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Theorem xaddeq0 28405
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 11439 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 simpll 768 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  e.  RR )
32rexrd 9708 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  e.  RR* )
4 xnegneg 11530 . . . . . . 7  |-  ( A  e.  RR*  ->  -e  -e A  =  A )
53, 4syl 17 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e  -e A  =  A
)
63xnegcld 11611 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e A  e.  RR* )
7 xaddid2 11557 . . . . . . . . 9  |-  (  -e A  e.  RR*  ->  ( 0 +e  -e A )  = 
-e A )
86, 7syl 17 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( 0 +e  -e
A )  =  -e A )
9 simplr 770 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  e.  RR* )
10 xaddcom 11555 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  ( B +e A ) )
113, 9, 10syl2anc 673 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  ( B +e A ) )
1211oveq1d 6323 . . . . . . . . 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 468 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  0 )
1413oveq1d 6323 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( ( A +e B ) +e  -e
A )  =  ( 0 +e  -e A ) )
15 xpncan 11562 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  A  e.  RR )  ->  (
( B +e
A ) +e  -e A )  =  B )
1615ancoms 460 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( B +e A ) +e  -e A )  =  B )
1716adantr 472 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( ( B +e A ) +e  -e
A )  =  B )
1812, 14, 173eqtr3d 2513 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( 0 +e  -e
A )  =  B )
198, 18eqtr3d 2507 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e A  =  B )
20 xnegeq 11523 . . . . . . 7  |-  (  -e A  =  B  -> 
-e  -e
A  =  -e
B )
2119, 20syl 17 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e  -e A  =  -e
B )
225, 21eqtr3d 2507 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  =  -e B )
2322ex 441 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A +e B )  =  0  ->  A  =  -e B ) )
24 simpll 768 . . . . . 6  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  = +oo )
25 simplr 770 . . . . . . . . . 10  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  e.  RR* )
2624oveq1d 6323 . . . . . . . . . . . . 13  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  ( +oo +e B ) )
27 simpr 468 . . . . . . . . . . . . 13  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  0 )
2826, 27eqtr3d 2507 . . . . . . . . . . . 12  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( +oo +e B )  =  0 )
29 0re 9661 . . . . . . . . . . . . 13  |-  0  e.  RR
30 renepnf 9706 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  0  =/= +oo )
3129, 30mp1i 13 . . . . . . . . . . . 12  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  0  =/= +oo )
3228, 31eqnetrd 2710 . . . . . . . . . . 11  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( +oo +e B )  =/= +oo )
3332neneqd 2648 . . . . . . . . . 10  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  ( +oo +e B )  = +oo )
34 xaddpnf2 11543 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
3534stoic1a 1663 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  -.  ( +oo +e B )  = +oo )  ->  -.  B  =/= -oo )
3625, 33, 35syl2anc 673 . . . . . . . . 9  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  B  =/= -oo )
37 nne 2647 . . . . . . . . 9  |-  ( -.  B  =/= -oo  <->  B  = -oo )
3836, 37sylib 201 . . . . . . . 8  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  = -oo )
39 xnegeq 11523 . . . . . . . 8  |-  ( B  = -oo  ->  -e
B  =  -e -oo )
4038, 39syl 17 . . . . . . 7  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e B  =  -e -oo )
41 xnegmnf 11526 . . . . . . 7  |-  -e -oo  = +oo
4240, 41syl6req 2522 . . . . . 6  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  -> +oo  =  -e B )
4324, 42eqtrd 2505 . . . . 5  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  =  -e B )
4443ex 441 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( ( A +e B )  =  0  ->  A  =  -e B ) )
45 simpll 768 . . . . . 6  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  = -oo )
46 simplr 770 . . . . . . . . . 10  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  e.  RR* )
4745oveq1d 6323 . . . . . . . . . . . . 13  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  ( -oo +e B ) )
48 simpr 468 . . . . . . . . . . . . 13  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  0 )
4947, 48eqtr3d 2507 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( -oo +e B )  =  0 )
50 renemnf 9707 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  0  =/= -oo )
5129, 50mp1i 13 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  0  =/= -oo )
5249, 51eqnetrd 2710 . . . . . . . . . . 11  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( -oo +e B )  =/= -oo )
5352neneqd 2648 . . . . . . . . . 10  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  ( -oo +e B )  = -oo )
54 xaddmnf2 11545 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
5554stoic1a 1663 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  -.  ( -oo +e B )  = -oo )  ->  -.  B  =/= +oo )
5646, 53, 55syl2anc 673 . . . . . . . . 9  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  B  =/= +oo )
57 nne 2647 . . . . . . . . 9  |-  ( -.  B  =/= +oo  <->  B  = +oo )
5856, 57sylib 201 . . . . . . . 8  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  = +oo )
59 xnegeq 11523 . . . . . . . 8  |-  ( B  = +oo  ->  -e
B  =  -e +oo )
6058, 59syl 17 . . . . . . 7  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e B  =  -e +oo )
61 xnegpnf 11525 . . . . . . 7  |-  -e +oo  = -oo
6260, 61syl6req 2522 . . . . . 6  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  -> -oo  =  -e B )
6345, 62eqtrd 2505 . . . . 5  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  =  -e B )
6463ex 441 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( ( A +e B )  =  0  ->  A  =  -e B ) )
6523, 44, 643jaoian 1359 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  ( ( A +e B )  =  0  ->  A  =  -e B ) )
661, 65sylanb 480 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A +e
B )  =  0  ->  A  =  -e B ) )
67 simpr 468 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  A  =  -e B )
6867oveq1d 6323 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  ( A +e B )  =  (  -e
B +e B ) )
69 xnegcl 11529 . . . . . 6  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
7069ad2antlr 741 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  -e
B  e.  RR* )
71 simplr 770 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  B  e.  RR* )
72 xaddcom 11555 . . . . 5  |-  ( ( 
-e B  e. 
RR*  /\  B  e.  RR* )  ->  (  -e
B +e B )  =  ( B +e  -e
B ) )
7370, 71, 72syl2anc 673 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  (  -e B +e
B )  =  ( B +e  -e B ) )
74 xnegid 11553 . . . . 5  |-  ( B  e.  RR*  ->  ( B +e  -e
B )  =  0 )
7574ad2antlr 741 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  ( B +e  -e
B )  =  0 )
7668, 73, 753eqtrd 2509 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  ( A +e B )  =  0 )
7776ex 441 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  -e B  ->  ( A +e B )  =  0 ) )
7866, 77impbid 195 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 189    /\ wa 376    \/ w3o 1006    = wceq 1452    e. wcel 1904    =/= wne 2641  (class class class)co 6308   RRcr 9556   0cc0 9557   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    -ecxne 11429   +ecxad 11430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-sub 9882  df-neg 9883  df-xneg 11432  df-xadd 11433
This theorem is referenced by:  xrsinvgval  28514
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