ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-addass GIF version

Axiom ax-addass 7044
Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 7004. Proofs should normally use addass 7069 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-addass ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Detailed syntax breakdown of Axiom ax-addass
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 6945 . . . 4 class
31, 2wcel 1409 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1409 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1409 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 896 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 caddc 6950 . . . . 5 class +
101, 4, 9co 5540 . . . 4 class (𝐴 + 𝐵)
1110, 6, 9co 5540 . . 3 class ((𝐴 + 𝐵) + 𝐶)
124, 6, 9co 5540 . . . 4 class (𝐵 + 𝐶)
131, 12, 9co 5540 . . 3 class (𝐴 + (𝐵 + 𝐶))
1411, 13wceq 1259 . 2 wff ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
158, 14wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Colors of variables: wff set class
This axiom is referenced by:  addass  7069
  Copyright terms: Public domain W3C validator