Metamath Proof Explorer


Theorem xnegneg

Description: Extended real version of negneg . (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xnegneg A * A = A

Proof

Step Hyp Ref Expression
1 elxr A * A A = +∞ A = −∞
2 rexneg A A = A
3 xnegeq A = A A = A
4 2 3 syl A A = A
5 renegcl A A
6 rexneg A A = A
7 5 6 syl A A = A
8 recn A A
9 8 negnegd A A = A
10 4 7 9 3eqtrd A A = A
11 xnegmnf −∞ = +∞
12 xnegeq A = +∞ A = +∞
13 xnegpnf +∞ = −∞
14 12 13 eqtrdi A = +∞ A = −∞
15 xnegeq A = −∞ A = −∞
16 14 15 syl A = +∞ A = −∞
17 id A = +∞ A = +∞
18 11 16 17 3eqtr4a A = +∞ A = A
19 xnegeq A = −∞ A = −∞
20 19 11 eqtrdi A = −∞ A = +∞
21 xnegeq A = +∞ A = +∞
22 20 21 syl A = −∞ A = +∞
23 id A = −∞ A = −∞
24 13 22 23 3eqtr4a A = −∞ A = A
25 10 18 24 3jaoi A A = +∞ A = −∞ A = A
26 1 25 sylbi A * A = A