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Theorem xnegneg 11402
Description: Extended real version of negneg 9858. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegneg  |-  ( A  e.  RR*  ->  -e  -e A  =  A )

Proof of Theorem xnegneg
StepHypRef Expression
1 elxr 11314 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 rexneg 11399 . . . . 5  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
3 xnegeq 11395 . . . . 5  |-  (  -e A  =  -u A  -> 
-e  -e
A  =  -e -u A )
42, 3syl 16 . . . 4  |-  ( A  e.  RR  ->  -e  -e A  =  -e -u A )
5 renegcl 9871 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
6 rexneg 11399 . . . . 5  |-  ( -u A  e.  RR  ->  -e -u A  =  -u -u A )
75, 6syl 16 . . . 4  |-  ( A  e.  RR  ->  -e -u A  =  -u -u A
)
8 recn 9571 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
98negnegd 9910 . . . 4  |-  ( A  e.  RR  ->  -u -u A  =  A )
104, 7, 93eqtrd 2505 . . 3  |-  ( A  e.  RR  ->  -e  -e A  =  A )
11 xnegmnf 11398 . . . 4  |-  -e -oo  = +oo
12 xnegeq 11395 . . . . . 6  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
13 xnegpnf 11397 . . . . . 6  |-  -e +oo  = -oo
1412, 13syl6eq 2517 . . . . 5  |-  ( A  = +oo  ->  -e
A  = -oo )
15 xnegeq 11395 . . . . 5  |-  (  -e A  = -oo  -> 
-e  -e
A  =  -e -oo )
1614, 15syl 16 . . . 4  |-  ( A  = +oo  ->  -e  -e A  =  -e -oo )
17 id 22 . . . 4  |-  ( A  = +oo  ->  A  = +oo )
1811, 16, 173eqtr4a 2527 . . 3  |-  ( A  = +oo  ->  -e  -e A  =  A )
19 xnegeq 11395 . . . . . 6  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
2019, 11syl6eq 2517 . . . . 5  |-  ( A  = -oo  ->  -e
A  = +oo )
21 xnegeq 11395 . . . . 5  |-  (  -e A  = +oo  -> 
-e  -e
A  =  -e +oo )
2220, 21syl 16 . . . 4  |-  ( A  = -oo  ->  -e  -e A  =  -e +oo )
23 id 22 . . . 4  |-  ( A  = -oo  ->  A  = -oo )
2413, 22, 233eqtr4a 2527 . . 3  |-  ( A  = -oo  ->  -e  -e A  =  A )
2510, 18, 243jaoi 1286 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  -e  -e A  =  A )
261, 25sylbi 195 1  |-  ( A  e.  RR*  ->  -e  -e A  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 967    = wceq 1374    e. wcel 1762   RRcr 9480   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616   -ucneg 9795    -ecxne 11304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-sub 9796  df-neg 9797  df-xneg 11307
This theorem is referenced by:  xneg11  11403  xltneg  11405  xnegdi  11429  xnpcan  11433  xposdif  11443  xrsxmet  21042  xrhmeo  21174  xaddeq0  27227  xrge0npcan  27332
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