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Mirrors > Home > ILE Home > Th. List > mulcomli | GIF version |
Description: Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
axi.1 | ⊢ 𝐴 ∈ ℂ |
axi.2 | ⊢ 𝐵 ∈ ℂ |
mulcomli.3 | ⊢ (𝐴 · 𝐵) = 𝐶 |
Ref | Expression |
---|---|
mulcomli | ⊢ (𝐵 · 𝐴) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axi.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
2 | axi.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
3 | 1, 2 | mulcomi 7772 | . 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵) |
4 | mulcomli.3 | . 2 ⊢ (𝐴 · 𝐵) = 𝐶 | |
5 | 3, 4 | eqtri 2160 | 1 ⊢ (𝐵 · 𝐴) = 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7618 · cmul 7625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-4 1487 ax-17 1506 ax-ext 2121 ax-mulcom 7721 |
This theorem depends on definitions: df-bi 116 df-cleq 2132 |
This theorem is referenced by: nummul2c 9231 halfthird 9324 5recm6rec 9325 sq4e2t8 10390 cos2bnd 11467 ex-exp 12939 ex-fac 12940 |
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