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Theorem xaddf 9749
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 7925 . . . . . . 7  |-  0  e.  RR*
21a1i 9 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  0  e.  RR* )
3 pnfxr 7931 . . . . . . 7  |- +oo  e.  RR*
43a1i 9 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  -> +oo  e.  RR* )
5 xrmnfdc 9748 . . . . . . 7  |-  ( y  e.  RR*  -> DECID  y  = -oo )
65adantl 275 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  -> DECID  y  = -oo )
72, 4, 6ifcldcd 3540 . . . . 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 7935 . . . . . . 7  |- -oo  e.  RR*
1110a1i 9 . . . . . 6  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  x  = -oo )  -> -oo  e.  RR* )
12 xrpnfdc 9747 . . . . . . 7  |-  ( y  e.  RR*  -> DECID  y  = +oo )
1312ad3antlr 485 . . . . . 6  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  x  = -oo )  -> DECID  y  = +oo )
149, 11, 13ifcldcd 3540 . . . . 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 532 . . . . . . . . . 10  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  -.  x  = +oo )
18 simp-5l 533 . . . . . . . . . . 11  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  x  e.  RR* )
19 simpllr 524 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  -.  x  = -oo )
2019neqned 2334 . . . . . . . . . . 11  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  x  =/= -oo )
21 xrnemnf 9685 . . . . . . . . . . . 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 409 . . . . . . . . . 10  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  ( x  e.  RR  \/  x  = +oo ) )
2417, 23ecased 1331 . . . . . . . . 9  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  x  e.  RR )
25 simplr 520 . . . . . . . . . 10  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  -.  y  = +oo )
26 simp-5r 534 . . . . . . . . . . 11  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  y  e.  RR* )
27 neqne 2335 . . . . . . . . . . . 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 9685 . . . . . . . . . . . 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 409 . . . . . . . . . 10  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  ( y  e.  RR  \/  y  = +oo ) )
3225, 31ecased 1331 . . . . . . . . 9  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  y  e.  RR )
3324, 32readdcld 7908 . . . . . . . 8  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  ( x  +  y )  e.  RR )
3433rexrd 7928 . . . . . . 7  |-  ( ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  ( x  +  y )  e.  RR* )
356ad3antrrr 484 . . . . . . 7  |-  ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  -> DECID  y  = -oo )
3616, 34, 35ifcldadc 3534 . . . . . 6  |-  ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  /\  -.  y  = +oo )  ->  if ( y  = -oo , -oo ,  ( x  +  y ) )  e.  RR* )
3712ad3antlr 485 . . . . . 6  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  -> DECID  y  = +oo )
3815, 36, 37ifcldadc 3534 . . . . 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 9748 . . . . . 6  |-  ( x  e.  RR*  -> DECID  x  = -oo )
4039ad2antrr 480 . . . . 5  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  -> DECID 
x  = -oo )
4114, 38, 40ifcldadc 3534 . . . 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 9747 . . . . 5  |-  ( x  e.  RR*  -> DECID  x  = +oo )
4342adantr 274 . . . 4  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  -> DECID  x  = +oo )
448, 41, 43ifcldadc 3534 . . 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 2511 . 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 9681 . . 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 6150 . 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 698  DECID wdc 820    = wceq 1335    e. wcel 2128    =/= wne 2327   A.wral 2435   ifcif 3505    X. cxp 4585   -->wf 5167  (class class class)co 5825   RRcr 7732   0cc0 7733    + caddc 7736   +oocpnf 7910   -oocmnf 7911   RR*cxr 7912   +ecxad 9678
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-cnex 7824  ax-resscn 7825  ax-1re 7827  ax-addrcl 7830  ax-rnegex 7842
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-fv 5179  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-pnf 7915  df-mnf 7916  df-xr 7917  df-xadd 9681
This theorem is referenced by:  xaddcl  9765
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