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Theorem xmulpnf1 11585
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 11435 . . . 4  |- +oo  e.  RR*
2 xmulval 11541 . . . 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 685 . . 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 472 . 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 9705 . . . . . 6  |-  0  e.  RR*
6 xrltne 11483 . . . . . 6  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  < 
A )  ->  A  =/=  0 )
75, 6mp3an1 1377 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  A  =/=  0 )
8 0re 9661 . . . . . . 7  |-  0  e.  RR
9 renepnf 9706 . . . . . . 7  |-  ( 0  e.  RR  ->  0  =/= +oo )
108, 9ax-mp 5 . . . . . 6  |-  0  =/= +oo
1110necomi 2697 . . . . 5  |- +oo  =/=  0
127, 11jctir 547 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A  =/=  0  /\ +oo  =/=  0 ) )
13 neanior 2735 . . . 4  |-  ( ( A  =/=  0  /\ +oo  =/=  0 )  <->  -.  ( A  =  0  \/ +oo  =  0 ) )
1412, 13sylib 201 . . 3  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  -.  ( A  =  0  \/ +oo  =  0 ) )
1514iffalsed 3883 . 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 ) ) ) )
16 simpr 468 . . . . . 6  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  0  <  A )
17 eqid 2471 . . . . . 6  |- +oo  = +oo
1816, 17jctir 547 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
0  <  A  /\ +oo  = +oo ) )
1918orcd 399 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
( 0  <  A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo )
) )
2019olcd 400 . . 3  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo ) ) ) )
2120iftrued 3880 . 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 )
224, 15, 213eqtrd 2509 1  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   ifcif 3872   class class class wbr 4395  (class class class)co 6308   RRcr 9556   0cc0 9557    x. cmul 9562   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    < clt 9693   xecxmu 11431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-i2m1 9625  ax-1ne0 9626  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-xmul 11434
This theorem is referenced by:  xmulpnf2  11586  xmulmnf1  11587  xmulpnf1n  11589  xmulgt0  11594  xmulasslem3  11597  xlemul1a  11599  xadddilem  11605  xdivpnfrp  28477  xrge0adddir  28529  esumcst  28958  esumpinfval  28968
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