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Theorem xmulpnf1 11351
Description: Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmulpnf1  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  = +oo )

Proof of Theorem xmulpnf1
StepHypRef Expression
1 pnfxr 11206 . . . 4  |- +oo  e.  RR*
2 xmulval 11309 . . . 4  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A xe +oo )  =  if ( ( A  =  0  \/ +oo  =  0 ) ,  0 ,  if ( ( ( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  < +oo  /\  A  = -oo )  \/  ( +oo  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\ +oo  = -oo )  \/  ( A  <  0  /\ +oo  = +oo ) ) ) , -oo ,  ( A  x. +oo ) ) ) ) )
31, 2mpan2 671 . . 3  |-  ( A  e.  RR*  ->  ( A xe +oo )  =  if ( ( A  =  0  \/ +oo  =  0 ) ,  0 ,  if ( ( ( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  < +oo  /\  A  = -oo )  \/  ( +oo  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\ +oo  = -oo )  \/  ( A  <  0  /\ +oo  = +oo ) ) ) , -oo ,  ( A  x. +oo ) ) ) ) )
43adantr 465 . 2  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  =  if ( ( A  =  0  \/ +oo  =  0 ) ,  0 ,  if ( ( ( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  < +oo  /\  A  = -oo )  \/  ( +oo  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\ +oo  = -oo )  \/  ( A  <  0  /\ +oo  = +oo ) ) ) , -oo ,  ( A  x. +oo ) ) ) ) )
5 0xr 9544 . . . . . 6  |-  0  e.  RR*
6 xrltne 11251 . . . . . 6  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  < 
A )  ->  A  =/=  0 )
75, 6mp3an1 1302 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  A  =/=  0 )
8 0re 9500 . . . . . . 7  |-  0  e.  RR
9 renepnf 9545 . . . . . . 7  |-  ( 0  e.  RR  ->  0  =/= +oo )
108, 9ax-mp 5 . . . . . 6  |-  0  =/= +oo
1110necomi 2722 . . . . 5  |- +oo  =/=  0
127, 11jctir 538 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A  =/=  0  /\ +oo  =/=  0 ) )
13 neanior 2777 . . . 4  |-  ( ( A  =/=  0  /\ +oo  =/=  0 )  <->  -.  ( A  =  0  \/ +oo  =  0 ) )
1412, 13sylib 196 . . 3  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  -.  ( A  =  0  \/ +oo  =  0 ) )
15 iffalse 3910 . . 3  |-  ( -.  ( A  =  0  \/ +oo  =  0 )  ->  if (
( A  =  0  \/ +oo  =  0 ) ,  0 ,  if ( ( ( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo )
) ) , +oo ,  if ( ( ( ( 0  < +oo  /\  A  = -oo )  \/  ( +oo  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\ +oo  = -oo )  \/  ( A  <  0  /\ +oo  = +oo )
) ) , -oo ,  ( A  x. +oo ) ) ) )  =  if ( ( ( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  < +oo  /\  A  = -oo )  \/  ( +oo  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\ +oo  = -oo )  \/  ( A  <  0  /\ +oo  = +oo ) ) ) , -oo ,  ( A  x. +oo ) ) ) )
1614, 15syl 16 . 2  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  if ( ( A  =  0  \/ +oo  = 
0 ) ,  0 ,  if ( ( ( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  < +oo  /\  A  = -oo )  \/  ( +oo  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\ +oo  = -oo )  \/  ( A  <  0  /\ +oo  = +oo ) ) ) , -oo ,  ( A  x. +oo ) ) ) )  =  if ( ( ( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo )
) ) , +oo ,  if ( ( ( ( 0  < +oo  /\  A  = -oo )  \/  ( +oo  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\ +oo  = -oo )  \/  ( A  <  0  /\ +oo  = +oo )
) ) , -oo ,  ( A  x. +oo ) ) ) )
17 simpr 461 . . . . . 6  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  0  <  A )
18 eqid 2454 . . . . . 6  |- +oo  = +oo
1917, 18jctir 538 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
0  <  A  /\ +oo  = +oo ) )
2019orcd 392 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
( 0  <  A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo )
) )
2120olcd 393 . . 3  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo ) ) ) )
22 iftrue 3908 . . 3  |-  ( ( ( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo ) ) )  ->  if ( ( ( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo )
) ) , +oo ,  if ( ( ( ( 0  < +oo  /\  A  = -oo )  \/  ( +oo  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\ +oo  = -oo )  \/  ( A  <  0  /\ +oo  = +oo )
) ) , -oo ,  ( A  x. +oo ) ) )  = +oo )
2321, 22syl 16 . 2  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  if ( ( ( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo )
) ) , +oo ,  if ( ( ( ( 0  < +oo  /\  A  = -oo )  \/  ( +oo  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\ +oo  = -oo )  \/  ( A  <  0  /\ +oo  = +oo )
) ) , -oo ,  ( A  x. +oo ) ) )  = +oo )
244, 16, 233eqtrd 2499 1  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   ifcif 3902   class class class wbr 4403  (class class class)co 6203   RRcr 9395   0cc0 9396    x. cmul 9401   +oocpnf 9529   -oocmnf 9530   RR*cxr 9531    < clt 9532   xecxmu 11202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-i2m1 9464  ax-1ne0 9465  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-xmul 11205
This theorem is referenced by:  xmulpnf2  11352  xmulmnf1  11353  xmulpnf1n  11355  xmulgt0  11360  xmulasslem3  11363  xlemul1a  11365  xadddilem  11371  xdivpnfrp  26273  xrge0adddir  26320  esumcst  26679  esumpinfval  26687
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