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Mirrors > Home > MPE Home > Th. List > adddiri | Structured version Visualization version GIF version |
Description: Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.) |
Ref | Expression |
---|---|
axi.1 | ⊢ 𝐴 ∈ ℂ |
axi.2 | ⊢ 𝐵 ∈ ℂ |
axi.3 | ⊢ 𝐶 ∈ ℂ |
Ref | Expression |
---|---|
adddiri | ⊢ ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | axi.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | axi.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
4 | adddir 10620 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1452 | 1 ⊢ ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 + caddc 10528 · 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-addcl 10585 ax-mulcom 10589 ax-distr 10592 |
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: numma 12130 binom2i 13562 3dvdsdec 15669 3dvds2dec 15670 dec5nprm 16390 dec2nprm 16391 mod2xnegi 16395 karatsuba 16408 sincosq3sgn 25013 sincosq4sgn 25014 ang180lem2 25315 1cubrlem 25346 bposlem8 25794 2lgsoddprmlem3c 25915 2lgsoddprmlem3d 25916 normlem3 28816 dpmul100 30500 dpmul1000 30502 dpadd3 30515 dpmul4 30517 problem2 32806 areaquad 39701 tgoldbachlt 43858 |
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