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Theorem xpindir 4572
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 4570 . 2 ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) = ((𝐴𝐵) × (𝐶𝐶))
2 inidm 3209 . . 3 (𝐶𝐶) = 𝐶
32xpeq2i 4459 . 2 ((𝐴𝐵) × (𝐶𝐶)) = ((𝐴𝐵) × 𝐶)
41, 3eqtr2i 2109 1 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cin 2998   × cxp 4436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-xp 4444  df-rel 4445
This theorem is referenced by:  resres  4725  resindi  4728  imainrect  4876  resdmres  4922
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