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Database REAL AND COMPLEX NUMBERS Construction and axiomatization of real and complex numbers Real and complex number postulates restated as axioms ax-addass
Description: Addition of complex numbers is associative. Axiom 9 of 22 for real and
complex numbers, justified by Theorem axaddass . Proofs should normally
use addass instead. (New usage is discouraged.) (Contributed by NM , 22-Nov-1994)
Ref
Expression
Assertion
ax-addass ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( 𝐴 + ( 𝐵 + 𝐶 ) ) )
Detailed syntax breakdown
Step
Hyp
Ref
Expression
0
cA ⊢ 𝐴
1
cc ⊢ ℂ
2
0 1
wcel ⊢ 𝐴 ∈ ℂ
3
cB ⊢ 𝐵
4
3 1
wcel ⊢ 𝐵 ∈ ℂ
5
cC ⊢ 𝐶
6
5 1
wcel ⊢ 𝐶 ∈ ℂ
7
2 4 6
w3a ⊢ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ )
8
caddc ⊢ +
9
0 3 8
co ⊢ ( 𝐴 + 𝐵 )
10
9 5 8
co ⊢ ( ( 𝐴 + 𝐵 ) + 𝐶 )
11
3 5 8
co ⊢ ( 𝐵 + 𝐶 )
12
0 11 8
co ⊢ ( 𝐴 + ( 𝐵 + 𝐶 ) )
13
10 12
wceq ⊢ ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( 𝐴 + ( 𝐵 + 𝐶 ) )
14
7 13
wi ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( 𝐴 + ( 𝐵 + 𝐶 ) ) )