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Theorem xaddeq0 27396
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 11337 . . 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 9655 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  e.  RR* )
4 xnegneg 11425 . . . . . . 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 11504 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e A  e.  RR* )
7 xaddid2 11451 . . . . . . . . 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 11449 . . . . . . . . . . 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 6310 . . . . . . . . 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 6310 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( ( A +e B ) +e  -e
A )  =  ( 0 +e  -e A ) )
15 xpncan 11455 . . . . . . . . . . 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 2516 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( 0 +e  -e
A )  =  B )
198, 18eqtr3d 2510 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e A  =  B )
20 xnegeq 11418 . . . . . . 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 2510 . . . . 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 6310 . . . . . . . . . . . . 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 2510 . . . . . . . . . . . 12  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( +oo +e B )  =  0 )
29 0re 9608 . . . . . . . . . . . . 13  |-  0  e.  RR
30 renepnf 9653 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  0  =/= +oo )
3129, 30mp1i 12 . . . . . . . . . . . 12  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  0  =/= +oo )
3228, 31eqnetrd 2760 . . . . . . . . . . 11  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( +oo +e B )  =/= +oo )
3332neneqd 2669 . . . . . . . . . 10  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  ( +oo +e B )  = +oo )
34 xaddpnf2 11438 . . . . . . . . . . . 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 2668 . . . . . . . . 9  |-  ( -.  B  =/= -oo  <->  B  = -oo )
3937, 38sylib 196 . . . . . . . 8  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  = -oo )
40 xnegeq 11418 . . . . . . . 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 11421 . . . . . . 7  |-  -e -oo  = +oo
4341, 42syl6req 2525 . . . . . 6  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  -> +oo  =  -e B )
4424, 43eqtrd 2508 . . . . 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 6310 . . . . . . . . . . . . 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 2510 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( -oo +e B )  =  0 )
51 renemnf 9654 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  0  =/= -oo )
5229, 51mp1i 12 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  0  =/= -oo )
5350, 52eqnetrd 2760 . . . . . . . . . . 11  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( -oo +e B )  =/= -oo )
5453neneqd 2669 . . . . . . . . . 10  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  ( -oo +e B )  = -oo )
55 xaddmnf2 11440 . . . . . . . . . . . 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 2668 . . . . . . . . 9  |-  ( -.  B  =/= +oo  <->  B  = +oo )
6058, 59sylib 196 . . . . . . . 8  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  = +oo )
61 xnegeq 11418 . . . . . . . 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 11420 . . . . . . 7  |-  -e +oo  = -oo
6462, 63syl6req 2525 . . . . . 6  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  -> -oo  =  -e B )
6546, 64eqtrd 2508 . . . . 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 1293 . . 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 6310 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  ( A +e B )  =  (  -e
B +e B ) )
71 xnegcl 11424 . . . . . 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 11449 . . . . 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 11447 . . . . 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 2512 . . 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 972    = wceq 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6295   RRcr 9503   0cc0 9504   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639    -ecxne 11327   +ecxad 11328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-sub 9819  df-neg 9820  df-xneg 11330  df-xadd 11331
This theorem is referenced by:  xrsinvgval  27489
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