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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 10632 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1457 | 1 ⊢ ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 + caddc 10540 · cmul 10542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-addcl 10597 ax-mulcom 10601 ax-distr 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: numma 12143 binom2i 13575 3dvdsdec 15681 3dvds2dec 15682 dec5nprm 16402 dec2nprm 16403 mod2xnegi 16407 karatsuba 16420 sincosq3sgn 25086 sincosq4sgn 25087 ang180lem2 25388 1cubrlem 25419 bposlem8 25867 2lgsoddprmlem3c 25988 2lgsoddprmlem3d 25989 normlem3 28889 dpmul100 30573 dpmul1000 30575 dpadd3 30588 dpmul4 30590 problem2 32909 areaquad 39843 tgoldbachlt 44001 |
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