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Mirrors > Home > ILE Home > Th. List > xpindir | GIF version |
Description: Distributive law for cross 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 4773 | . 2 ⊢ ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) = ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐶)) | |
2 | inidm 3356 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
3 | 2 | xpeq2i 4659 | . 2 ⊢ ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) × 𝐶) |
4 | 1, 3 | eqtr2i 2209 | 1 ⊢ ((𝐴 ∩ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ∩ cin 3140 × cxp 4636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-opab 4077 df-xp 4644 df-rel 4645 |
This theorem is referenced by: resres 4931 resindi 4934 imainrect 5086 resdmres 5132 |
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