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Theorem xpindir 4635
 Description: Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
Assertion
Ref Expression
xpindir ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))

Proof of Theorem xpindir
StepHypRef Expression
1 inxp 4633 . 2 ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) = ((𝐴𝐵) × (𝐶𝐶))
2 inidm 3251 . . 3 (𝐶𝐶) = 𝐶
32xpeq2i 4520 . 2 ((𝐴𝐵) × (𝐶𝐶)) = ((𝐴𝐵) × 𝐶)
41, 3eqtr2i 2136 1 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1314   ∩ cin 3036   × cxp 4497 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-opab 3950  df-xp 4505  df-rel 4506 This theorem is referenced by:  resres  4789  resindi  4792  imainrect  4942  resdmres  4988
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