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Theorem xaddmnf1 11310
Description: Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddmnf1  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )

Proof of Theorem xaddmnf1
StepHypRef Expression
1 mnfxr 11206 . . 3  |- -oo  e.  RR*
2 xaddval 11305 . . 3  |-  ( ( A  e.  RR*  /\ -oo  e.  RR* )  ->  ( A +e -oo )  =  if ( A  = +oo ,  if ( -oo  = -oo , 
0 , +oo ) ,  if ( A  = -oo ,  if ( -oo  = +oo , 
0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo ) ) ) ) ) )
31, 2mpan2 671 . 2  |-  ( A  e.  RR*  ->  ( A +e -oo )  =  if ( A  = +oo ,  if ( -oo  = -oo , 
0 , +oo ) ,  if ( A  = -oo ,  if ( -oo  = +oo , 
0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo ) ) ) ) ) )
4 ifnefalse 3910 . . 3  |-  ( A  =/= +oo  ->  if ( A  = +oo ,  if ( -oo  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( -oo  = +oo , 
0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo ) ) ) ) )  =  if ( A  = -oo ,  if ( -oo  = +oo ,  0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo )
) ) ) )
5 mnfnepnf 11210 . . . . . 6  |- -oo  =/= +oo
6 ifnefalse 3910 . . . . . 6  |-  ( -oo  =/= +oo  ->  if ( -oo  = +oo ,  0 , -oo )  = -oo )
75, 6ax-mp 5 . . . . 5  |-  if ( -oo  = +oo , 
0 , -oo )  = -oo
8 ifnefalse 3910 . . . . . . 7  |-  ( -oo  =/= +oo  ->  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo )
) )  =  if ( -oo  = -oo , -oo ,  ( A  + -oo ) ) )
95, 8ax-mp 5 . . . . . 6  |-  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo , 
( A  + -oo ) ) )  =  if ( -oo  = -oo , -oo ,  ( A  + -oo )
)
10 eqid 2454 . . . . . . 7  |- -oo  = -oo
1110iftruei 3907 . . . . . 6  |-  if ( -oo  = -oo , -oo ,  ( A  + -oo ) )  = -oo
129, 11eqtri 2483 . . . . 5  |-  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo , 
( A  + -oo ) ) )  = -oo
13 ifeq12 3915 . . . . 5  |-  ( ( if ( -oo  = +oo ,  0 , -oo )  = -oo  /\  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo , 
( A  + -oo ) ) )  = -oo )  ->  if ( A  = -oo ,  if ( -oo  = +oo ,  0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo )
) ) )  =  if ( A  = -oo , -oo , -oo ) )
147, 12, 13mp2an 672 . . . 4  |-  if ( A  = -oo ,  if ( -oo  = +oo ,  0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo )
) ) )  =  if ( A  = -oo , -oo , -oo )
15 ifid 3935 . . . 4  |-  if ( A  = -oo , -oo , -oo )  = -oo
1614, 15eqtri 2483 . . 3  |-  if ( A  = -oo ,  if ( -oo  = +oo ,  0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo )
) ) )  = -oo
174, 16syl6eq 2511 . 2  |-  ( A  =/= +oo  ->  if ( A  = +oo ,  if ( -oo  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( -oo  = +oo , 
0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo ) ) ) ) )  = -oo )
183, 17sylan9eq 2515 1  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   ifcif 3900  (class class class)co 6201   0cc0 9394    + caddc 9397   +oocpnf 9527   -oocmnf 9528   RR*cxr 9529   +ecxad 11199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-mulcl 9456  ax-i2m1 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-pnf 9532  df-mnf 9533  df-xr 9534  df-xadd 11202
This theorem is referenced by:  xaddnepnf  11317  xaddcom  11320  xnegdi  11323  xleadd1a  11328  xsubge0  11336  xlesubadd  11338  xadddilem  11369  xblss2ps  20109  xblss2  20110
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