Metamath Proof Explorer


Theorem 6p5lem

Description: Lemma for 6p5e11 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015)

Ref Expression
Hypotheses 6p5lem.1 ⊢ A ∈ ℕ 0
6p5lem.2 ⊢ D ∈ ℕ 0
6p5lem.3 ⊢ E ∈ ℕ 0
6p5lem.4 ⊢ B = D + 1
6p5lem.5 ⊢ C = E + 1
6p5lem.6 No typesetting found for |- ( A + D ) = ; 1 E with typecode |-
Assertion 6p5lem Could not format assertion : No typesetting found for |- ( A + B ) = ; 1 C with typecode |-

Proof

Step Hyp Ref Expression
1 6p5lem.1 ⊢ A ∈ ℕ 0
2 6p5lem.2 ⊢ D ∈ ℕ 0
3 6p5lem.3 ⊢ E ∈ ℕ 0
4 6p5lem.4 ⊢ B = D + 1
5 6p5lem.5 ⊢ C = E + 1
6 6p5lem.6 Could not format ( A + D ) = ; 1 E : No typesetting found for |- ( A + D ) = ; 1 E with typecode |-
7 4 oveq2i ⊢ A + B = A + D + 1
8 1 nn0cni ⊢ A ∈ ℂ
9 2 nn0cni ⊢ D ∈ ℂ
10 ax-1cn ⊢ 1 ∈ ℂ
11 8 9 10 addassi ⊢ A + D + 1 = A + D + 1
12 1nn0 ⊢ 1 ∈ ℕ 0
13 5 eqcomi ⊢ E + 1 = C
14 12 3 13 6 decsuc Could not format ( ( A + D ) + 1 ) = ; 1 C : No typesetting found for |- ( ( A + D ) + 1 ) = ; 1 C with typecode |-
15 7 11 14 3eqtr2i Could not format ( A + B ) = ; 1 C : No typesetting found for |- ( A + B ) = ; 1 C with typecode |-