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Theorem xnegid 11438
Description: Extended real version of negid 9857. (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 11328 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 rexneg 11413 . . . . 5  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
32oveq2d 6286 . . . 4  |-  ( A  e.  RR  ->  ( A +e  -e
A )  =  ( A +e -u A ) )
4 renegcl 9873 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
5 rexadd 11434 . . . . 5  |-  ( ( A  e.  RR  /\  -u A  e.  RR )  ->  ( A +e -u A )  =  ( A  +  -u A ) )
64, 5mpdan 666 . . . 4  |-  ( A  e.  RR  ->  ( A +e -u A
)  =  ( A  +  -u A ) )
7 recn 9571 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
87negidd 9912 . . . 4  |-  ( A  e.  RR  ->  ( A  +  -u A )  =  0 )
93, 6, 83eqtrd 2499 . . 3  |-  ( A  e.  RR  ->  ( A +e  -e
A )  =  0 )
10 id 22 . . . . 5  |-  ( A  = +oo  ->  A  = +oo )
11 xnegeq 11409 . . . . . 6  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
12 xnegpnf 11411 . . . . . 6  |-  -e +oo  = -oo
1311, 12syl6eq 2511 . . . . 5  |-  ( A  = +oo  ->  -e
A  = -oo )
1410, 13oveq12d 6288 . . . 4  |-  ( A  = +oo  ->  ( A +e  -e
A )  =  ( +oo +e -oo ) )
15 pnfaddmnf 11432 . . . 4  |-  ( +oo +e -oo )  =  0
1614, 15syl6eq 2511 . . 3  |-  ( A  = +oo  ->  ( A +e  -e
A )  =  0 )
17 id 22 . . . . 5  |-  ( A  = -oo  ->  A  = -oo )
18 xnegeq 11409 . . . . . 6  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
19 xnegmnf 11412 . . . . . 6  |-  -e -oo  = +oo
2018, 19syl6eq 2511 . . . . 5  |-  ( A  = -oo  ->  -e
A  = +oo )
2117, 20oveq12d 6288 . . . 4  |-  ( A  = -oo  ->  ( A +e  -e
A )  =  ( -oo +e +oo ) )
22 mnfaddpnf 11433 . . . 4  |-  ( -oo +e +oo )  =  0
2321, 22syl6eq 2511 . . 3  |-  ( A  = -oo  ->  ( A +e  -e
A )  =  0 )
249, 16, 233jaoi 1289 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( A +e  -e
A )  =  0 )
251, 24sylbi 195 1  |-  ( A  e.  RR*  ->  ( A +e  -e
A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 970    = wceq 1398    e. wcel 1823  (class class class)co 6270   RRcr 9480   0cc0 9481    + caddc 9484   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616   -ucneg 9797    -ecxne 11318   +ecxad 11319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-sub 9798  df-neg 9799  df-xneg 11321  df-xadd 11322
This theorem is referenced by:  xrsxmet  21480  xaddeq0  27804  xlt2addrd  27809  xrge0npcan  27918  carsgclctunlem2  28527
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