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Theorem xmulpnf1 11567
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 11419 . . . 4  |- +oo  e.  RR*
2 xmulval 11525 . . . 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 675 . . 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 466 . 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 9694 . . . . . 6  |-  0  e.  RR*
6 xrltne 11467 . . . . . 6  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  < 
A )  ->  A  =/=  0 )
75, 6mp3an1 1347 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  A  =/=  0 )
8 0re 9650 . . . . . . 7  |-  0  e.  RR
9 renepnf 9695 . . . . . . 7  |-  ( 0  e.  RR  ->  0  =/= +oo )
108, 9ax-mp 5 . . . . . 6  |-  0  =/= +oo
1110necomi 2690 . . . . 5  |- +oo  =/=  0
127, 11jctir 540 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A  =/=  0  /\ +oo  =/=  0 ) )
13 neanior 2745 . . . 4  |-  ( ( A  =/=  0  /\ +oo  =/=  0 )  <->  -.  ( A  =  0  \/ +oo  =  0 ) )
1412, 13sylib 199 . . 3  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  -.  ( A  =  0  \/ +oo  =  0 ) )
1514iffalsed 3922 . 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 462 . . . . . 6  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  0  <  A )
17 eqid 2422 . . . . . 6  |- +oo  = +oo
1816, 17jctir 540 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
0  <  A  /\ +oo  = +oo ) )
1918orcd 393 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
( 0  <  A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo )
) )
2019olcd 394 . . 3  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo ) ) ) )
2120iftrued 3919 . 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 2467 1  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   ifcif 3911   class class class wbr 4423  (class class class)co 6305   RRcr 9545   0cc0 9546    x. cmul 9551   +oocpnf 9679   -oocmnf 9680   RR*cxr 9681    < clt 9682   xecxmu 11415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-i2m1 9614  ax-1ne0 9615  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-xmul 11418
This theorem is referenced by:  xmulpnf2  11568  xmulmnf1  11569  xmulpnf1n  11571  xmulgt0  11576  xmulasslem3  11579  xlemul1a  11581  xadddilem  11587  xdivpnfrp  28409  xrge0adddir  28462  esumcst  28892  esumpinfval  28902
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