Theorem xpindi 4489

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 × (𝐵 ∩ 𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))

Proof of Theorem xpindi

Step	Hyp	Ref	Expression
1		inxp 4488	. 2 ⊢ ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) = ((𝐴 ∩ 𝐴) × (𝐵 ∩ 𝐶))
2		inidm 3175	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	xpeq1i 4383	. 2 ⊢ ((𝐴 ∩ 𝐴) × (𝐵 ∩ 𝐶)) = (𝐴 × (𝐵 ∩ 𝐶))
4	1, 3	eqtr2i 2102	1 ⊢ (𝐴 × (𝐵 ∩ 𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))

Syntax hints:   = wceq 1284   ∩ cin 2972   × cxp 4361

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-xp 4369  df-rel 4370

This theorem is referenced by:  xpriindim  4492