Mathbox for Stanislas Polu |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > int-mulassocd | Structured version Visualization version GIF version |
Description: MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
Ref | Expression |
---|---|
int-mulassocd.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
int-mulassocd.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
int-mulassocd.3 | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
int-mulassocd.4 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
int-mulassocd | ⊢ (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | int-mulassocd.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | 1 | recnd 10657 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
3 | int-mulassocd.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | recnd 10657 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
5 | int-mulassocd.3 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
6 | 5 | recnd 10657 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
7 | 2, 4, 6 | mulassd 10652 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐶) · 𝐷) = (𝐵 · (𝐶 · 𝐷))) |
8 | int-mulassocd.4 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
9 | 8 | eqcomd 2824 | . . . 4 ⊢ (𝜑 → 𝐵 = 𝐴) |
10 | 9 | oveq1d 7160 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐴 · 𝐶)) |
11 | 10 | oveq1d 7160 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐶) · 𝐷) = ((𝐴 · 𝐶) · 𝐷)) |
12 | 7, 11 | eqtr3d 2855 | 1 ⊢ (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℝcr 10524 · cmul 10530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-resscn 10582 ax-mulass 10591 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |