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Theorem xnegid 11310
Description: Extended real version of negid 9760. (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 11200 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 rexneg 11285 . . . . 5  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
32oveq2d 6209 . . . 4  |-  ( A  e.  RR  ->  ( A +e  -e
A )  =  ( A +e -u A ) )
4 renegcl 9776 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
5 rexadd 11306 . . . . 5  |-  ( ( A  e.  RR  /\  -u A  e.  RR )  ->  ( A +e -u A )  =  ( A  +  -u A ) )
64, 5mpdan 668 . . . 4  |-  ( A  e.  RR  ->  ( A +e -u A
)  =  ( A  +  -u A ) )
7 recn 9476 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
87negidd 9813 . . . 4  |-  ( A  e.  RR  ->  ( A  +  -u A )  =  0 )
93, 6, 83eqtrd 2496 . . 3  |-  ( A  e.  RR  ->  ( A +e  -e
A )  =  0 )
10 id 22 . . . . 5  |-  ( A  = +oo  ->  A  = +oo )
11 xnegeq 11281 . . . . . 6  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
12 xnegpnf 11283 . . . . . 6  |-  -e +oo  = -oo
1311, 12syl6eq 2508 . . . . 5  |-  ( A  = +oo  ->  -e
A  = -oo )
1410, 13oveq12d 6211 . . . 4  |-  ( A  = +oo  ->  ( A +e  -e
A )  =  ( +oo +e -oo ) )
15 pnfaddmnf 11304 . . . 4  |-  ( +oo +e -oo )  =  0
1614, 15syl6eq 2508 . . 3  |-  ( A  = +oo  ->  ( A +e  -e
A )  =  0 )
17 id 22 . . . . 5  |-  ( A  = -oo  ->  A  = -oo )
18 xnegeq 11281 . . . . . 6  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
19 xnegmnf 11284 . . . . . 6  |-  -e -oo  = +oo
2018, 19syl6eq 2508 . . . . 5  |-  ( A  = -oo  ->  -e
A  = +oo )
2117, 20oveq12d 6211 . . . 4  |-  ( A  = -oo  ->  ( A +e  -e
A )  =  ( -oo +e +oo ) )
22 mnfaddpnf 11305 . . . 4  |-  ( -oo +e +oo )  =  0
2321, 22syl6eq 2508 . . 3  |-  ( A  = -oo  ->  ( A +e  -e
A )  =  0 )
249, 16, 233jaoi 1282 . 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 964    = wceq 1370    e. wcel 1758  (class class class)co 6193   RRcr 9385   0cc0 9386    + caddc 9389   +oocpnf 9519   -oocmnf 9520   RR*cxr 9521   -ucneg 9700    -ecxne 11190   +ecxad 11191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-sub 9701  df-neg 9702  df-xneg 11193  df-xadd 11194
This theorem is referenced by:  xrsxmet  20511  xaddeq0  26190  xlt2addrd  26195  xrge0npcan  26295
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