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

 Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 6736. Proofs should normally use addass 6789 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-addass ((A B 𝐶 ℂ) → ((A + B) + 𝐶) = (A + (B + 𝐶)))

Detailed syntax breakdown of Axiom ax-addass
StepHypRef Expression
1 cA . . . 4 class A
2 cc 6689 . . . 4 class
31, 2wcel 1390 . . 3 wff A
4 cB . . . 4 class B
54, 2wcel 1390 . . 3 wff B
6 cC . . . 4 class 𝐶
76, 2wcel 1390 . . 3 wff 𝐶
83, 5, 7w3a 884 . 2 wff (A B 𝐶 ℂ)
9 caddc 6694 . . . . 5 class +
101, 4, 9co 5455 . . . 4 class (A + B)
1110, 6, 9co 5455 . . 3 class ((A + B) + 𝐶)
124, 6, 9co 5455 . . . 4 class (B + 𝐶)
131, 12, 9co 5455 . . 3 class (A + (B + 𝐶))
1411, 13wceq 1242 . 2 wff ((A + B) + 𝐶) = (A + (B + 𝐶))
158, 14wi 4 1 wff ((A B 𝐶 ℂ) → ((A + B) + 𝐶) = (A + (B + 𝐶)))
 Colors of variables: wff set class This axiom is referenced by:  addass  6789
 Copyright terms: Public domain W3C validator