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Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxnegex 11501 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  -e A  e.  _V
 
Theoremxnegpnf 11502 Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  -e +oo  = -oo
 
Theoremxnegmnf 11503 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 11504 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 11505 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  -e 0  =  0
 
Theoremxnegcl 11506 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e A  e.  RR* )
 
Theoremxnegneg 11507 Extended real version of negneg 9923. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e  -e A  =  A )
 
Theoremxneg11 11508 Extended real version of neg11 9924. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A  =  -e B  <->  A  =  B )
 )
 
Theoremxltnegi 11509 Forward direction of xltneg 11510. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A )
 
Theoremxltneg 11510 Extended real version of ltneg 10113. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -e B  <  -e A ) )
 
Theoremxleneg 11511 Extended real version of leneg 10116. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -e B  <_  -e A ) )
 
Theoremxlt0neg1 11512 Extended real version of lt0neg1 10119. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
 
Theoremxlt0neg2 11513 Extended real version of lt0neg2 10120. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <  A  <->  -e A  <  0 ) )
 
Theoremxle0neg1 11514 Extended real version of le0neg1 10121. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( A  <_  0  <->  0  <_  -e A ) )
 
Theoremxle0neg2 11515 Extended real version of le0neg2 10122. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  -e A  <_  0 ) )
 
Theoremxaddval 11516 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 11517 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 +e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxmulval 11518 Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe 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 11519 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
 
Theoremxaddpnf2 11520 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( +oo +e A )  = +oo )
 
Theoremxaddmnf1 11521 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
 
Theoremxaddmnf2 11522 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( -oo +e A )  = -oo )
 
Theorempnfaddmnf 11523 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 11524 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 11525 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 11526 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 11527 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 11528 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 11529 Extended real version of negid 9920. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A +e  -e A )  =  0 )
 
Theoremxaddcl 11530 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 11531 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 11532 Extended real version of addid1 9812. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
 
Theoremxaddid2 11533 Extended real version of addid2 9815. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 +e A )  =  A )
 
Theoremxnegdi 11534 Extended real version of xnegdi 11534. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> 
 -e ( A +e B )  =  (  -e A +e  -e B ) )
 
Theoremxaddass 11535 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 11536, 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 11536 Associativity of extended real addition. See xaddass 11535 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 11537 Extended real version of pncan 9880. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A +e B ) +e  -e B )  =  A )
 
Theoremxnpcan 11538 Extended real version of npcan 9883. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A +e  -e B ) +e B )  =  A )
 
Theoremxleadd1a 11539 Extended real version of leadd1 10081; 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 11540 Commuted form of xleadd1a 11539. (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 11541 Weakened version of xleadd1a 11539 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 11542 Extended real version of ltadd1 10080. (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 11543 Extended real version of ltadd2 9737. (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 11544 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 11545 Extended real version of le2add 10095. (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 11546 Extended real version of lt2add 10098. Note that ltleadd 10096, 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 11547 Extended real version of subge0 10126. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( 0  <_  ( A +e  -e B )  <->  B  <_  A ) )
 
Theoremxposdif 11548 Extended real version of posdif 10106. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
 
Theoremxlesubadd 11549 Under certain conditions, the conclusion of lesubadd 10085 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 11550 Lemma for rexmul 11557. (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 11551 Lemma for xmulneg1 11555. (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 11552 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  =  ( B xe A ) )
 
Theoremxmul01 11553 Extended real version of mul01 9811. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A xe 0 )  =  0 )
 
Theoremxmul02 11554 Extended real version of mul02 9810. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 xe A )  =  0
 )
 
Theoremxmulneg1 11555 Extended real version of mulneg1 10054. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A xe B )  =  -e ( A xe B ) )
 
Theoremxmulneg2 11556 Extended real version of mulneg2 10055. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe  -e B )  =  -e ( A xe B ) )
 
Theoremrexmul 11557 The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A xe B )  =  ( A  x.  B ) )
 
Theoremxmulf 11558 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  xe : (
 RR*  X.  RR* ) --> RR*
 
Theoremxmulcl 11559 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  e.  RR* )
 
Theoremxmulpnf1 11560 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( A xe +oo )  = +oo )
 
Theoremxmulpnf2 11561 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( +oo xe A )  = +oo )
 
Theoremxmulmnf1 11562 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( A xe -oo )  = -oo )
 
Theoremxmulmnf2 11563 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( -oo xe A )  = -oo )
 
Theoremxmulpnf1n 11564 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  <  0 ) 
 ->  ( A xe +oo )  = -oo )
 
Theoremxmulid1 11565 Extended real version of mulid1 9639. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A xe 1 )  =  A )
 
Theoremxmulid2 11566 Extended real version of mulid2 9640. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 1 xe A )  =  A )
 
Theoremxmulm1 11567 Extended real version of mulm1 10059. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( -u 1 xe A )  =  -e A )
 
Theoremxmulasslem2 11568 Lemma for xmulass 11573. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( 0  <  A  /\  A  = -oo )  ->  ph )
 
Theoremxmulgt0 11569 Extended real version of mulgt0 9710. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  -> 
 0  <  ( A xe B ) )
 
Theoremxmulge0 11570 Extended real version of mulge0 10131. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  -> 
 0  <_  ( A xe B ) )
 
Theoremxmulasslem 11571* Lemma for xmulass 11573. (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 11572 Lemma for xmulass 11573. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B )  /\  ( C  e.  RR*  /\  0  <  C ) )  ->  ( ( A xe B ) xe C )  =  ( A xe ( B xe C ) ) )
 
Theoremxmulass 11573 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 11535 which has to avoid the "undefined" combinations +oo +e -oo and -oo +e +oo. The equivalent "undefined" expression here would be  0 xe +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 xe B ) xe C )  =  ( A xe ( B xe C ) ) )
 
Theoremxlemul1a 11574 Extended real version of lemul1a 10458. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR*  /\  0  <_  C )
 )  /\  A  <_  B )  ->  ( A xe C )  <_  ( B xe C ) )
 
Theoremxlemul2a 11575 Extended real version of lemul2a 10459. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR*  /\  0  <_  C )
 )  /\  A  <_  B )  ->  ( C xe A )  <_  ( C xe B ) )
 
Theoremxlemul1 11576 Extended real version of lemul1 10456. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( A xe C )  <_  ( B xe C ) ) )
 
Theoremxlemul2 11577 Extended real version of lemul2 10457. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( C xe A )  <_  ( C xe B ) ) )
 
Theoremxltmul1 11578 Extended real version of ltmul1 10454. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A  <  B  <->  ( A xe C )  <  ( B xe C ) ) )
 
Theoremxltmul2 11579 Extended real version of ltmul2 10455. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A  <  B  <->  ( C xe A )  <  ( C xe B ) ) )
 
Theoremxadddilem 11580 Lemma for xadddi 11581. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
 
Theoremxadddi 11581 Distributive property for extended real addition and multiplication. Like xaddass 11535, 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 xe
 ( B +e C ) )  =  ( ( A xe B ) +e
 ( A xe C ) ) )
 
Theoremxadddir 11582 Commuted version of xadddi 11581. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
 ( A +e B ) xe C )  =  (
 ( A xe C ) +e
 ( B xe C ) ) )
 
Theoremxadddi2 11583 The assumption that the multiplier be real in xadddi 11581 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 xe ( B +e C ) )  =  ( ( A xe B ) +e
 ( A xe C ) ) )
 
Theoremxadddi2r 11584 Commuted version of xadddi2 11583. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )  /\  C  e.  RR* )  ->  (
 ( A +e B ) xe C )  =  (
 ( A xe C ) +e
 ( B xe C ) ) )
 
Theoremx2times 11585 Extended real version of 2times 10728. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 2 xe A )  =  ( A +e A ) )
 
Theoremxnegcld 11586 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  -e A  e.  RR* )
 
Theoremxaddcld 11587 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 11588 Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   =>    |-  ( ph  ->  ( A xe B )  e.  RR* )
 
Theoremxadd4d 11589 Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 9857. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
 |-  ( ph  ->  ( A  e.  RR*  /\  A  =/= -oo ) )   &    |-  ( ph  ->  ( B  e.  RR*  /\  B  =/= -oo )
 )   &    |-  ( ph  ->  ( C  e.  RR*  /\  C  =/= -oo ) )   &    |-  ( ph  ->  ( D  e.  RR*  /\  D  =/= -oo )
 )   =>    |-  ( ph  ->  (
 ( A +e B ) +e
 ( C +e D ) )  =  ( ( A +e C ) +e
 ( B +e D ) ) )
 
5.5.3  Supremum and infimum on the extended reals
 
Theoremxrsupexmnf 11590* Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)
 |-  ( E. x  e.  RR*  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e.  RR*  ( y  <  x  ->  E. z  e.  A  y  <  z ) ) 
 ->  E. x  e.  RR*  ( A. y  e.  ( A  u.  { -oo }
 )  -.  x  <  y 
 /\  A. y  e.  RR*  ( y  <  x  ->  E. z  e.  ( A  u.  { -oo }
 ) y  <  z
 ) ) )
 
Theoremxrinfmexpnf 11591* Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)
 |-  ( E. x  e.  RR*  ( A. y  e.  A  -.  y  < 
 x  /\  A. y  e.  RR*  ( x  <  y  ->  E. z  e.  A  z  <  y ) ) 
 ->  E. x  e.  RR*  ( A. y  e.  ( A  u.  { +oo }
 )  -.  y  <  x 
 /\  A. y  e.  RR*  ( x  <  y  ->  E. z  e.  ( A  u.  { +oo }
 ) z  <  y
 ) ) )
 
Theoremxrsupsslem 11592* Lemma for xrsupss 11594. (Contributed by NM, 25-Oct-2005.)
 |-  ( ( A  C_  RR*  /\  ( A  C_  RR  \/ +oo  e.  A ) )  ->  E. x  e.  RR*  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e.  RR*  ( y  <  x  ->  E. z  e.  A  y  <  z ) ) )
 
Theoremxrinfmsslem 11593* Lemma for xrinfmss 11595. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  C_  RR*  /\  ( A  C_  RR  \/ -oo  e.  A ) )  ->  E. x  e.  RR*  ( A. y  e.  A  -.  y  < 
 x  /\  A. y  e.  RR*  ( x  <  y  ->  E. z  e.  A  z  <  y ) ) )
 
Theoremxrsupss 11594* Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.)
 |-  ( A  C_  RR*  ->  E. x  e.  RR*  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR*  (
 y  <  x  ->  E. z  e.  A  y  <  z ) ) )
 
Theoremxrinfmss 11595* Any subset of extended reals has an infimum. (Contributed by NM, 25-Oct-2005.)
 |-  ( A  C_  RR*  ->  E. x  e.  RR*  ( A. y  e.  A  -.  y  <  x  /\  A. y  e.  RR*  ( x  <  y  ->  E. z  e.  A  z  <  y
 ) ) )
 
Theoremxrinfmss2 11596* Any subset of extended reals has an infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( A  C_  RR*  ->  E. x  e.  RR*  ( A. y  e.  A  -.  x `'  <  y  /\  A. y  e.  RR*  ( y `'  <  x 
 ->  E. z  e.  A  y `'  <  z ) ) )
 
Theoremxrub 11597* By quantifying only over reals, we can specify any extended real upper bound for any set of extended reals. (Contributed by NM, 9-Apr-2006.)
 |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( A. x  e. 
 RR  ( x  <  B  ->  E. y  e.  A  x  <  y )  <->  A. x  e.  RR*  ( x  <  B  ->  E. y  e.  A  x  <  y ) ) )
 
Theoremsupxr 11598* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.) (Revised by Mario Carneiro, 21-Apr-2015.)
 |-  ( ( ( A 
 C_  RR*  /\  B  e.  RR* )  /\  ( A. x  e.  A  -.  B  <  x  /\  A. x  e.  RR  ( x  <  B  ->  E. y  e.  A  x  <  y
 ) ) )  ->  sup ( A ,  RR* ,  <  )  =  B )
 
Theoremsupxr2 11599* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.)
 |-  ( ( ( A 
 C_  RR*  /\  B  e.  RR* )  /\  ( A. x  e.  A  x  <_  B  /\  A. x  e.  RR  ( x  <  B  ->  E. y  e.  A  x  <  y ) ) )  ->  sup ( A ,  RR* ,  <  )  =  B )
 
Theoremsupxrcl 11600 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005.)
 |-  ( A  C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
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