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Mirrors > Home > ILE Home > Th. List > mulassi | GIF version |
Description: Associative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
axi.1 | ⊢ 𝐴 ∈ ℂ |
axi.2 | ⊢ 𝐵 ∈ ℂ |
axi.3 | ⊢ 𝐶 ∈ ℂ |
Ref | Expression |
---|---|
mulassi | ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | axi.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | axi.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
4 | mulass 8003 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1348 | 1 ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2164 (class class class)co 5918 ℂcc 7870 · cmul 7877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-mulass 7975 |
This theorem depends on definitions: df-bi 117 df-3an 982 |
This theorem is referenced by: 8th4div3 9201 numma 9491 decbin0 9587 sq4e2t8 10708 3dec 10785 ef01bndlem 11899 3dvdsdec 12006 3dvds2dec 12007 sincos4thpi 14975 sincos6thpi 14977 2lgsoddprmlem3d 15198 |
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