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Axiom ax-distr 6767
 Description: Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by theorem axdistr 6738. Proofs should normally use adddi 6791 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-distr ((A B 𝐶 ℂ) → (A · (B + 𝐶)) = ((A · B) + (A · 𝐶)))

Detailed syntax breakdown of Axiom ax-distr
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 +
104, 6, 9co 5455 . . . 4 class (B + 𝐶)
11 cmul 6696 . . . 4 class ·
121, 10, 11co 5455 . . 3 class (A · (B + 𝐶))
131, 4, 11co 5455 . . . 4 class (A · B)
141, 6, 11co 5455 . . . 4 class (A · 𝐶)
1513, 14, 9co 5455 . . 3 class ((A · B) + (A · 𝐶))
1612, 15wceq 1242 . 2 wff (A · (B + 𝐶)) = ((A · B) + (A · 𝐶))
178, 16wi 4 1 wff ((A B 𝐶 ℂ) → (A · (B + 𝐶)) = ((A · B) + (A · 𝐶)))
 Colors of variables: wff set class This axiom is referenced by:  adddi  6791
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