Metamath Proof Explorer


Theorem xaddeq0

Description: Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017)

Ref Expression
Assertion xaddeq0 A * B * A + 𝑒 B = 0 A = B

Proof

Step Hyp Ref Expression
1 elxr A * A A = +∞ A = −∞
2 simpll A B * A + 𝑒 B = 0 A
3 2 rexrd A B * A + 𝑒 B = 0 A *
4 xnegneg A * A = A
5 3 4 syl A B * A + 𝑒 B = 0 A = A
6 3 xnegcld A B * A + 𝑒 B = 0 A *
7 xaddid2 A * 0 + 𝑒 A = A
8 6 7 syl A B * A + 𝑒 B = 0 0 + 𝑒 A = A
9 simplr A B * A + 𝑒 B = 0 B *
10 xaddcom A * B * A + 𝑒 B = B + 𝑒 A
11 3 9 10 syl2anc A B * A + 𝑒 B = 0 A + 𝑒 B = B + 𝑒 A
12 11 oveq1d A B * A + 𝑒 B = 0 A + 𝑒 B + 𝑒 A = B + 𝑒 A + 𝑒 A
13 simpr A B * A + 𝑒 B = 0 A + 𝑒 B = 0
14 13 oveq1d A B * A + 𝑒 B = 0 A + 𝑒 B + 𝑒 A = 0 + 𝑒 A
15 xpncan B * A B + 𝑒 A + 𝑒 A = B
16 15 ancoms A B * B + 𝑒 A + 𝑒 A = B
17 16 adantr A B * A + 𝑒 B = 0 B + 𝑒 A + 𝑒 A = B
18 12 14 17 3eqtr3d A B * A + 𝑒 B = 0 0 + 𝑒 A = B
19 8 18 eqtr3d A B * A + 𝑒 B = 0 A = B
20 xnegeq A = B A = B
21 19 20 syl A B * A + 𝑒 B = 0 A = B
22 5 21 eqtr3d A B * A + 𝑒 B = 0 A = B
23 22 ex A B * A + 𝑒 B = 0 A = B
24 simpll A = +∞ B * A + 𝑒 B = 0 A = +∞
25 simplr A = +∞ B * A + 𝑒 B = 0 B *
26 24 oveq1d A = +∞ B * A + 𝑒 B = 0 A + 𝑒 B = +∞ + 𝑒 B
27 simpr A = +∞ B * A + 𝑒 B = 0 A + 𝑒 B = 0
28 26 27 eqtr3d A = +∞ B * A + 𝑒 B = 0 +∞ + 𝑒 B = 0
29 0re 0
30 renepnf 0 0 +∞
31 29 30 mp1i A = +∞ B * A + 𝑒 B = 0 0 +∞
32 28 31 eqnetrd A = +∞ B * A + 𝑒 B = 0 +∞ + 𝑒 B +∞
33 32 neneqd A = +∞ B * A + 𝑒 B = 0 ¬ +∞ + 𝑒 B = +∞
34 xaddpnf2 B * B −∞ +∞ + 𝑒 B = +∞
35 34 stoic1a B * ¬ +∞ + 𝑒 B = +∞ ¬ B −∞
36 25 33 35 syl2anc A = +∞ B * A + 𝑒 B = 0 ¬ B −∞
37 nne ¬ B −∞ B = −∞
38 36 37 sylib A = +∞ B * A + 𝑒 B = 0 B = −∞
39 xnegeq B = −∞ B = −∞
40 38 39 syl A = +∞ B * A + 𝑒 B = 0 B = −∞
41 xnegmnf −∞ = +∞
42 40 41 eqtr2di A = +∞ B * A + 𝑒 B = 0 +∞ = B
43 24 42 eqtrd A = +∞ B * A + 𝑒 B = 0 A = B
44 43 ex A = +∞ B * A + 𝑒 B = 0 A = B
45 simpll A = −∞ B * A + 𝑒 B = 0 A = −∞
46 simplr A = −∞ B * A + 𝑒 B = 0 B *
47 45 oveq1d A = −∞ B * A + 𝑒 B = 0 A + 𝑒 B = −∞ + 𝑒 B
48 simpr A = −∞ B * A + 𝑒 B = 0 A + 𝑒 B = 0
49 47 48 eqtr3d A = −∞ B * A + 𝑒 B = 0 −∞ + 𝑒 B = 0
50 renemnf 0 0 −∞
51 29 50 mp1i A = −∞ B * A + 𝑒 B = 0 0 −∞
52 49 51 eqnetrd A = −∞ B * A + 𝑒 B = 0 −∞ + 𝑒 B −∞
53 52 neneqd A = −∞ B * A + 𝑒 B = 0 ¬ −∞ + 𝑒 B = −∞
54 xaddmnf2 B * B +∞ −∞ + 𝑒 B = −∞
55 54 stoic1a B * ¬ −∞ + 𝑒 B = −∞ ¬ B +∞
56 46 53 55 syl2anc A = −∞ B * A + 𝑒 B = 0 ¬ B +∞
57 nne ¬ B +∞ B = +∞
58 56 57 sylib A = −∞ B * A + 𝑒 B = 0 B = +∞
59 xnegeq B = +∞ B = +∞
60 58 59 syl A = −∞ B * A + 𝑒 B = 0 B = +∞
61 xnegpnf +∞ = −∞
62 60 61 eqtr2di A = −∞ B * A + 𝑒 B = 0 −∞ = B
63 45 62 eqtrd A = −∞ B * A + 𝑒 B = 0 A = B
64 63 ex A = −∞ B * A + 𝑒 B = 0 A = B
65 23 44 64 3jaoian A A = +∞ A = −∞ B * A + 𝑒 B = 0 A = B
66 1 65 sylanb A * B * A + 𝑒 B = 0 A = B
67 simpr A * B * A = B A = B
68 67 oveq1d A * B * A = B A + 𝑒 B = B + 𝑒 B
69 xnegcl B * B *
70 69 ad2antlr A * B * A = B B *
71 simplr A * B * A = B B *
72 xaddcom B * B * B + 𝑒 B = B + 𝑒 B
73 70 71 72 syl2anc A * B * A = B B + 𝑒 B = B + 𝑒 B
74 xnegid B * B + 𝑒 B = 0
75 74 ad2antlr A * B * A = B B + 𝑒 B = 0
76 68 73 75 3eqtrd A * B * A = B A + 𝑒 B = 0
77 76 ex A * B * A = B A + 𝑒 B = 0
78 66 77 impbid A * B * A + 𝑒 B = 0 A = B