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Theorem xmulpnf1n 11474
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 457 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  A  e.  RR* )
2 pnfxr 11325 . . . . 5  |- +oo  e.  RR*
3 xmulneg1 11465 . . . . 5  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  (  -e A xe +oo )  =  -e ( A xe +oo ) )
41, 2, 3sylancl 662 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (  -e A xe +oo )  =  -e ( A xe +oo ) )
5 xnegcl 11416 . . . . . 6  |-  ( A  e.  RR*  ->  -e
A  e.  RR* )
65adantr 465 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  -e
A  e.  RR* )
7 xlt0neg1 11422 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
87biimpa 484 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  0  <  -e A )
9 xmulpnf1 11470 . . . . 5  |-  ( ( 
-e A  e. 
RR*  /\  0  <  -e A )  -> 
(  -e A xe +oo )  = +oo )
106, 8, 9syl2anc 661 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (  -e A xe +oo )  = +oo )
114, 10eqtr3d 2484 . . 3  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  -e
( A xe +oo )  = +oo )
12 xnegmnf 11413 . . 3  |-  -e -oo  = +oo
1311, 12syl6eqr 2500 . 2  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  -e
( A xe +oo )  =  -e -oo )
14 xmulcl 11469 . . . 4  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A xe +oo )  e.  RR* )
151, 2, 14sylancl 662 . . 3  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  ( A xe +oo )  e.  RR* )
16 mnfxr 11327 . . 3  |- -oo  e.  RR*
17 xneg11 11418 . . 3  |-  ( ( ( A xe +oo )  e.  RR*  /\ -oo  e.  RR* )  ->  (  -e ( A xe +oo )  = 
-e -oo  <->  ( A xe +oo )  = -oo ) )
1815, 16, 17sylancl 662 . 2  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (  -e ( A xe +oo )  = 
-e -oo  <->  ( A xe +oo )  = -oo ) )
1913, 18mpbid 210 1  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  ( A xe +oo )  = -oo )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   class class class wbr 4433  (class class class)co 6277   0cc0 9490   +oocpnf 9623   -oocmnf 9624   RR*cxr 9625    < clt 9626    -ecxne 11319   xecxmu 11321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-po 4786  df-so 4787  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-xneg 11322  df-xmul 11324
This theorem is referenced by:  xlemul1a  11484
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