Theorem xpindir 5773

Description: Distributive law for Cartesian product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Assertion

Ref	Expression
xpindir	⊢ ((𝐴 ∩ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))

Proof of Theorem xpindir

Step	Hyp	Ref	Expression
1		inxp 5770	. 2 ⊢ ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) = ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐶))
2		inidm 4174	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
3	2	xpeq2i 5641	. 2 ⊢ ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) × 𝐶)
4	1, 3	eqtr2i 2755	1 ⊢ ((𝐴 ∩ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))

Syntax hints:   = wceq 1541   ∩ cin 3896   × cxp 5612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-opab 5152  df-xp 5620  df-rel 5621

This theorem is referenced by:  resres  5940  resindi  5943  imainrect  6128  resdmres  6179  txhaus  23562  ustund  24137