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Mirrors > Home > MPE Home > Th. List > Mathboxes > int-mulcomd | Structured version Visualization version GIF version |
Description: MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
Ref | Expression |
---|---|
int-mulcomd.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
int-mulcomd.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
int-mulcomd.3 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
int-mulcomd | ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | int-mulcomd.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | 1 | recnd 10657 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
3 | int-mulcomd.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | recnd 10657 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
5 | 2, 4 | mulcomd 10650 | . 2 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
6 | int-mulcomd.3 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
7 | 6 | eqcomd 2824 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐴) |
8 | 7 | oveq2d 7161 | . 2 ⊢ (𝜑 → (𝐶 · 𝐵) = (𝐶 · 𝐴)) |
9 | 5, 8 | eqtrd 2853 | 1 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴)) |
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-mulcom 10589 |
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) |
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