![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 10838 | . 2 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
5 | 2, 1, 3 | mulassd 10653 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶))) |
6 | 2, 1 | mulcld 10650 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐴) ∈ ℂ) |
7 | 6, 3 | mulcomd 10651 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐶 · (𝐵 · 𝐴))) |
8 | 4, 5, 7 | 3eqtr2d 2839 | 1 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 · cmul 10531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-mulcl 10588 ax-mulcom 10590 ax-mulass 10592 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 |
This theorem is referenced by: dirkertrigeqlem3 42742 fourierdlem83 42831 |
Copyright terms: Public domain | W3C validator |