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Mirrors > Home > MPE Home > Th. List > Mathboxes > int-rightdistd | Structured version Visualization version GIF version |
Description: AdditionMultiplicationRightDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
Ref | Expression |
---|---|
int-rightdistd.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
int-rightdistd.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
int-rightdistd.3 | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
int-rightdistd.4 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
int-rightdistd | ⊢ (𝜑 → (𝐵 · (𝐶 + 𝐷)) = ((𝐴 · 𝐶) + (𝐴 · 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | int-rightdistd.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | 1 | recnd 10668 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
3 | int-rightdistd.2 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | recnd 10668 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
5 | int-rightdistd.3 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
6 | 5 | recnd 10668 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
7 | 4, 6 | addcld 10659 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℂ) |
8 | 2, 7 | mulcomd 10661 | . 2 ⊢ (𝜑 → (𝐵 · (𝐶 + 𝐷)) = ((𝐶 + 𝐷) · 𝐵)) |
9 | 4, 2 | mulcomd 10661 | . . . . 5 ⊢ (𝜑 → (𝐶 · 𝐵) = (𝐵 · 𝐶)) |
10 | int-rightdistd.4 | . . . . . . 7 ⊢ (𝜑 → 𝐴 = 𝐵) | |
11 | 10 | eqcomd 2827 | . . . . . 6 ⊢ (𝜑 → 𝐵 = 𝐴) |
12 | 11 | oveq1d 7170 | . . . . 5 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐴 · 𝐶)) |
13 | 9, 12 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝐵) = (𝐴 · 𝐶)) |
14 | 6, 2 | mulcomd 10661 | . . . . 5 ⊢ (𝜑 → (𝐷 · 𝐵) = (𝐵 · 𝐷)) |
15 | 11 | oveq1d 7170 | . . . . 5 ⊢ (𝜑 → (𝐵 · 𝐷) = (𝐴 · 𝐷)) |
16 | 14, 15 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝐵) = (𝐴 · 𝐷)) |
17 | 13, 16 | oveq12d 7173 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) + (𝐷 · 𝐵)) = ((𝐴 · 𝐶) + (𝐴 · 𝐷))) |
18 | 4, 2, 6, 17 | joinlmuladdmuld 10667 | . 2 ⊢ (𝜑 → ((𝐶 + 𝐷) · 𝐵) = ((𝐴 · 𝐶) + (𝐴 · 𝐷))) |
19 | 8, 18 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝐵 · (𝐶 + 𝐷)) = ((𝐴 · 𝐶) + (𝐴 · 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 (class class class)co 7155 ℝcr 10535 + caddc 10539 · cmul 10541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-resscn 10593 ax-addcl 10596 ax-mulcom 10600 ax-distr 10603 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-ov 7158 |
This theorem is referenced by: (None) |
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