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Mirrors > Home > MPE Home > Th. List > mul32i | Structured version Visualization version GIF version |
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 10794 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) | |
5 | 1, 2, 3, 4 | mp3an 1452 | 1 ⊢ ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 · cmul 10530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-mulcom 10589 ax-mulass 10591 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: 8th4div3 11845 faclbnd4lem1 13641 bpoly4 15401 dec5nprm 16390 dec2nprm 16391 karatsuba 16408 quart1lem 25360 log2ublem2 25452 log2ub 25454 normlem3 28816 bcseqi 28824 dpmul100 30500 dpmul1000 30502 |
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