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Theorem xaddf 9620
Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
xaddf  |-  +e : ( RR*  X.  RR* )
--> RR*

Proof of Theorem xaddf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0xr 7805 . . . . . . 7  |-  0  e.  RR*
21a1i 9 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  0  e.  RR* )
3 pnfxr 7811 . . . . . . 7  |- +oo  e.  RR*
43a1i 9 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  -> +oo  e.  RR* )
5 xrmnfdc 9619 . . . . . . 7  |-  ( y  e.  RR*  -> DECID  y  = -oo )
65adantl 275 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  -> DECID  y  = -oo )
72, 4, 6ifcldcd 3502 . . . . 5  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  if ( y  = -oo ,  0 , +oo )  e.  RR* )
87adantr 274 . . . 4  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  x  = +oo )  ->  if ( y  = -oo ,  0 , +oo )  e. 
RR* )
91a1i 9 . . . . . 6  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  x  = -oo )  ->  0  e.  RR* )
10 mnfxr 7815 . . . . . . 7  |- -oo  e.  RR*
1110a1i 9 . . . . . 6  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  x  = -oo )  -> -oo  e.  RR* )
12 xrpnfdc 9618 . . . . . . 7  |-  ( y  e.  RR*  -> DECID  y  = +oo )
1312ad3antlr 484 . . . . . 6  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  x  = -oo )  -> DECID  y  = +oo )
149, 11, 13ifcldcd 3502 . . . . 5  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  x  = -oo )  ->  if ( y  = +oo ,  0 , -oo )  e.  RR* )
153a1i 9 . . . . . 6  |-  ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  y  = +oo )  -> +oo  e.  RR* )
1610a1i 9 . . . . . . 7  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  y  = -oo )  -> -oo  e.  RR* )
17 simp-4r 531 . . . . . . . . . 10  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  -.  x  = +oo )
18 simp-5l 532 . . . . . . . . . . 11  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  x  e.  RR* )
19 simpllr 523 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  -.  x  = -oo )
2019neqned 2313 . . . . . . . . . . 11  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  x  =/= -oo )
21 xrnemnf 9557 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  x  =/= -oo )  <->  ( x  e.  RR  \/  x  = +oo ) )
2221biimpi 119 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  x  =/= -oo )  ->  (
x  e.  RR  \/  x  = +oo )
)
2318, 20, 22syl2anc 408 . . . . . . . . . 10  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  ( x  e.  RR  \/  x  = +oo ) )
2417, 23ecased 1327 . . . . . . . . 9  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  x  e.  RR )
25 simplr 519 . . . . . . . . . 10  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  -.  y  = +oo )
26 simp-5r 533 . . . . . . . . . . 11  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  y  e.  RR* )
27 neqne 2314 . . . . . . . . . . . 12  |-  ( -.  y  = -oo  ->  y  =/= -oo )
2827adantl 275 . . . . . . . . . . 11  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  y  =/= -oo )
29 xrnemnf 9557 . . . . . . . . . . . 12  |-  ( ( y  e.  RR*  /\  y  =/= -oo )  <->  ( y  e.  RR  \/  y  = +oo ) )
3029biimpi 119 . . . . . . . . . . 11  |-  ( ( y  e.  RR*  /\  y  =/= -oo )  ->  (
y  e.  RR  \/  y  = +oo )
)
3126, 28, 30syl2anc 408 . . . . . . . . . 10  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  ( y  e.  RR  \/  y  = +oo ) )
3225, 31ecased 1327 . . . . . . . . 9  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  y  e.  RR )
3324, 32readdcld 7788 . . . . . . . 8  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  ( x  +  y )  e.  RR )
3433rexrd 7808 . . . . . . 7  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  ( x  +  y )  e.  RR* )
356ad3antrrr 483 . . . . . . 7  |-  ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  -> DECID  y  = -oo )
3616, 34, 35ifcldadc 3496 . . . . . 6  |-  ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  ->  if ( y  = -oo , -oo ,  ( x  +  y ) )  e.  RR* )
3712ad3antlr 484 . . . . . 6  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  -> DECID  y  = +oo )
3815, 36, 37ifcldadc 3496 . . . . 5  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  ->  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) )  e.  RR* )
39 xrmnfdc 9619 . . . . . 6  |-  ( x  e.  RR*  -> DECID  x  = -oo )
4039ad2antrr 479 . . . . 5  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  -> DECID 
x  = -oo )
4114, 38, 40ifcldadc 3496 . . . 4  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  ->  if ( x  = -oo ,  if ( y  = +oo ,  0 , -oo ) ,  if (
y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) )  e.  RR* )
42 xrpnfdc 9618 . . . . 5  |-  ( x  e.  RR*  -> DECID  x  = +oo )
4342adantr 274 . . . 4  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  -> DECID  x  = +oo )
448, 41, 43ifcldadc 3496 . . 3  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  if ( x  = +oo ,  if ( y  = -oo ,  0 , +oo ) ,  if ( x  = -oo ,  if ( y  = +oo ,  0 , -oo ) ,  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) ) )  e.  RR* )
4544rgen2a 2484 . 2  |-  A. x  e.  RR*  A. y  e. 
RR*  if ( x  = +oo ,  if ( y  = -oo , 
0 , +oo ) ,  if ( x  = -oo ,  if ( y  = +oo , 
0 , -oo ) ,  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) ) )  e. 
RR*
46 df-xadd 9553 . . 3  |-  +e 
=  ( x  e. 
RR* ,  y  e.  RR*  |->  if ( x  = +oo ,  if ( y  = -oo , 
0 , +oo ) ,  if ( x  = -oo ,  if ( y  = +oo , 
0 , -oo ) ,  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) ) ) )
4746fmpo 6092 . 2  |-  ( A. x  e.  RR*  A. y  e.  RR*  if ( x  = +oo ,  if ( y  = -oo ,  0 , +oo ) ,  if (
x  = -oo ,  if ( y  = +oo ,  0 , -oo ) ,  if (
y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) ) )  e.  RR*  <->  +e : ( RR*  X.  RR* )
--> RR* )
4845, 47mpbi 144 1  |-  +e : ( RR*  X.  RR* )
--> RR*
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 697  DECID wdc 819    = wceq 1331    e. wcel 1480    =/= wne 2306   A.wral 2414   ifcif 3469    X. cxp 4532   -->wf 5114  (class class class)co 5767   RRcr 7612   0cc0 7613    + caddc 7616   +oocpnf 7790   -oocmnf 7791   RR*cxr 7792   +ecxad 9550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1re 7707  ax-addrcl 7710  ax-rnegex 7722
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-pnf 7795  df-mnf 7796  df-xr 7797  df-xadd 9553
This theorem is referenced by:  xaddcl  9636
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