Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xaddeq0 Structured version   Unicode version

Theorem xaddeq0 28280
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 11367 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 simpll 758 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  e.  RR )
32rexrd 9641 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  e.  RR* )
4 xnegneg 11458 . . . . . . 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 11537 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e A  e.  RR* )
7 xaddid2 11484 . . . . . . . . 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 760 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  e.  RR* )
10 xaddcom 11482 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  ( B +e A ) )
113, 9, 10syl2anc 665 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  ( B +e A ) )
1211oveq1d 6264 . . . . . . . . 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 462 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  0 )
1413oveq1d 6264 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( ( A +e B ) +e  -e
A )  =  ( 0 +e  -e A ) )
15 xpncan 11488 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  A  e.  RR )  ->  (
( B +e
A ) +e  -e A )  =  B )
1615ancoms 454 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( B +e A ) +e  -e A )  =  B )
1716adantr 466 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( ( B +e A ) +e  -e
A )  =  B )
1812, 14, 173eqtr3d 2470 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( 0 +e  -e
A )  =  B )
198, 18eqtr3d 2464 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -e A  =  B )
20 xnegeq 11451 . . . . . . 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 2464 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  =  -e B )
2322ex 435 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A +e B )  =  0  ->  A  =  -e B ) )
24 simpll 758 . . . . . 6  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  = +oo )
25 simplr 760 . . . . . . . . . 10  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  e.  RR* )
2624oveq1d 6264 . . . . . . . . . . . . 13  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  ( +oo +e B ) )
27 simpr 462 . . . . . . . . . . . . 13  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  0 )
2826, 27eqtr3d 2464 . . . . . . . . . . . 12  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( +oo +e B )  =  0 )
29 0re 9594 . . . . . . . . . . . . 13  |-  0  e.  RR
30 renepnf 9639 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  0  =/= +oo )
3129, 30mp1i 13 . . . . . . . . . . . 12  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  0  =/= +oo )
3228, 31eqnetrd 2668 . . . . . . . . . . 11  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( +oo +e B )  =/= +oo )
3332neneqd 2606 . . . . . . . . . 10  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  ( +oo +e B )  = +oo )
34 xaddpnf2 11471 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
3534stoic1a 1649 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  -.  ( +oo +e B )  = +oo )  ->  -.  B  =/= -oo )
3625, 33, 35syl2anc 665 . . . . . . . . 9  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  B  =/= -oo )
37 nne 2605 . . . . . . . . 9  |-  ( -.  B  =/= -oo  <->  B  = -oo )
3836, 37sylib 199 . . . . . . . 8  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  = -oo )
39 xnegeq 11451 . . . . . . . 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 11454 . . . . . . 7  |-  -e -oo  = +oo
4240, 41syl6req 2479 . . . . . 6  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  -> +oo  =  -e B )
4324, 42eqtrd 2462 . . . . 5  |-  ( ( ( A  = +oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  =  -e B )
4443ex 435 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( ( A +e B )  =  0  ->  A  =  -e B ) )
45 simpll 758 . . . . . 6  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  = -oo )
46 simplr 760 . . . . . . . . . 10  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  e.  RR* )
4745oveq1d 6264 . . . . . . . . . . . . 13  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  ( -oo +e B ) )
48 simpr 462 . . . . . . . . . . . . 13  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( A +e B )  =  0 )
4947, 48eqtr3d 2464 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( -oo +e B )  =  0 )
50 renemnf 9640 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  0  =/= -oo )
5129, 50mp1i 13 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  0  =/= -oo )
5249, 51eqnetrd 2668 . . . . . . . . . . 11  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  ( -oo +e B )  =/= -oo )
5352neneqd 2606 . . . . . . . . . 10  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  ( -oo +e B )  = -oo )
54 xaddmnf2 11473 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
5554stoic1a 1649 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  -.  ( -oo +e B )  = -oo )  ->  -.  B  =/= +oo )
5646, 53, 55syl2anc 665 . . . . . . . . 9  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  -.  B  =/= +oo )
57 nne 2605 . . . . . . . . 9  |-  ( -.  B  =/= +oo  <->  B  = +oo )
5856, 57sylib 199 . . . . . . . 8  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  B  = +oo )
59 xnegeq 11451 . . . . . . . 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 11453 . . . . . . 7  |-  -e +oo  = -oo
6260, 61syl6req 2479 . . . . . 6  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  -> -oo  =  -e B )
6345, 62eqtrd 2462 . . . . 5  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( A +e
B )  =  0 )  ->  A  =  -e B )
6463ex 435 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( ( A +e B )  =  0  ->  A  =  -e B ) )
6523, 44, 643jaoian 1329 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  ( ( A +e B )  =  0  ->  A  =  -e B ) )
661, 65sylanb 474 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A +e
B )  =  0  ->  A  =  -e B ) )
67 simpr 462 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  A  =  -e B )
6867oveq1d 6264 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  ( A +e B )  =  (  -e
B +e B ) )
69 xnegcl 11457 . . . . . 6  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
7069ad2antlr 731 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  -e
B  e.  RR* )
71 simplr 760 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  B  e.  RR* )
72 xaddcom 11482 . . . . 5  |-  ( ( 
-e B  e. 
RR*  /\  B  e.  RR* )  ->  (  -e
B +e B )  =  ( B +e  -e
B ) )
7370, 71, 72syl2anc 665 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  (  -e B +e
B )  =  ( B +e  -e B ) )
74 xnegid 11480 . . . . 5  |-  ( B  e.  RR*  ->  ( B +e  -e
B )  =  0 )
7574ad2antlr 731 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  ( B +e  -e
B )  =  0 )
7668, 73, 753eqtrd 2466 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  -e
B )  ->  ( A +e B )  =  0 )
7776ex 435 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  -e B  ->  ( A +e B )  =  0 ) )
7866, 77impbid 193 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 187    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1872    =/= wne 2599  (class class class)co 6249   RRcr 9489   0cc0 9490   +oocpnf 9623   -oocmnf 9624   RR*cxr 9625    -ecxne 11357   +ecxad 11358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-1st 6751  df-2nd 6752  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-sub 9813  df-neg 9814  df-xneg 11360  df-xadd 11361
This theorem is referenced by:  xrsinvgval  28390
  Copyright terms: Public domain W3C validator