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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 10124 | . 2 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
5 | 2, 1, 3 | mulassd 9942 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶))) |
6 | 2, 1 | mulcld 9939 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐴) ∈ ℂ) |
7 | 6, 3 | mulcomd 9940 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐶 · (𝐵 · 𝐴))) |
8 | 4, 5, 7 | 3eqtr2d 2650 | 1 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 · cmul 9820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-mulcl 9877 ax-mulcom 9879 ax-mulass 9881 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: dirkertrigeqlem3 38993 fourierdlem83 39082 |
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