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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 9947 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
3 | int-mulcomd.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | recnd 9947 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
5 | 2, 4 | mulcomd 9940 | . 2 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
6 | int-mulcomd.3 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
7 | 6 | eqcomd 2616 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐴) |
8 | 7 | oveq2d 6565 | . 2 ⊢ (𝜑 → (𝐶 · 𝐵) = (𝐶 · 𝐴)) |
9 | 5, 8 | eqtrd 2644 | 1 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℝcr 9814 · cmul 9820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-resscn 9872 ax-mulcom 9879 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: (None) |
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