Theorem xpindi 4895

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 4894	. 2 ⊢ ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) = ((𝐴 ∩ 𝐴) × (𝐵 ∩ 𝐶))
2		inidm 3434	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	xpeq1i 4774	. 2 ⊢ ((𝐴 ∩ 𝐴) × (𝐵 ∩ 𝐶)) = (𝐴 × (𝐵 ∩ 𝐶))
4	1, 3	eqtr2i 2256	1 ⊢ (𝐴 × (𝐵 ∩ 𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))

Syntax hints:   = wceq 1398   ∩ cin 3213   × cxp 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-opab 4177  df-xp 4760  df-rel 4761

This theorem is referenced by:  xpriindim  4898  djuassen  7537  xpdjuen  7538