Metamath Proof Explorer


Theorem problem3

Description: Practice problem 3. Clues: eqcomi eqtri subaddrii recni 4re 3re 1re df-4 addcomi . (Contributed by Filip Cernatescu, 16-Mar-2019) (Proof modification is discouraged.)

Ref Expression
Hypotheses problem3.1 ⊢ A ∈ ℂ
problem3.2 ⊢ A + 3 = 4
Assertion problem3 ⊢ A = 1

Proof

Step Hyp Ref Expression
1 problem3.1 ⊢ A ∈ ℂ
2 problem3.2 ⊢ A + 3 = 4
3 4re ⊢ 4 ∈ ℝ
4 3 recni ⊢ 4 ∈ ℂ
5 3re ⊢ 3 ∈ ℝ
6 5 recni ⊢ 3 ∈ ℂ
7 1re ⊢ 1 ∈ ℝ
8 7 recni ⊢ 1 ∈ ℂ
9 df-4 ⊢ 4 = 3 + 1
10 9 eqcomi ⊢ 3 + 1 = 4
11 4 6 8 10 subaddrii ⊢ 4 − 3 = 1
12 11 eqcomi ⊢ 1 = 4 − 3
13 6 1 addcomi ⊢ 3 + A = A + 3
14 13 2 eqtri ⊢ 3 + A = 4
15 4 6 1 14 subaddrii ⊢ 4 − 3 = A
16 12 15 eqtri ⊢ 1 = A
17 16 eqcomi ⊢ A = 1