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Theorem xnegneg 11465
Description: Extended real version of negneg 9904. (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 11377 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 rexneg 11462 . . . . 5  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
3 xnegeq 11458 . . . . 5  |-  (  -e A  =  -u A  -> 
-e  -e
A  =  -e -u A )
42, 3syl 17 . . . 4  |-  ( A  e.  RR  ->  -e  -e A  =  -e -u A )
5 renegcl 9917 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
6 rexneg 11462 . . . . 5  |-  ( -u A  e.  RR  ->  -e -u A  =  -u -u A )
75, 6syl 17 . . . 4  |-  ( A  e.  RR  ->  -e -u A  =  -u -u A
)
8 recn 9611 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
98negnegd 9957 . . . 4  |-  ( A  e.  RR  ->  -u -u A  =  A )
104, 7, 93eqtrd 2447 . . 3  |-  ( A  e.  RR  ->  -e  -e A  =  A )
11 xnegmnf 11461 . . . 4  |-  -e -oo  = +oo
12 xnegeq 11458 . . . . . 6  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
13 xnegpnf 11460 . . . . . 6  |-  -e +oo  = -oo
1412, 13syl6eq 2459 . . . . 5  |-  ( A  = +oo  ->  -e
A  = -oo )
15 xnegeq 11458 . . . . 5  |-  (  -e A  = -oo  -> 
-e  -e
A  =  -e -oo )
1614, 15syl 17 . . . 4  |-  ( A  = +oo  ->  -e  -e A  =  -e -oo )
17 id 22 . . . 4  |-  ( A  = +oo  ->  A  = +oo )
1811, 16, 173eqtr4a 2469 . . 3  |-  ( A  = +oo  ->  -e  -e A  =  A )
19 xnegeq 11458 . . . . . 6  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
2019, 11syl6eq 2459 . . . . 5  |-  ( A  = -oo  ->  -e
A  = +oo )
21 xnegeq 11458 . . . . 5  |-  (  -e A  = +oo  -> 
-e  -e
A  =  -e +oo )
2220, 21syl 17 . . . 4  |-  ( A  = -oo  ->  -e  -e A  =  -e +oo )
23 id 22 . . . 4  |-  ( A  = -oo  ->  A  = -oo )
2413, 22, 233eqtr4a 2469 . . 3  |-  ( A  = -oo  ->  -e  -e A  =  A )
2510, 18, 243jaoi 1293 . 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 973    = wceq 1405    e. wcel 1842   RRcr 9520   +oocpnf 9654   -oocmnf 9655   RR*cxr 9656   -ucneg 9841    -ecxne 11367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-sub 9842  df-neg 9843  df-xneg 11370
This theorem is referenced by:  xneg11  11466  xltneg  11468  xnegdi  11492  xnpcan  11496  xposdif  11506  xrsxmet  21604  xrhmeo  21736  xaddeq0  28000  xrge0npcan  28122  carsgclctunlem2  28753
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