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Theorem xaddeq0 23319
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 10474 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
2 simpll 730 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  e.  RR )
32rexrd 8897 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  e.  RR* )
4 xnegneg 10557 . . . . . . 7  |-  ( A  e.  RR*  ->  - e  - e A  =  A )
53, 4syl 15 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  - e  - e A  =  A
)
63xnegcld 10636 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  - e A  e.  RR* )
7 xaddid2 10583 . . . . . . . . 9  |-  (  - e A  e.  RR*  ->  ( 0 + e  - e A )  =  - e A )
86, 7syl 15 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( 0 + e  - e A )  =  - e A )
9 simplr 731 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  e.  RR* )
10 xaddcom 10581 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e B )  =  ( B + e A ) )
113, 9, 10syl2anc 642 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  ( B + e A ) )
1211oveq1d 5889 . . . . . . . . 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 447 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  0 )
1413oveq1d 5889 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( ( A + e B ) + e  - e A )  =  ( 0 + e  - e A ) )
15 xpncan 10587 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  A  e.  RR )  ->  (
( B + e A ) + e  - e A )  =  B )
1615ancoms 439 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( B + e A ) + e  - e A )  =  B )
1716adantr 451 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( ( B + e A ) + e  - e A )  =  B )
1812, 14, 173eqtr3d 2336 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( 0 + e  - e A )  =  B )
198, 18eqtr3d 2330 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  - e A  =  B )
20 xnegeq 10550 . . . . . . 7  |-  (  - e A  =  B  -> 
- e  - e A  =  - e B )
2119, 20syl 15 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  - e  - e A  =  - e B )
225, 21eqtr3d 2330 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  =  - e B )
2322ex 423 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A + e B )  =  0  ->  A  =  - e B ) )
24 simpll 730 . . . . . 6  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  =  +oo )
25 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  e.  RR* )
2624oveq1d 5889 . . . . . . . . . . . . 13  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  (  +oo + e B ) )
27 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  0 )
2826, 27eqtr3d 2330 . . . . . . . . . . . 12  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  +oo + e B )  =  0 )
29 0re 8854 . . . . . . . . . . . . . 14  |-  0  e.  RR
30 renepnf 8895 . . . . . . . . . . . . . 14  |-  ( 0  e.  RR  ->  0  =/=  +oo )
3129, 30ax-mp 8 . . . . . . . . . . . . 13  |-  0  =/=  +oo
3231a1i 10 . . . . . . . . . . . 12  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  0  =/=  +oo )
3328, 32eqnetrd 2477 . . . . . . . . . . 11  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  +oo + e B )  =/= 
+oo )
3433neneqd 2475 . . . . . . . . . 10  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -.  (  +oo + e B )  =  +oo )
35 xaddpnf2 10570 . . . . . . . . . . . 12  |-  ( ( B  e.  RR*  /\  B  =/=  -oo )  ->  (  +oo + e B )  =  +oo )
3635ex 423 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  ( B  =/=  -oo  ->  (  +oo + e B )  = 
+oo ) )
3736con3and 428 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  -.  (  +oo + e B )  =  +oo )  ->  -.  B  =/=  -oo )
3825, 34, 37syl2anc 642 . . . . . . . . 9  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -.  B  =/=  -oo )
39 nne 2463 . . . . . . . . 9  |-  ( -.  B  =/=  -oo  <->  B  =  -oo )
4038, 39sylib 188 . . . . . . . 8  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  =  -oo )
41 xnegeq 10550 . . . . . . . 8  |-  ( B  =  -oo  ->  - e B  =  - e  -oo )
4240, 41syl 15 . . . . . . 7  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  - e B  =  - e  -oo )
43 xnegmnf 10553 . . . . . . 7  |-  - e  -oo  =  +oo
4442, 43syl6req 2345 . . . . . 6  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  +oo  =  - e B )
4524, 44eqtrd 2328 . . . . 5  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  =  - e B )
4645ex 423 . . . 4  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( ( A + e B )  =  0  ->  A  =  - e B ) )
47 simpll 730 . . . . . 6  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  =  -oo )
48 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  e.  RR* )
4947oveq1d 5889 . . . . . . . . . . . . 13  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  (  -oo + e B ) )
50 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( A + e B )  =  0 )
5149, 50eqtr3d 2330 . . . . . . . . . . . 12  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  -oo + e B )  =  0 )
52 renemnf 8896 . . . . . . . . . . . . . 14  |-  ( 0  e.  RR  ->  0  =/=  -oo )
5329, 52ax-mp 8 . . . . . . . . . . . . 13  |-  0  =/=  -oo
5453a1i 10 . . . . . . . . . . . 12  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  0  =/=  -oo )
5551, 54eqnetrd 2477 . . . . . . . . . . 11  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  -oo + e B )  =/= 
-oo )
5655neneqd 2475 . . . . . . . . . 10  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -.  (  -oo + e B )  =  -oo )
57 xaddmnf2 10572 . . . . . . . . . . . 12  |-  ( ( B  e.  RR*  /\  B  =/=  +oo )  ->  (  -oo + e B )  =  -oo )
5857ex 423 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  ( B  =/=  +oo  ->  (  -oo + e B )  = 
-oo ) )
5958con3and 428 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  -.  (  -oo + e B )  =  -oo )  ->  -.  B  =/=  +oo )
6048, 56, 59syl2anc 642 . . . . . . . . 9  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -.  B  =/=  +oo )
61 nne 2463 . . . . . . . . 9  |-  ( -.  B  =/=  +oo  <->  B  =  +oo )
6260, 61sylib 188 . . . . . . . 8  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  =  +oo )
63 xnegeq 10550 . . . . . . . 8  |-  ( B  =  +oo  ->  - e B  =  - e  +oo )
6462, 63syl 15 . . . . . . 7  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  - e B  =  - e  +oo )
65 xnegpnf 10552 . . . . . . 7  |-  - e  +oo  =  -oo
6664, 65syl6req 2345 . . . . . 6  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -oo  =  - e B )
6747, 66eqtrd 2328 . . . . 5  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  =  - e B )
6867ex 423 . . . 4  |-  ( ( A  =  -oo  /\  B  e.  RR* )  -> 
( ( A + e B )  =  0  ->  A  =  - e B ) )
6923, 46, 683jaoian 1247 . . 3  |-  ( ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  /\  B  e.  RR* )  ->  ( ( A + e B )  =  0  ->  A  =  - e B ) )
701, 69sylanb 458 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A + e B )  =  0  ->  A  =  - e B ) )
71 simpr 447 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  A  =  - e B )
7271oveq1d 5889 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  ( A + e B )  =  (  - e B + e B ) )
73 xnegcl 10556 . . . . . 6  |-  ( B  e.  RR*  ->  - e B  e.  RR* )
7473ad2antlr 707 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  - e B  e.  RR* )
75 simplr 731 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  B  e.  RR* )
76 xaddcom 10581 . . . . 5  |-  ( ( 
- e B  e. 
RR*  /\  B  e.  RR* )  ->  (  - e B + e B )  =  ( B + e  - e B ) )
7774, 75, 76syl2anc 642 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  (  - e B + e B )  =  ( B + e  - e B ) )
78 xnegid 10579 . . . . 5  |-  ( B  e.  RR*  ->  ( B + e  - e B )  =  0 )
7975, 78syl 15 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  ( B + e  - e B )  =  0 )
8072, 77, 793eqtrd 2332 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  ( A + e B )  =  0 )
8180ex 423 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  - e B  ->  ( A + e B )  =  0 ) )
8270, 81impbid 183 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 176    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696    =/= wne 2459  (class class class)co 5874   RRcr 8752   0cc0 8753    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    - ecxne 10465   + ecxad 10466
This theorem is referenced by:  xrsinvgval  23321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-sub 9055  df-neg 9056  df-xneg 10468  df-xadd 10469
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