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Theorem xaddeq0 23306
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 10460 . . 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 8883 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  A  e.  RR* )
4 xnegneg 10543 . . . . . . 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 10622 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  - e A  e.  RR* )
7 xaddid2 10569 . . . . . . . . 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 10567 . . . . . . . . . . 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 5875 . . . . . . . . 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 5875 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( ( A + e B ) + e  - e A )  =  ( 0 + e  - e A ) )
15 xpncan 10573 . . . . . . . . . . 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 2325 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  ( 0 + e  - e A )  =  B )
198, 18eqtr3d 2319 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  - e A  =  B )
20 xnegeq 10536 . . . . . . 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 2319 . . . . 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 5875 . . . . . . . . . . . . 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 2319 . . . . . . . . . . . 12  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  +oo + e B )  =  0 )
29 0re 8840 . . . . . . . . . . . . . 14  |-  0  e.  RR
30 renepnf 8881 . . . . . . . . . . . . . 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 2466 . . . . . . . . . . 11  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  +oo + e B )  =/= 
+oo )
3433neneqd 2464 . . . . . . . . . 10  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -.  (  +oo + e B )  =  +oo )
35 xaddpnf2 10556 . . . . . . . . . . . 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 2452 . . . . . . . . 9  |-  ( -.  B  =/=  -oo  <->  B  =  -oo )
4038, 39sylib 188 . . . . . . . 8  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  =  -oo )
41 xnegeq 10536 . . . . . . . 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 10539 . . . . . . 7  |-  - e  -oo  =  +oo
4442, 43syl6req 2334 . . . . . 6  |-  ( ( ( A  =  +oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  +oo  =  - e B )
4524, 44eqtrd 2317 . . . . 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 5875 . . . . . . . . . . . . 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 2319 . . . . . . . . . . . 12  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  -oo + e B )  =  0 )
52 renemnf 8882 . . . . . . . . . . . . . 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 2466 . . . . . . . . . . 11  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  (  -oo + e B )  =/= 
-oo )
5655neneqd 2464 . . . . . . . . . 10  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -.  (  -oo + e B )  =  -oo )
57 xaddmnf2 10558 . . . . . . . . . . . 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 2452 . . . . . . . . 9  |-  ( -.  B  =/=  +oo  <->  B  =  +oo )
6260, 61sylib 188 . . . . . . . 8  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  B  =  +oo )
63 xnegeq 10536 . . . . . . . 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 10538 . . . . . . 7  |-  - e  +oo  =  -oo
6664, 65syl6req 2334 . . . . . 6  |-  ( ( ( A  =  -oo  /\  B  e.  RR* )  /\  ( A + e B )  =  0 )  ->  -oo  =  - e B )
6747, 66eqtrd 2317 . . . . 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 5875 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  =  - e B )  ->  ( A + e B )  =  (  - e B + e B ) )
73 xnegcl 10542 . . . . . 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 10567 . . . . 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 10565 . . . . 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 2321 . . 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 1625    e. wcel 1686    =/= wne 2448  (class class class)co 5860   RRcr 8738   0cc0 8739    +oocpnf 8866    -oocmnf 8867   RR*cxr 8868    - ecxne 10451   + ecxad 10452
This theorem is referenced by:  xrsinvgval  23308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-sub 9041  df-neg 9042  df-xneg 10454  df-xadd 10455
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