended real version of lt0neg2 9161. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <  A  <->  - e A  <  0 ) )
 
Theoremxle0neg1 10426 Extended real version of le0neg1 9162. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( A  <_  0  <->  0  <_  - e A ) )
 
Theoremxle0neg2 10427 Extended real version of le0neg2 9163. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  - e A  <_  0 ) )
 
Theoremxaddval 10428 Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e B )  =  if ( A  =  +oo ,  if ( B  =  -oo ,  0 ,  +oo ) ,  if ( A  =  -oo ,  if ( B  =  +oo ,  0 ,  -oo ) ,  if ( B  =  +oo ,  +oo ,  if ( B  =  -oo , 
 -oo ,  ( A  +  B ) ) ) ) ) )
 
Theoremxaddf 10429 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 + e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxmulval 10430 Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e B )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo )
 )  \/  ( ( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo )
 ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo )
 )  \/  ( ( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo )
 ) ) ,  -oo ,  ( A  x.  B ) ) ) ) )
 
Theoremxaddpnf1 10431 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  ->  ( A + e  +oo )  =  +oo )
 
Theoremxaddpnf2 10432 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  ->  (  +oo + e A )  =  +oo )
 
Theoremxaddmnf1 10433 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  ( A + e  -oo )  =  -oo )
 
Theoremxaddmnf2 10434 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  (  -oo + e A )  =  -oo )
 
Theorempnfaddmnf 10435 Addition of positive and negative infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  (  +oo + e  -oo )  =  0
 
Theoremmnfaddpnf 10436 Addition of negative and positive infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  (  -oo + e  +oo )  =  0
 
Theoremrexadd 10437 The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A + e B )  =  ( A  +  B ) )
 
Theoremrexsub 10438 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A + e  - e B )  =  ( A  -  B ) )
 
Theoremxaddnemnf 10439 Closure of extended real addition in the subset  RR*  /  {  -oo }. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/=  -oo )  /\  ( B  e.  RR*  /\  B  =/=  -oo ) )  ->  ( A + e B )  =/=  -oo )
 
Theoremxaddnepnf 10440 Closure of extended real addition in the subset  RR*  /  {  +oo }. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/=  +oo )  /\  ( B  e.  RR*  /\  B  =/=  +oo ) )  ->  ( A + e B )  =/=  +oo )
 
Theoremxnegid 10441 Extended real version of negid 8974. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A + e  - e A )  =  0 )
 
Theoremxaddcl 10442 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e B )  e.  RR* )
 
Theoremxaddcom 10443 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e B )  =  ( B + e A ) )
 
Theoremxaddid1 10444 Extended real version of addid1 8872. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A + e 0 )  =  A )
 
Theoremxaddid2 10445 Extended real version of addid2 8875. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 + e A )  =  A )
 
Theoremxnegdi 10446 Extended real version of xnegdi 10446. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> 
 - e ( A + e B )  =  (  - e A + e  - e B ) )
 
Theoremxaddass 10447 Associativity of extended real addition. The correct condition here is "it is not the case that both  +oo and  -oo appear as one of  A ,  B ,  C, i.e.  -.  {  +oo , 
-oo }  C_  { A ,  B ,  C }", but this condition is difficult to work with, so we break the theorem into two parts: this one, where  -oo is not present in  A ,  B ,  C, and xaddass2 10448, where  +oo is not present. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/=  -oo )  /\  ( B  e.  RR*  /\  B  =/=  -oo )  /\  ( C  e.  RR*  /\  C  =/=  -oo ) )  ->  (
 ( A + e B ) + e C )  =  ( A + e ( B + e C ) ) )
 
Theoremxaddass2 10448 Associativity of extended real addition. See xaddass 10447 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/=  +oo )  /\  ( B  e.  RR*  /\  B  =/=  +oo )  /\  ( C  e.  RR*  /\  C  =/=  +oo ) )  ->  (
 ( A + e B ) + e C )  =  ( A + e ( B + e C ) ) )
 
Theoremxpncan 10449 Extended real version of pncan 8937. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A + e B ) + e  - e B )  =  A )
 
Theoremxnpcan 10450 Extended real version of npcan 8940. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A + e  - e B ) + e B )  =  A )
 
Theoremxleadd1a 10451 Extended real version of leadd1 9122; note that the converse implication is not true, unlike the real version (for example  0  <  1 but  ( 1 + e  +oo )  <_ 
( 0 + e  +oo )). (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  ( A + e C )  <_  ( B + e C ) )
 
Theoremxleadd2a 10452 Commuted form of xleadd1a 10451. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  ( C + e A )  <_  ( C + e B ) )
 
Theoremxleadd1 10453 Weakened version of xleadd1a 10451 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( A + e C )  <_  ( B + e C ) ) )
 
Theoremxltadd1 10454 Extended real version of ltadd1 9121. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A + e C )  <  ( B + e C ) ) )
 
Theoremxltadd2 10455 Extended real version of ltadd2 8804. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C + e A )  <  ( C + e B ) ) )
 
Theoremxaddge0 10456 The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 0 
 <_  A  /\  0  <_  B ) )  -> 
 0  <_  ( A + e B ) )
 
Theoremxle2add 10457 Extended real version of le2add 9136. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  ->  (
 ( A  <_  C  /\  B  <_  D )  ->  ( A + e B )  <_  ( C + e D ) ) )
 
Theoremxlt2add 10458 Extended real version of lt2add 9139. Note that ltleadd 9137, which has weaker assumptions, is not true for the extended reals (since  0  +  +oo  <  1  +  +oo fails). (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  ->  (
 ( A  <  C  /\  B  <  D ) 
 ->  ( A + e B )  <  ( C + e D ) ) )
 
Theoremxsubge0 10459 Extended real version of subge0 9167. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( 0  <_  ( A + e  - e B )  <->  B  <_  A ) )
 
Theoremxposdif 10460 Extended real version of posdif 9147. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  0  <  ( B + e  - e A ) ) )
 
Theoremxlesubadd 10461 Under certain conditions, the conclusion of lesubadd 9126 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( 0  <_  A  /\  B  =/=  -oo  /\  0  <_  C ) )  ->  ( ( A + e  - e B ) 
 <_  C  <->  A  <_  ( C + e B ) ) )
 
Theoremxmullem 10462 Lemma for rexmul 10469. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  /\  -.  ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  ( ( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0 
 /\  B  =  -oo ) ) ) ) 
 /\  -.  ( (
 ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo )
 )  \/  ( ( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo )
 ) ) )  ->  A  e.  RR )
 
Theoremxmullem2 10463 Lemma for xmulneg1 10467. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo )
 )  \/  ( ( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo )
 ) )  ->  -.  (
 ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0 
 /\  A  =  +oo ) )  \/  (
 ( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo )
 ) ) ) )
 
Theoremxmulcom 10464 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e B )  =  ( B x e A ) )
 
Theoremxmul01 10465 Extended real version of mul01 8871. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A x e 0 )  =  0 )
 
Theoremxmul02 10466 Extended real version of mul02 8870. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 x e A )  =  0
 )
 
Theoremxmulneg1 10467 Extended real version of mulneg1 9096. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  - e A x e B )  =  - e ( A x e B ) )
 
Theoremxmulneg2 10468 Extended real version of mulneg2 9097. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e  - e B )  =  - e ( A x e B ) )
 
Theoremrexmul 10469 The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A x e B )  =  ( A  x.  B ) )
 
Theoremxmulf 10470 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  x e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxmulcl 10471 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e B )  e.  RR* )
 
Theoremxmulpnf1 10472 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( A x e  +oo )  =  +oo )
 
Theoremxmulpnf2 10473 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  (  +oo x e A )  =  +oo )
 
Theoremxmulmnf1 10474 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( A x e  -oo )  =  -oo )
 
Theoremxmulmnf2 10475 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  (  -oo x e A )  =  -oo )
 
Theoremxmulpnf1n 10476 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  <  0 ) 
 ->  ( A x e  +oo )  =  -oo )
 
Theoremxmulid1 10477 Extended real version of mulid1 8714. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A x e 1 )  =  A )
 
Theoremxmulid2 10478 Extended real version of mulid2 8715. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 1 x e A )  =  A )
 
Theoremxmulm1 10479 Extended real version of mulm1 9101. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( -u 1 x e A )  =  - e A )
 
Theoremxmulasslem2 10480 Lemma for xmulass 10485. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( 0  <  A  /\  A  =  -oo )  ->  ph )
 
Theoremxmulgt0 10481 Extended real version of mulgt0 8780. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  -> 
 0  <  ( A x e B ) )
 
Theoremxmulge0 10482 Extended real version of mulge0 9171. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  -> 
 0  <_  ( A x e B ) )
 
Theoremxmulasslem 10483* Lemma for xmulass 10485. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( x  =  D  ->  ( ps  <->  X  =  Y ) )   &    |-  ( x  =  - e D  ->  ( ps 
 <->  E  =  F ) )   &    |-  ( ph  ->  X  e.  RR* )   &    |-  ( ph  ->  Y  e.  RR* )   &    |-  ( ph  ->  D  e.  RR* )   &    |-  ( ( ph  /\  ( x  e.  RR*  /\  0  <  x ) )  ->  ps )   &    |-  ( ph  ->  ( x  =  0  ->  ps )
 )   &    |-  ( ph  ->  E  =  - e X )   &    |-  ( ph  ->  F  =  - e Y )   =>    |-  ( ph  ->  X  =  Y )
 
Theoremxmulasslem3 10484 Lemma for xmulass 10485. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B )  /\  ( C  e.  RR*  /\  0  <  C ) )  ->  ( ( A x e B ) x e C )  =  ( A x e ( B x e C ) ) )
 
Theoremxmulass 10485 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 10447 which has to avoid the "undefined" combinations  +oo + e  -oo and  -oo + e  +oo. The equivalent "undefined" expression here would be  0 x e 
+oo, but since this is defined to equal  0 any zeroes in the expression make the whole thing evaluate to zero (on both sides), thus establishing the identity in this case. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A x e B ) x e C )  =  ( A x e ( B x e C ) ) )
 
Theoremxlemul1a 10486 Extended real version of lemul1a 9490. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR*  /\  0  <_  C )
 )  /\  A  <_  B )  ->  ( A x e C )  <_  ( B x e C ) )
 
Theoremxlemul2a 10487 Extended real version of lemul2a 9491. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR*  /\  0  <_  C )
 )  /\  A  <_  B )  ->  ( C x e A )  <_  ( C x e B ) )
 
Theoremxlemul1 10488 Extended real version of lemul1 9488. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( A x e C )  <_  ( B x e C ) ) )
 
Theoremxlemul2 10489 Extended real version of lemul2 9489. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( C x e A )  <_  ( C x e B ) ) )
 
Theoremxltmul1 10490 Extended real version of ltmul1 9486. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A  <  B  <->  ( A x e C )  <  ( B x e C ) ) )
 
Theoremxltmul2 10491 Extended real version of ltmul2 9487. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A  <  B  <->  ( C x e A )  <  ( C x e B ) ) )
 
Theoremxadddilem 10492 Lemma for xadddi 10493. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
 
Theoremxadddi 10493 Distributive property for extended real addition and multiplication. Like xaddass 10447, this has an unusual domain of correctness due to counterexamples like  (  +oo  x.  (
2  -  1 ) )  =  -oo  =/=  ( (  +oo  x.  2 )  -  (  +oo  x.  1 ) )  =  (  +oo  - 
+oo )  =  0. In this theorem we show that if the multiplier is real then everything works as expected. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( A x e
 ( B + e C ) )  =  ( ( A x e B ) + e
 ( A x e C ) ) )
 
Theoremxadddir 10494 Commuted version of xadddi 10493. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
 ( A + e B ) x e C )  =  (
 ( A x e C ) + e
 ( B x e C ) ) )
 
Theoremxadddi2 10495 The assumption that the multiplier be real in xadddi 10493 can be relaxed if the addends have the same sign. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  ( B  e.  RR*  /\  0  <_  B )  /\  ( C  e.  RR*  /\  0  <_  C )
 )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e
 ( A x e C ) ) )
 
Theoremxadddi2r 10496 Commuted version of xadddi2 10495. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )  /\  C  e.  RR* )  ->  (
 ( A + e B ) x e C )  =  (
 ( A x e C ) + e
 ( B x e C ) ) )
 
Theoremx2times 10497 Extended real version of 2times 9722. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 2 x e A )  =  ( A + e A ) )
 
Theoremxnegcld 10498 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  - e A  e.  RR* )
 
Theoremxaddcld 10499 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   =>    |-  ( ph  ->  ( A + e B )  e.  RR* )
 
Theoremxmulcld 10500 Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   =>    |-  ( ph  ->  ( A x e B )  e.  RR* )
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