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Metamath Proof Explorer

Theorem indi

Description: Distributive law for intersection over union. Exercise 10 of TakeutiZaring p. 17. (Contributed by NM, 30-Sep-2002) (Proof shortened by Andrew Salmon, 26-Jun-2011)

Ref Expression
Assertion indi 
|- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )

Proof

Step Hyp Ref Expression
1 andi 
 |-  ( ( x e. A /\ ( x e. B \/ x e. C ) ) <-> ( ( x e. A /\ x e. B ) \/ ( x e. A /\ x e. C ) ) )
2 elin 
 |-  ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
3 elin 
 |-  ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4 2 3 orbi12i 
 |-  ( ( x e. ( A i^i B ) \/ x e. ( A i^i C ) ) <-> ( ( x e. A /\ x e. B ) \/ ( x e. A /\ x e. C ) ) )
5 1 4 bitr4i 
 |-  ( ( x e. A /\ ( x e. B \/ x e. C ) ) <-> ( x e. ( A i^i B ) \/ x e. ( A i^i C ) ) )
6 elun 
 |-  ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
7 6 anbi2i 
 |-  ( ( x e. A /\ x e. ( B u. C ) ) <-> ( x e. A /\ ( x e. B \/ x e. C ) ) )
8 elun 
 |-  ( x e. ( ( A i^i B ) u. ( A i^i C ) ) <-> ( x e. ( A i^i B ) \/ x e. ( A i^i C ) ) )
9 5 7 8 3bitr4i 
 |-  ( ( x e. A /\ x e. ( B u. C ) ) <-> x e. ( ( A i^i B ) u. ( A i^i C ) ) )
10 9 ineqri 
 |-  ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )