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Mirrors > Home > MPE Home > Th. List > xpindi | Structured version Visualization version GIF version |
Description: Distributive law for Cartesian product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.) |
Ref | Expression |
---|---|
xpindi | ⊢ (𝐴 × (𝐵 ∩ 𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inxp 5287 | . 2 ⊢ ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) = ((𝐴 ∩ 𝐴) × (𝐵 ∩ 𝐶)) | |
2 | inidm 3855 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
3 | 2 | xpeq1i 5169 | . 2 ⊢ ((𝐴 ∩ 𝐴) × (𝐵 ∩ 𝐶)) = (𝐴 × (𝐵 ∩ 𝐶)) |
4 | 1, 3 | eqtr2i 2674 | 1 ⊢ (𝐴 × (𝐵 ∩ 𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∩ cin 3606 × cxp 5141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-opab 4746 df-xp 5149 df-rel 5150 |
This theorem is referenced by: xpriindi 5291 xpcdaen 9043 fpwwe2lem13 9502 txhaus 21498 ustund 22072 |
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