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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 7918 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 (class class class)co 5868 ℂcc 7787 · cmul 7794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia3 108 ax-mulcom 7890 |
This theorem is referenced by: mulcomli 7942 8th4div3 9114 numma2c 9405 nummul2c 9409 9t11e99 9489 binom2i 10601 fac3 10683 tanval2ap 11692 pockthi 12326 sincosq4sgn 13883 2logb9irrALT 14025 |
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