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Mirrors > Home > MPE Home > Th. List > indir | Structured version Visualization version GIF version |
Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.) |
Ref | Expression |
---|---|
indir | ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indi 4253 | . 2 ⊢ (𝐶 ∩ (𝐴 ∪ 𝐵)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵)) | |
2 | incom 4181 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∪ 𝐵)) | |
3 | incom 4181 | . . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴) | |
4 | incom 4181 | . . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵) | |
5 | 3, 4 | uneq12i 4140 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵)) |
6 | 1, 2, 5 | 3eqtr4i 2857 | 1 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∪ cun 3937 ∩ cin 3938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-un 3944 df-in 3946 |
This theorem is referenced by: difundir 4260 undisj1 4414 disjpr2 4652 resundir 5871 predun 6175 djuassen 9607 fin23lem26 9750 fpwwe2lem13 10067 neitr 21791 fiuncmp 22015 connsuba 22031 trfil2 22498 tsmsres 22755 trust 22841 restmetu 23183 volun 24149 uniioombllem3 24189 itgsplitioo 24441 ppiprm 25731 chtprm 25733 chtdif 25738 ppidif 25743 cycpmco2f1 30770 carsgclctunlem1 31579 ballotlemfp1 31753 ballotlemgun 31786 mrsubvrs 32773 mthmpps 32833 fixun 33374 mbfposadd 34943 iunrelexp0 40053 31prm 43767 |
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