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Theorem xmulpnf1n 11589
Description: Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmulpnf1n  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  ( A xe +oo )  = -oo )

Proof of Theorem xmulpnf1n
StepHypRef Expression
1 simpl 464 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  A  e.  RR* )
2 pnfxr 11435 . . . . 5  |- +oo  e.  RR*
3 xmulneg1 11580 . . . . 5  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  (  -e A xe +oo )  =  -e ( A xe +oo ) )
41, 2, 3sylancl 675 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (  -e A xe +oo )  =  -e ( A xe +oo ) )
5 xnegcl 11529 . . . . . 6  |-  ( A  e.  RR*  ->  -e
A  e.  RR* )
65adantr 472 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  -e
A  e.  RR* )
7 xlt0neg1 11535 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
87biimpa 492 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  0  <  -e A )
9 xmulpnf1 11585 . . . . 5  |-  ( ( 
-e A  e. 
RR*  /\  0  <  -e A )  -> 
(  -e A xe +oo )  = +oo )
106, 8, 9syl2anc 673 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (  -e A xe +oo )  = +oo )
114, 10eqtr3d 2507 . . 3  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  -e
( A xe +oo )  = +oo )
12 xnegmnf 11526 . . 3  |-  -e -oo  = +oo
1311, 12syl6eqr 2523 . 2  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  -e
( A xe +oo )  =  -e -oo )
14 xmulcl 11584 . . . 4  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A xe +oo )  e.  RR* )
151, 2, 14sylancl 675 . . 3  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  ( A xe +oo )  e.  RR* )
16 mnfxr 11437 . . 3  |- -oo  e.  RR*
17 xneg11 11531 . . 3  |-  ( ( ( A xe +oo )  e.  RR*  /\ -oo  e.  RR* )  ->  (  -e ( A xe +oo )  = 
-e -oo  <->  ( A xe +oo )  = -oo ) )
1815, 16, 17sylancl 675 . 2  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (  -e ( A xe +oo )  = 
-e -oo  <->  ( A xe +oo )  = -oo ) )
1913, 18mpbid 215 1  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  ( A xe +oo )  = -oo )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   class class class wbr 4395  (class class class)co 6308   0cc0 9557   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    < clt 9693    -ecxne 11429   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-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
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-reu 2763  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-iun 4271  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-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  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-le 9699  df-sub 9882  df-neg 9883  df-xneg 11432  df-xmul 11434
This theorem is referenced by:  xlemul1a  11599
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