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Theorem xpindi 4674
Description: Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
Assertion
Ref Expression
xpindi (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))

Proof of Theorem xpindi
StepHypRef Expression
1 inxp 4673 . 2 ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) = ((𝐴𝐴) × (𝐵𝐶))
2 inidm 3285 . . 3 (𝐴𝐴) = 𝐴
32xpeq1i 4559 . 2 ((𝐴𝐴) × (𝐵𝐶)) = (𝐴 × (𝐵𝐶))
41, 3eqtr2i 2161 1 (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1331  cin 3070   × cxp 4537
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545  df-rel 4546
This theorem is referenced by:  xpriindim  4677  djuassen  7080  xpdjuen  7081
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