Theorem xpindir 5732

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 5730	. 2 ⊢ ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) = ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐶))
2		inidm 4149	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
3	2	xpeq2i 5607	. 2 ⊢ ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) × 𝐶)
4	1, 3	eqtr2i 2767	1 ⊢ ((𝐴 ∩ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))

Syntax hints:   = wceq 1539   ∩ cin 3882   × cxp 5578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-xp 5586  df-rel 5587

This theorem is referenced by:  resres  5893  resindi  5896  imainrect  6073  resdmres  6124  txhaus  22706  ustund  23281