Metamath Proof Explorer


Theorem dnibndlem8

Description: Lemma for dnibnd . (Contributed by Asger C. Ipsen, 4-Apr-2021)

Ref Expression
Hypothesis dnibndlem8.1 φ A
Assertion dnibndlem8 φ 1 2 A + 1 2 A A + 1 2 + 1 2 - A

Proof

Step Hyp Ref Expression
1 dnibndlem8.1 φ A
2 halfre 1 2
3 2 a1i φ 1 2
4 1 3 jca φ A 1 2
5 simpl A 1 2 A
6 2 a1i A 1 2 1 2
7 5 6 readdcld A 1 2 A + 1 2
8 4 7 syl φ A + 1 2
9 reflcl A + 1 2 A + 1 2
10 8 9 syl φ A + 1 2
11 1 10 resubcld φ A A + 1 2
12 1 dnicld1 φ A + 1 2 A
13 11 leabsd φ A A + 1 2 A A + 1 2
14 1 recnd φ A
15 10 recnd φ A + 1 2
16 14 15 abssubd φ A A + 1 2 = A + 1 2 A
17 13 16 breqtrd φ A A + 1 2 A + 1 2 A
18 11 12 3 17 lesub2dd φ 1 2 A + 1 2 A 1 2 A A + 1 2
19 3 recnd φ 1 2
20 19 14 15 subsub3d φ 1 2 A A + 1 2 = 1 2 + A + 1 2 - A
21 19 15 addcomd φ 1 2 + A + 1 2 = A + 1 2 + 1 2
22 21 oveq1d φ 1 2 + A + 1 2 - A = A + 1 2 + 1 2 - A
23 20 22 eqtrd φ 1 2 A A + 1 2 = A + 1 2 + 1 2 - A
24 18 23 breqtrd φ 1 2 A + 1 2 A A + 1 2 + 1 2 - A