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Theorem xaddmnf1 10571
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 10472 . . 3  |-  -oo  e.  RR*
2 xaddval 10566 . . 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 652 . 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 3586 . . 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 pnfnemnf 10475 . . . . . . 7  |-  +oo  =/=  -oo
65necomi 2541 . . . . . 6  |-  -oo  =/=  +oo
7 ifnefalse 3586 . . . . . 6  |-  (  -oo  =/=  +oo  ->  if (  -oo  =  +oo ,  0 ,  -oo )  = 
-oo )
86, 7ax-mp 8 . . . . 5  |-  if ( 
-oo  =  +oo , 
0 ,  -oo )  =  -oo
9 ifnefalse 3586 . . . . . . 7  |-  (  -oo  =/=  +oo  ->  if (  -oo  =  +oo ,  +oo ,  if (  -oo  =  -oo ,  -oo ,  ( A  +  -oo )
) )  =  if (  -oo  =  -oo , 
-oo ,  ( A  +  -oo ) ) )
106, 9ax-mp 8 . . . . . 6  |-  if ( 
-oo  =  +oo ,  +oo ,  if (  -oo  =  -oo ,  -oo , 
( A  +  -oo ) ) )  =  if (  -oo  =  -oo ,  -oo ,  ( A  +  -oo )
)
11 eqid 2296 . . . . . . 7  |-  -oo  =  -oo
12 iftrue 3584 . . . . . . 7  |-  (  -oo  =  -oo  ->  if (  -oo  =  -oo ,  -oo ,  ( A  +  -oo ) )  =  -oo )
1311, 12ax-mp 8 . . . . . 6  |-  if ( 
-oo  =  -oo ,  -oo ,  ( A  +  -oo ) )  =  -oo
1410, 13eqtri 2316 . . . . 5  |-  if ( 
-oo  =  +oo ,  +oo ,  if (  -oo  =  -oo ,  -oo , 
( A  +  -oo ) ) )  = 
-oo
15 ifeq12 3591 . . . . 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 ) )
168, 14, 15mp2an 653 . . . 4  |-  if ( A  =  -oo ,  if (  -oo  =  +oo ,  0 ,  -oo ) ,  if (  -oo  =  +oo ,  +oo ,  if (  -oo  =  -oo ,  -oo ,  ( A  +  -oo ) ) ) )  =  if ( A  =  -oo ,  -oo , 
-oo )
17 ifid 3610 . . . 4  |-  if ( A  =  -oo ,  -oo ,  -oo )  = 
-oo
1816, 17eqtri 2316 . . 3  |-  if ( A  =  -oo ,  if (  -oo  =  +oo ,  0 ,  -oo ) ,  if (  -oo  =  +oo ,  +oo ,  if (  -oo  =  -oo ,  -oo ,  ( A  +  -oo ) ) ) )  =  -oo
194, 18syl6eq 2344 . 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 )
203, 19sylan9eq 2348 1  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  ( A + e  -oo )  =  -oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   ifcif 3578  (class class class)co 5874   0cc0 8753    + caddc 8756    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882   + ecxad 10466
This theorem is referenced by:  xaddnepnf  10578  xaddcom  10581  xnegdi  10584  xleadd1a  10589  xsubge0  10597  xlesubadd  10599  xadddilem  10630  xblss2  17974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-mulcl 8815  ax-i2m1 8821
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pnf 8885  df-mnf 8886  df-xr 8887  df-xadd 10469
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