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Theorem xaddmnf1 11423
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 11319 . . 3  |- -oo  e.  RR*
2 xaddval 11418 . . 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 3951 . . 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 11323 . . . . . 6  |- -oo  =/= +oo
6 ifnefalse 3951 . . . . . 6  |-  ( -oo  =/= +oo  ->  if ( -oo  = +oo ,  0 , -oo )  = -oo )
75, 6ax-mp 5 . . . . 5  |-  if ( -oo  = +oo , 
0 , -oo )  = -oo
8 ifnefalse 3951 . . . . . . 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 2467 . . . . . . 7  |- -oo  = -oo
1110iftruei 3946 . . . . . 6  |-  if ( -oo  = -oo , -oo ,  ( A  + -oo ) )  = -oo
129, 11eqtri 2496 . . . . 5  |-  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo , 
( A  + -oo ) ) )  = -oo
13 ifeq12 3956 . . . . 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 3976 . . . 4  |-  if ( A  = -oo , -oo , -oo )  = -oo
1614, 15eqtri 2496 . . 3  |-  if ( A  = -oo ,  if ( -oo  = +oo ,  0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo )
) ) )  = -oo
174, 16syl6eq 2524 . 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 2528 1  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3939  (class class class)co 6282   0cc0 9488    + caddc 9491   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623   +ecxad 11312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-mulcl 9550  ax-i2m1 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-pnf 9626  df-mnf 9627  df-xr 9628  df-xadd 11315
This theorem is referenced by:  xaddnepnf  11430  xaddcom  11433  xnegdi  11436  xleadd1a  11441  xsubge0  11449  xlesubadd  11451  xadddilem  11482  xblss2ps  20636  xblss2  20637
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