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| Description: Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.) | 
| Ref | Expression | 
|---|---|
| mul.1 | ⊢ 𝐴 ∈ ℂ | 
| mul.2 | ⊢ 𝐵 ∈ ℂ | 
| mul.3 | ⊢ 𝐶 ∈ ℂ | 
| Ref | Expression | 
|---|---|
| mul32i | ⊢ ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mul.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
| 2 | mul.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
| 3 | mul.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
| 4 | mul32 11428 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) | |
| 5 | 1, 2, 3, 4 | mp3an 1462 | 1 ⊢ ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 (class class class)co 7432 ℂcc 11154 · cmul 11161 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-mulcom 11220 ax-mulass 11222 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 | 
| This theorem is referenced by: 8th4div3 12488 faclbnd4lem1 14333 bpoly4 16096 dec5nprm 17105 dec2nprm 17106 karatsuba 17122 quart1lem 26899 log2ublem2 26991 log2ub 26993 normlem3 31132 bcseqi 31140 dpmul100 32880 dpmul1000 32882 | 
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