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Mathbox for Stanislas Polu |
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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 11314 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
3 | int-mulcomd.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | recnd 11314 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
5 | 2, 4 | mulcomd 11307 | . 2 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
6 | int-mulcomd.3 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
7 | 6 | eqcomd 2740 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐴) |
8 | 7 | oveq2d 7461 | . 2 ⊢ (𝜑 → (𝐶 · 𝐵) = (𝐶 · 𝐴)) |
9 | 5, 8 | eqtrd 2774 | 1 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 (class class class)co 7445 ℝcr 11179 · cmul 11185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 ax-resscn 11237 ax-mulcom 11244 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-iota 6524 df-fv 6580 df-ov 7448 |
This theorem is referenced by: (None) |
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