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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 10245 | . 2 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
5 | 2, 1, 3 | mulassd 10063 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶))) |
6 | 2, 1 | mulcld 10060 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐴) ∈ ℂ) |
7 | 6, 3 | mulcomd 10061 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐶 · (𝐵 · 𝐴))) |
8 | 4, 5, 7 | 3eqtr2d 2662 | 1 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 · cmul 9941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-mulcl 9998 ax-mulcom 10000 ax-mulass 10002 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: dirkertrigeqlem3 40317 fourierdlem83 40406 |
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