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Mirrors > Home > ILE Home > Th. List > mul12d | GIF version |
Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
muld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addcomd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
mul12d.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
mul12d | ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | addcomd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | mul12d.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | mul12 7237 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) | |
5 | 1, 2, 3, 4 | syl3anc 1169 | 1 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 (class class class)co 5532 ℂcc 6979 · cmul 6986 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-mulcom 7077 ax-mulass 7079 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: mulreim 7704 divrecap 7776 remullem 9758 dvdscmulr 10224 bezoutlemnewy 10385 dvdsmulgcd 10414 lcmgcdlem 10459 cncongr1 10485 |
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