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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 4720 | . 2 ⊢ ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) = ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐶)) | |
2 | inidm 3316 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
3 | 2 | xpeq2i 4607 | . 2 ⊢ ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) × 𝐶) |
4 | 1, 3 | eqtr2i 2179 | 1 ⊢ ((𝐴 ∩ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∩ cin 3101 × cxp 4584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-opab 4026 df-xp 4592 df-rel 4593 |
This theorem is referenced by: resres 4878 resindi 4881 imainrect 5031 resdmres 5077 |
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