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Theorem xaddeq0 26046
Description: Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)
Assertion
Ref Expression
xaddeq0  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A +e
B )  =  0  <-> 
A  =  -e
B ) )

Proof of Theorem xaddeq0
StepHypRef Expression
1 elxr 11096 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 simpll 753 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  e.  RR )
32rexrd 9433 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  e.  RR* )
4 xnegneg 11184 . . . . . . 7  |-  ( A  e.  RR*  ->  -e  -e A  =  A )
53, 4syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e  -e A  =  A
)
63xnegcld 11263 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e A  e.  RR* )
7 xaddid2 11210 . . . . . . . . 9  |-  (  -e A  e.  RR*  ->  ( 0 +e  -e A )  = 
-e A )
86, 7syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( 0 +e  -e
A )  =  -e A )
9 simplr 754 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  e.  RR* )
10 xaddcom 11208 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  ( B +e A ) )
113, 9, 10syl2anc 661 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  ( B +e A ) )
1211oveq1d 6106 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( ( A +e B ) +e  -e
A )  =  ( ( B +e
A ) +e  -e A ) )
13 simpr 461 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  0 )
1413oveq1d 6106 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( ( A +e B ) +e  -e
A )  =  ( 0 +e  -e A ) )
15 xpncan 11214 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  A  e.  RR )  ->  (
( B +e
A ) +e  -e A )  =  B )
1615ancoms 453 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( B +e A ) +e  -e A )  =  B )
1716adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( ( B +e A ) +e  -e
A )  =  B )
1812, 14, 173eqtr3d 2483 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( 0 +e  -e
A )  =  B )
198, 18eqtr3d 2477 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e A  =  B )
20 xnegeq 11177 . . . . . . 7  |-  (  -e A  =  B  -> 
-e  -e
A  =  -e
B )
2119, 20syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e  -e A  =  -e
B )
225, 21eqtr3d 2477 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  =  -e B )
2322ex 434 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A +e B )  =  0  ->  A  =  -e B ) )
24 simpll 753 . . . . . 6  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  = +oo )
25 simplr 754 . . . . . . . . . 10  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  e.  RR* )
2624oveq1d 6106 . . . . . . . . . . . . 13  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  ( +oo +e B ) )
27 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  0 )
2826, 27eqtr3d 2477 . . . . . . . . . . . 12  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( +oo +e B )  =  0 )
29 0re 9386 . . . . . . . . . . . . 13  |-  0  e.  RR
30 renepnf 9431 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  0  =/= +oo )
3129, 30mp1i 12 . . . . . . . . . . . 12  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  0  =/= +oo )
3228, 31eqnetrd 2626 . . . . . . . . . . 11  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( +oo +e B )  =/= +oo )
3332neneqd 2624 . . . . . . . . . 10  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  ( +oo +e B )  = +oo )
34 xaddpnf2 11197 . . . . . . . . . . . 12  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
3534ex 434 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  ( B  =/= -oo  ->  ( +oo +e B )  = +oo ) )
3635con3dimp 441 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  -.  ( +oo +e B )  = +oo )  ->  -.  B  =/= -oo )
3725, 33, 36syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  B  =/= -oo )
38 nne 2612 . . . . . . . . 9  |-  ( -.  B  =/= -oo  <->  B  = -oo )
3937, 38sylib 196 . . . . . . . 8  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  = -oo )
40 xnegeq 11177 . . . . . . . 8  |-  ( B  = -oo  ->  -e
B  =  -e -oo )
4139, 40syl 16 . . . . . . 7  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e B  =  -e -oo )
42 xnegmnf 11180 . . . . . . 7  |-  -e -oo  = +oo
4341, 42syl6req 2492 . . . . . 6  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  -> +oo  =  -e B )
4424, 43eqtrd 2475 . . . . 5  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  =  -e B )
4544ex 434 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( ( A +e B )  =  0  ->  A  =  -e B ) )
46 simpll 753 . . . . . 6  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  = -oo )
47 simplr 754 . . . . . . . . . 10  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  e.  RR* )
4846oveq1d 6106 . . . . . . . . . . . . 13  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  ( -oo +e B ) )
49 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  0 )
5048, 49eqtr3d 2477 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( -oo +e B )  =  0 )
51 renemnf 9432 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  0  =/= -oo )
5229, 51mp1i 12 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  0  =/= -oo )
5350, 52eqnetrd 2626 . . . . . . . . . . 11  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( -oo +e B )  =/= -oo )
5453neneqd 2624 . . . . . . . . . 10  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  ( -oo +e B )  = -oo )
55 xaddmnf2 11199 . . . . . . . . . . . 12  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
5655ex 434 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  ( B  =/= +oo  ->  ( -oo +e B )  = -oo ) )
5756con3dimp 441 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  -.  ( -oo +e B )  = -oo )  ->  -.  B  =/= +oo )
5847, 54, 57syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  B  =/= +oo )
59 nne 2612 . . . . . . . . 9  |-  ( -.  B  =/= +oo  <->  B  = +oo )
6058, 59sylib 196 . . . . . . . 8  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  = +oo )
61 xnegeq 11177 . . . . . . . 8  |-  ( B  = +oo  ->  -e
B  =  -e +oo )
6260, 61syl 16 . . . . . . 7  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e B  =  -e +oo )
63 xnegpnf 11179 . . . . . . 7  |-  -e +oo  = -oo
6462, 63syl6req 2492 . . . . . 6  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  -> -oo  =  -e B )
6546, 64eqtrd 2475 . . . . 5  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  =  -e B )
6665ex 434 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( ( A +e B )  =  0  ->  A  =  -e B ) )
6723, 45, 663jaoian 1283 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  ( ( A +e B )  =  0  ->  A  =  -e B ) )
681, 67sylanb 472 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A +e
B )  =  0  ->  A  =  -e B ) )
69 simpr 461 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  A  =  -e B )
7069oveq1d 6106 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  ( A +e B )  =  (  -e
B +e B ) )
71 xnegcl 11183 . . . . . 6  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
7271ad2antlr 726 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  -e
B  e.  RR* )
73 simplr 754 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  B  e.  RR* )
74 xaddcom 11208 . . . . 5  |-  ( ( 
-e B  e. 
RR*  /\  B  e.  RR* )  ->  (  -e
B +e B )  =  ( B +e  -e
B ) )
7572, 73, 74syl2anc 661 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  (  -e B +e
B )  =  ( B +e  -e B ) )
76 xnegid 11206 . . . . 5  |-  ( B  e.  RR*  ->  ( B +e  -e
B )  =  0 )
7776ad2antlr 726 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  ( B +e  -e
B )  =  0 )
7870, 75, 773eqtrd 2479 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  ( A +e B )  =  0 )
7978ex 434 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  -e B  ->  ( A +e B )  =  0 ) )
8068, 79impbid 191 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A +e
B )  =  0  <-> 
A  =  -e
B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 964    = wceq 1369    e. wcel 1756    =/= wne 2606  (class class class)co 6091   RRcr 9281   0cc0 9282   +oocpnf 9415   -oocmnf 9416   RR*cxr 9417    -ecxne 11086   +ecxad 11087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-sub 9597  df-neg 9598  df-xneg 11089  df-xadd 11090
This theorem is referenced by:  xrsinvgval  26138
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