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Theorem List for Metamath Proof Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxle0neg2 10501 Extended real version of le0neg2 9237. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  - e A  <_  0 ) )
 
Theoremxaddval 10502 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 10503 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 + e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxmulval 10504 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 10505 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  ->  ( A + e  +oo )  =  +oo )
 
Theoremxaddpnf2 10506 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  ->  (  +oo + e A )  =  +oo )
 
Theoremxaddmnf1 10507 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  ( A + e  -oo )  =  -oo )
 
Theoremxaddmnf2 10508 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  (  -oo + e A )  =  -oo )
 
Theorempnfaddmnf 10509 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 10510 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 10511 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 10512 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 10513 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 10514 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 10515 Extended real version of negid 9048. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A + e  - e A )  =  0 )
 
Theoremxaddcl 10516 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 10517 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 10518 Extended real version of addid1 8946. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A + e 0 )  =  A )
 
Theoremxaddid2 10519 Extended real version of addid2 8949. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 + e A )  =  A )
 
Theoremxnegdi 10520 Extended real version of xnegdi 10520. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> 
 - e ( A + e B )  =  (  - e A + e  - e B ) )
 
Theoremxaddass 10521 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 10522, 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 10522 Associativity of extended real addition. See xaddass 10521 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 10523 Extended real version of pncan 9011. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A + e B ) + e  - e B )  =  A )
 
Theoremxnpcan 10524 Extended real version of npcan 9014. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A + e  - e B ) + e B )  =  A )
 
Theoremxleadd1a 10525 Extended real version of leadd1 9196; 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 10526 Commuted form of xleadd1a 10525. (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 10527 Weakened version of xleadd1a 10525 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 10528 Extended real version of ltadd1 9195. (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 10529 Extended real version of ltadd2 8878. (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 10530 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 10531 Extended real version of le2add 9210. (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 10532 Extended real version of lt2add 9213. Note that ltleadd 9211, 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 10533 Extended real version of subge0 9241. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( 0  <_  ( A + e  - e B )  <->  B  <_  A ) )
 
Theoremxposdif 10534 Extended real version of posdif 9221. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  0  <  ( B + e  - e A ) ) )
 
Theoremxlesubadd 10535 Under certain conditions, the conclusion of lesubadd 9200 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 10536 Lemma for rexmul 10543. (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 10537 Lemma for xmulneg1 10541. (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 10538 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 10539 Extended real version of mul01 8945. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A x e 0 )  =  0 )
 
Theoremxmul02 10540 Extended real version of mul02 8944. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 x e A )  =  0
 )
 
Theoremxmulneg1 10541 Extended real version of mulneg1 9170. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  - e A x e B )  =  - e ( A x e B ) )
 
Theoremxmulneg2 10542 Extended real version of mulneg2 9171. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e  - e B )  =  - e ( A x e B ) )
 
Theoremrexmul 10543 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 10544 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  x e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxmulcl 10545 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 10546 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 10547 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 10548 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 10549 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 10550 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 10551 Extended real version of mulid1 8788. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A x e 1 )  =  A )
 
Theoremxmulid2 10552 Extended real version of mulid2 8789. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 1 x e A )  =  A )
 
Theoremxmulm1 10553 Extended real version of mulm1 9175. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( -u 1 x e A )  =  - e A )
 
Theoremxmulasslem2 10554 Lemma for xmulass 10559. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( 0  <  A  /\  A  =  -oo )  ->  ph )
 
Theoremxmulgt0 10555 Extended real version of mulgt0 8854. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  -> 
 0  <  ( A x e B ) )
 
Theoremxmulge0 10556 Extended real version of mulge0 9245. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  -> 
 0  <_  ( A x e B ) )
 
Theoremxmulasslem 10557* Lemma for xmulass 10559. (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 10558 Lemma for xmulass 10559. (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 10559 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 10521 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 10560 Extended real version of lemul1a 9564. (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 10561 Extended real version of lemul2a 9565. (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 10562 Extended real version of lemul1 9562. (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 10563 Extended real version of lemul2 9563. (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 10564 Extended real version of ltmul1 9560. (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 10565 Extended real version of ltmul2 9561. (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 10566 Lemma for xadddi 10567. (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 10567 Distributive property for extended real addition and multiplication. Like xaddass 10521, 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 10568 Commuted version of xadddi 10567. (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 10569 The assumption that the multiplier be real in xadddi 10567 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 10570 Commuted version of xadddi2 10569. (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 10571 Extended real version of 2times 9796. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 2 x e A )  =  ( A + e A ) )
 
Theoremxnegcld 10572 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  - e A  e.  RR* )
 
Theoremxaddcld 10573 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 10574 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* )
 
5.5.3  Supremum on the extended reals
 
Theoremxrsupexmnf 10575* 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 10576* 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 10577* Lemma for xrsupss 10579. (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 10578* Lemma for xrinfmss 10580. (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 10579* 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 10580* 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 10581* 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 10582* 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 10583* 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 10584* 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 10585 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* )
 
Theoremsupxrun 10586 The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  C_  RR*  /\  B  C_  RR*  /\  sup ( A ,  RR* ,  <  ) 
 <_  sup ( B ,  RR*
 ,  <  ) )  ->  sup ( ( A  u.  B ) , 
 RR* ,  <  )  = 
 sup ( B ,  RR*
 ,  <  ) )
 
Theoreminfmxrcl 10587 The infimum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 19-Jan-2006.) (Revised by Mario Carneiro, 16-Mar-2014.)
 |-  ( A  C_  RR*  ->  sup ( A ,  RR* ,  `'  <  )  e.  RR* )
 
Theoremsupxrmnf 10588 Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  C_  RR*  ->  sup ( ( A  u.  { 
 -oo } ) ,  RR* ,  <  )  =  sup ( A ,  RR* ,  <  ) )
 
Theoremsupxrpnf 10589 The supremum of a set of extended reals containing plus infnity is plus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( ( A  C_  RR*  /\  +oo  e.  A ) 
 ->  sup ( A ,  RR*
 ,  <  )  =  +oo )
 
Theoremsupxrunb1 10590* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  C_  RR*  ->  (
 A. x  e.  RR  E. y  e.  A  x  <_  y  <->  sup ( A ,  RR*
 ,  <  )  =  +oo ) )
 
Theoremsupxrunb2 10591* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  C_  RR*  ->  (
 A. x  e.  RR  E. y  e.  A  x  <  y  <->  sup ( A ,  RR*
 ,  <  )  =  +oo ) )
 
Theoremsupxrbnd1 10592* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  C_  RR*  ->  ( E. x  e.  RR  A. y  e.  A  y  <  x  <->  sup ( A ,  RR*
 ,  <  )  <  +oo ) )
 
Theoremsupxrbnd2 10593* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  C_  RR*  ->  ( E. x  e.  RR  A. y  e.  A  y 
 <_  x  <->  sup ( A ,  RR*
 ,  <  )  <  +oo ) )
 
Theoremxrsup0 10594 The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.)
 |- 
 sup ( (/) ,  RR* ,  <  )  =  -oo
 
Theoremsupxrub 10595 A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by NM, 7-Feb-2006.)
 |-  ( ( A  C_  RR*  /\  B  e.  A ) 
 ->  B  <_  sup ( A ,  RR* ,  <  )
 )
 
Theoremsupxrlub 10596* The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by Mario Carneiro, 13-Sep-2015.)
 |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B  <  sup ( A ,  RR* ,  <  )  <->  E. x  e.  A  B  <  x ) )
 
Theoremsupxrleub 10597* The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by NM, 22-Feb-2006.) (Revised by Mario Carneiro, 6-Sep-2014.)
 |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( sup ( A ,  RR* ,  <  )  <_  B  <->  A. x  e.  A  x  <_  B ) )
 
Theoremsupxrre 10598* The real and extended real suprema match when the real supremum exists. (Contributed by NM, 18-Oct-2005.) (Proof shortened by Mario Carneiro, 7-Sep-2014.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  ->  sup ( A ,  RR*
 ,  <  )  =  sup ( A ,  RR ,  <  ) )
 
Theoremsupxrbnd 10599 The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  sup ( A ,  RR* ,  <  )  e.  RR )
 
Theoremsupxrgtmnf 10600 The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  -oo  <  sup ( A ,  RR*
 ,  <  ) )
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