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Theorem xaddeq0 24119
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 10716 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
2 simpll 731 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  e.  RR )
32rexrd 9134 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  e.  RR* )
4 xnegneg 10800 . . . . . . 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 10879 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  - e A  e.  RR* )
7 xaddid2 10826 . . . . . . . . 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 732 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  e.  RR* )
10 xaddcom 10824 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e B )  =  ( B + e A ) )
113, 9, 10syl2anc 643 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  ( B + e A ) )
1211oveq1d 6096 . . . . . . . . 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 448 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  0 )
1413oveq1d 6096 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( ( A + e B ) + e  - e A )  =  ( 0 + e  - e A ) )
15 xpncan 10830 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  A  e.  RR )  ->  (
( B + e A ) + e  - e A )  =  B )
1615ancoms 440 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( B + e A ) + e  - e A )  =  B )
1716adantr 452 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( ( B + e A ) + e  - e A )  =  B )
1812, 14, 173eqtr3d 2476 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( 0 + e  - e A )  =  B )
198, 18eqtr3d 2470 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  - e A  =  B )
20 xnegeq 10793 . . . . . . 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 2470 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  =  - e B )
2322ex 424 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A + e B )  =  0  ->  A  =  - e B ) )
24 simpll 731 . . . . . 6  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  =  +oo )
25 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  e.  RR* )
2624oveq1d 6096 . . . . . . . . . . . . 13  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  (  +oo + e B ) )
27 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  0 )
2826, 27eqtr3d 2470 . . . . . . . . . . . 12  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  +oo + e B )  =  0 )
29 0re 9091 . . . . . . . . . . . . 13  |-  0  e.  RR
30 renepnf 9132 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  0  =/=  +oo )
3129, 30mp1i 12 . . . . . . . . . . . 12  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  0  =/=  +oo )
3228, 31eqnetrd 2619 . . . . . . . . . . 11  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  +oo + e B )  =/= 
+oo )
3332neneqd 2617 . . . . . . . . . 10  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -.  (  +oo + e B )  =  +oo )
34 xaddpnf2 10813 . . . . . . . . . . . 12  |-  ( ( B  e.  RR*  /\  B  =/=  -oo )  ->  (  +oo + e B )  =  +oo )
3534ex 424 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  ( B  =/=  -oo  ->  (  +oo + e B )  = 
+oo ) )
3635con3and 429 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  -.  (  +oo + e B )  =  +oo )  ->  -.  B  =/=  -oo )
3725, 33, 36syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -.  B  =/=  -oo )
38 nne 2605 . . . . . . . . 9  |-  ( -.  B  =/=  -oo  <->  B  =  -oo )
3937, 38sylib 189 . . . . . . . 8  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  =  -oo )
40 xnegeq 10793 . . . . . . . 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 10796 . . . . . . 7  |-  - e  -oo  =  +oo
4341, 42syl6req 2485 . . . . . 6  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  +oo  =  - e B )
4424, 43eqtrd 2468 . . . . 5  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  =  - e B )
4544ex 424 . . . 4  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( ( A + e B )  =  0  ->  A  =  - e B ) )
46 simpll 731 . . . . . 6  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  =  -oo )
47 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  e.  RR* )
4846oveq1d 6096 . . . . . . . . . . . . 13  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  (  -oo + e B ) )
49 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  0 )
5048, 49eqtr3d 2470 . . . . . . . . . . . 12  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  -oo + e B )  =  0 )
51 renemnf 9133 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  0  =/=  -oo )
5229, 51mp1i 12 . . . . . . . . . . . 12  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  0  =/=  -oo )
5350, 52eqnetrd 2619 . . . . . . . . . . 11  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  -oo + e B )  =/= 
-oo )
5453neneqd 2617 . . . . . . . . . 10  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -.  (  -oo + e B )  =  -oo )
55 xaddmnf2 10815 . . . . . . . . . . . 12  |-  ( ( B  e.  RR*  /\  B  =/=  +oo )  ->  (  -oo + e B )  =  -oo )
5655ex 424 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  ( B  =/=  +oo  ->  (  -oo + e B )  = 
-oo ) )
5756con3and 429 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  -.  (  -oo + e B )  =  -oo )  ->  -.  B  =/=  +oo )
5847, 54, 57syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -.  B  =/=  +oo )
59 nne 2605 . . . . . . . . 9  |-  ( -.  B  =/=  +oo  <->  B  =  +oo )
6058, 59sylib 189 . . . . . . . 8  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  =  +oo )
61 xnegeq 10793 . . . . . . . 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 10795 . . . . . . 7  |-  - e  +oo  =  -oo
6462, 63syl6req 2485 . . . . . 6  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -oo  =  - e B )
6546, 64eqtrd 2468 . . . . 5  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  =  - e B )
6665ex 424 . . . 4  |-  ( ( A  =  -oo  /\  B  e.  RR* )  -> 
( ( A + e B )  =  0  ->  A  =  - e B ) )
6723, 45, 663jaoian 1249 . . 3  |-  ( ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  /\  B  e.  RR* )  ->  ( ( A + e B )  =  0  ->  A  =  - e B ) )
681, 67sylanb 459 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A + e B )  =  0  ->  A  =  - e B ) )
69 simpr 448 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  A  =  - e B )
7069oveq1d 6096 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  ( A + e B )  =  (  - e B + e B ) )
71 xnegcl 10799 . . . . . 6  |-  ( B  e.  RR*  ->  - e B  e.  RR* )
7271ad2antlr 708 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  - e B  e.  RR* )
73 simplr 732 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  B  e.  RR* )
74 xaddcom 10824 . . . . 5  |-  ( ( 
- e B  e. 
RR*  /\  B  e.  RR* )  ->  (  - e B + e B )  =  ( B + e  - e B ) )
7572, 73, 74syl2anc 643 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  (  - e B + e B )  =  ( B + e  - e B ) )
76 xnegid 10822 . . . . 5  |-  ( B  e.  RR*  ->  ( B + e  - e B )  =  0 )
7776ad2antlr 708 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  ( B + e  - e B )  =  0 )
7870, 75, 773eqtrd 2472 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  ( A + e B )  =  0 )
7978ex 424 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  - e B  ->  ( A + e B )  =  0 ) )
8068, 79impbid 184 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A + e B )  =  0  <-> 
A  =  - e B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725    =/= wne 2599  (class class class)co 6081   RRcr 8989   0cc0 8990    +oocpnf 9117    -oocmnf 9118   RR*cxr 9119    - ecxne 10707   + ecxad 10708
This theorem is referenced by:  xrsinvgval  24199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-sub 9293  df-neg 9294  df-xneg 10710  df-xadd 10711
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