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Mirrors > Home > MPE Home > Th. List > Mathboxes > int-leftdistd | Structured version Visualization version GIF version |
Description: AdditionMultiplicationLeftDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
Ref | Expression |
---|---|
int-leftdistd.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
int-leftdistd.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
int-leftdistd.3 | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
int-leftdistd.4 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
int-leftdistd | ⊢ (𝜑 → ((𝐶 + 𝐷) · 𝐵) = ((𝐶 · 𝐴) + (𝐷 · 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | int-leftdistd.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
2 | 1 | recnd 10663 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
3 | int-leftdistd.3 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
4 | 3 | recnd 10663 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
5 | int-leftdistd.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 5 | recnd 10663 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
7 | 2, 4, 6 | adddird 10660 | . 2 ⊢ (𝜑 → ((𝐶 + 𝐷) · 𝐵) = ((𝐶 · 𝐵) + (𝐷 · 𝐵))) |
8 | 2, 6 | mulcld 10655 | . . 3 ⊢ (𝜑 → (𝐶 · 𝐵) ∈ ℂ) |
9 | 4, 6 | mulcld 10655 | . . 3 ⊢ (𝜑 → (𝐷 · 𝐵) ∈ ℂ) |
10 | 8, 9 | addcomd 10836 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐵) + (𝐷 · 𝐵)) = ((𝐷 · 𝐵) + (𝐶 · 𝐵))) |
11 | 9, 8 | addcomd 10836 | . . 3 ⊢ (𝜑 → ((𝐷 · 𝐵) + (𝐶 · 𝐵)) = ((𝐶 · 𝐵) + (𝐷 · 𝐵))) |
12 | int-leftdistd.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐵) | |
13 | 12 | eqcomd 2827 | . . . . 5 ⊢ (𝜑 → 𝐵 = 𝐴) |
14 | 13 | oveq2d 7166 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝐵) = (𝐶 · 𝐴)) |
15 | 13 | oveq2d 7166 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝐵) = (𝐷 · 𝐴)) |
16 | 14, 15 | oveq12d 7168 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) + (𝐷 · 𝐵)) = ((𝐶 · 𝐴) + (𝐷 · 𝐴))) |
17 | 11, 16 | eqtrd 2856 | . 2 ⊢ (𝜑 → ((𝐷 · 𝐵) + (𝐶 · 𝐵)) = ((𝐶 · 𝐴) + (𝐷 · 𝐴))) |
18 | 7, 10, 17 | 3eqtrd 2860 | 1 ⊢ (𝜑 → ((𝐶 + 𝐷) · 𝐵) = ((𝐶 · 𝐴) + (𝐷 · 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 (class class class)co 7150 ℝcr 10530 + caddc 10534 · cmul 10536 |
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-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 |
This theorem is referenced by: (None) |
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