Theorem xpun 2463

Description: The cross product of two unions.

Assertion

Ref	Expression
xpun	⊢ ((A ∪ B) × (C ∪ D)) = (((A × C) ∪ (A × D)) ∪ ((B × C) ∪ (B × D)))

Proof of Theorem xpun

Step	Hyp	Ref	Expression
1		xpundi 2461	. 2 ⊢ ((A ∪ B) × (C ∪ D)) = (((A ∪ B) × C) ∪ ((A ∪ B) × D))
2		xpundir 2462	. . 3 ⊢ ((A ∪ B) × C) = ((A × C) ∪ (B × C))
3		xpundir 2462	. . 3 ⊢ ((A ∪ B) × D) = ((A × D) ∪ (B × D))
4	2, 3	uneq12i 1609	. 2 ⊢ (((A ∪ B) × C) ∪ ((A ∪ B) × D)) = (((A × C) ∪ (B × C)) ∪ ((A × D) ∪ (B × D)))
5		un4 1618	. 2 ⊢ (((A × C) ∪ (B × C)) ∪ ((A × D) ∪ (B × D))) = (((A × C) ∪ (A × D)) ∪ ((B × C) ∪ (B × D)))
6	1, 4, 5	3eqtr 1123	1 ⊢ ((A ∪ B) × (C ∪ D)) = (((A × C) ∪ (A × D)) ∪ ((B × C) ∪ (B × D)))

Syntax hints:   = wceq 1091   ∪ cun 1485   × cxp 2408

This theorem is referenced by:  infxpidmlem11 4943

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-opab 2098  df-xp 2424

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