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Theorem xmulpnf1 11518
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 11373 . . . 4  |- +oo  e.  RR*
2 xmulval 11476 . . . 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 669 . . 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 463 . 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 9669 . . . . . 6  |-  0  e.  RR*
6 xrltne 11418 . . . . . 6  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  < 
A )  ->  A  =/=  0 )
75, 6mp3an1 1313 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  A  =/=  0 )
8 0re 9625 . . . . . . 7  |-  0  e.  RR
9 renepnf 9670 . . . . . . 7  |-  ( 0  e.  RR  ->  0  =/= +oo )
108, 9ax-mp 5 . . . . . 6  |-  0  =/= +oo
1110necomi 2673 . . . . 5  |- +oo  =/=  0
127, 11jctir 536 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A  =/=  0  /\ +oo  =/=  0 ) )
13 neanior 2728 . . . 4  |-  ( ( A  =/=  0  /\ +oo  =/=  0 )  <->  -.  ( A  =  0  \/ +oo  =  0 ) )
1412, 13sylib 196 . . 3  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  -.  ( A  =  0  \/ +oo  =  0 ) )
1514iffalsed 3895 . 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 459 . . . . . 6  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  0  <  A )
17 eqid 2402 . . . . . 6  |- +oo  = +oo
1816, 17jctir 536 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
0  <  A  /\ +oo  = +oo ) )
1918orcd 390 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
( 0  <  A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo )
) )
2019olcd 391 . . 3  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
( ( 0  < +oo  /\  A  = +oo )  \/  ( +oo  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\ +oo  = +oo )  \/  ( A  <  0  /\ +oo  = -oo ) ) ) )
2120iftrued 3892 . 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 2447 1  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   ifcif 3884   class class class wbr 4394  (class class class)co 6277   RRcr 9520   0cc0 9521    x. cmul 9526   +oocpnf 9654   -oocmnf 9655   RR*cxr 9656    < clt 9657   xecxmu 11369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-i2m1 9589  ax-1ne0 9590  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-xmul 11372
This theorem is referenced by:  xmulpnf2  11519  xmulmnf1  11520  xmulpnf1n  11522  xmulgt0  11527  xmulasslem3  11530  xlemul1a  11532  xadddilem  11538  xdivpnfrp  28067  xrge0adddir  28120  esumcst  28496  esumpinfval  28506
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