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Theorem xpindir 4500
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 4498 . 2 ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) = ((𝐴𝐵) × (𝐶𝐶))
2 inidm 3174 . . 3 (𝐶𝐶) = 𝐶
32xpeq2i 4394 . 2 ((𝐴𝐵) × (𝐶𝐶)) = ((𝐴𝐵) × 𝐶)
41, 3eqtr2i 2077 1 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cin 2944   × cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-xp 4379  df-rel 4380
This theorem is referenced by:  resres  4652  resindi  4655  imainrect  4794  resdmres  4840
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