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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mul13d | Structured version Visualization version GIF version | ||
| Description: Commutative/associative law that swaps the first and the third factor in a triple product. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| mul13d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| mul13d.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| mul13d.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
| Ref | Expression |
|---|---|
| mul13d | ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mul13d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | mul13d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 3 | mul13d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
| 4 | 1, 2, 3 | mul12d 11470 | . 2 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
| 5 | 2, 1, 3 | mulassd 11284 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶))) |
| 6 | 2, 1 | mulcld 11281 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐴) ∈ ℂ) |
| 7 | 6, 3 | mulcomd 11282 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐶 · (𝐵 · 𝐴))) |
| 8 | 4, 5, 7 | 3eqtr2d 2783 | 1 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 · cmul 11160 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-mulcl 11217 ax-mulcom 11219 ax-mulass 11221 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: dirkertrigeqlem3 46115 fourierdlem83 46204 |
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