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Theorem xnegid 11553
Description: Extended real version of negid 9941. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegid  |-  ( A  e.  RR*  ->  ( A +e  -e
A )  =  0 )

Proof of Theorem xnegid
StepHypRef Expression
1 elxr 11439 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 rexneg 11527 . . . . 5  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
32oveq2d 6324 . . . 4  |-  ( A  e.  RR  ->  ( A +e  -e
A )  =  ( A +e -u A ) )
4 renegcl 9957 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
5 rexadd 11548 . . . . 5  |-  ( ( A  e.  RR  /\  -u A  e.  RR )  ->  ( A +e -u A )  =  ( A  +  -u A ) )
64, 5mpdan 681 . . . 4  |-  ( A  e.  RR  ->  ( A +e -u A
)  =  ( A  +  -u A ) )
7 recn 9647 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
87negidd 9995 . . . 4  |-  ( A  e.  RR  ->  ( A  +  -u A )  =  0 )
93, 6, 83eqtrd 2509 . . 3  |-  ( A  e.  RR  ->  ( A +e  -e
A )  =  0 )
10 id 22 . . . . 5  |-  ( A  = +oo  ->  A  = +oo )
11 xnegeq 11523 . . . . . 6  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
12 xnegpnf 11525 . . . . . 6  |-  -e +oo  = -oo
1311, 12syl6eq 2521 . . . . 5  |-  ( A  = +oo  ->  -e
A  = -oo )
1410, 13oveq12d 6326 . . . 4  |-  ( A  = +oo  ->  ( A +e  -e
A )  =  ( +oo +e -oo ) )
15 pnfaddmnf 11546 . . . 4  |-  ( +oo +e -oo )  =  0
1614, 15syl6eq 2521 . . 3  |-  ( A  = +oo  ->  ( A +e  -e
A )  =  0 )
17 id 22 . . . . 5  |-  ( A  = -oo  ->  A  = -oo )
18 xnegeq 11523 . . . . . 6  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
19 xnegmnf 11526 . . . . . 6  |-  -e -oo  = +oo
2018, 19syl6eq 2521 . . . . 5  |-  ( A  = -oo  ->  -e
A  = +oo )
2117, 20oveq12d 6326 . . . 4  |-  ( A  = -oo  ->  ( A +e  -e
A )  =  ( -oo +e +oo ) )
22 mnfaddpnf 11547 . . . 4  |-  ( -oo +e +oo )  =  0
2321, 22syl6eq 2521 . . 3  |-  ( A  = -oo  ->  ( A +e  -e
A )  =  0 )
249, 16, 233jaoi 1357 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( A +e  -e
A )  =  0 )
251, 24sylbi 200 1  |-  ( A  e.  RR*  ->  ( A +e  -e
A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 1006    = wceq 1452    e. wcel 1904  (class class class)co 6308   RRcr 9556   0cc0 9557    + caddc 9560   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692   -ucneg 9881    -ecxne 11429   +ecxad 11430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-sub 9882  df-neg 9883  df-xneg 11432  df-xadd 11433
This theorem is referenced by:  xrsxmet  21905  xaddeq0  28405  xlt2addrd  28413  xrge0npcan  28531  carsgclctunlem2  29224
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