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Mirrors > Home > MPE Home > Th. List > xpindir | Structured version Visualization version GIF version |
Description: Distributive law for Cartesian product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.) |
Ref | Expression |
---|---|
xpindir | ⊢ ((𝐴 ∩ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inxp 5667 | . 2 ⊢ ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) = ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐶)) | |
2 | inidm 4145 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
3 | 2 | xpeq2i 5546 | . 2 ⊢ ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) × 𝐶) |
4 | 1, 3 | eqtr2i 2822 | 1 ⊢ ((𝐴 ∩ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∩ cin 3880 × cxp 5517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 df-rel 5526 |
This theorem is referenced by: resres 5831 resindi 5834 imainrect 6005 resdmres 6056 txhaus 22252 ustund 22827 |
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