MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xmulpnf1 Structured version   Unicode version

Theorem xmulpnf1 11229
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 11084 . . . 4  |- +oo  e.  RR*
2 xmulval 11187 . . . 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 9422 . . . . . 6  |-  0  e.  RR*
6 xrltne 11129 . . . . . 6  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  < 
A )  ->  A  =/=  0 )
75, 6mp3an1 1301 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  A  =/=  0 )
8 0re 9378 . . . . . . 7  |-  0  e.  RR
9 renepnf 9423 . . . . . . 7  |-  ( 0  e.  RR  ->  0  =/= +oo )
108, 9ax-mp 5 . . . . . 6  |-  0  =/= +oo
1110necomi 2689 . . . . 5  |- +oo  =/=  0
127, 11jctir 538 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A  =/=  0  /\ +oo  =/=  0 ) )
13 neanior 2692 . . . 4  |-  ( ( A  =/=  0  /\ +oo  =/=  0 )  <->  -.  ( A  =  0  \/ +oo  =  0 ) )
1412, 13sylib 196 . . 3  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  -.  ( A  =  0  \/ +oo  =  0 ) )
15 iffalse 3794 . . 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 2438 . . . . . 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 3792 . . 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 2474 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 1369    e. wcel 1756    =/= wne 2601   ifcif 3786   class class class wbr 4287  (class class class)co 6086   RRcr 9273   0cc0 9274    x. cmul 9279   +oocpnf 9407   -oocmnf 9408   RR*cxr 9409    < clt 9410   xecxmu 11080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-i2m1 9342  ax-1ne0 9343  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-xmul 11083
This theorem is referenced by:  xmulpnf2  11230  xmulmnf1  11231  xmulpnf1n  11233  xmulgt0  11238  xmulasslem3  11241  xlemul1a  11243  xadddilem  11249  xdivpnfrp  26059  xrge0adddir  26106  esumcst  26466  esumpinfval  26474
  Copyright terms: Public domain W3C validator