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Mirrors > Home > ILE Home > Th. List > mulcomi | GIF version |
Description: Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
axi.1 | ⊢ 𝐴 ∈ ℂ |
axi.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
mulcomi | ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | axi.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | mulcom 7773 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | |
4 | 1, 2, 3 | mp2an 423 | 1 ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 (class class class)co 5782 ℂcc 7642 · cmul 7649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia3 107 ax-mulcom 7745 |
This theorem is referenced by: mulcomli 7797 8th4div3 8963 numma2c 9251 nummul2c 9255 9t11e99 9335 binom2i 10432 fac3 10510 tanval2ap 11456 sincosq4sgn 12958 2logb9irrALT 13099 |
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