Theorem xpriindim 4805

Description: Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)

Assertion

Ref	Expression
xpriindim	|- ( E. y y e. A -> ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) )

Distinct variable groups:   x, y, A   x, C, y

Allowed substitution hints:   B( x, y)   D( x, y)

Proof of Theorem xpriindim

Step	Hyp	Ref	Expression
1		xpindi 4802	. 2 |- ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i ( C X. |^|_ x e. A B ) )
2		xpiindim 4804	. . 3 |- ( E. y y e. A -> ( C X. |^|_ x e. A B ) = |^|_ x e. A ( C X. B ) )
3	2	ineq2d 3365	. 2 |- ( E. y y e. A -> ( ( C X. D ) i^i ( C X. |^|_ x e. A B ) ) = ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) )
4	1, 3	eqtrid 2241	1 |- ( E. y y e. A -> ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) )

Syntax hints:   -> wi 4   = wceq 1364   E.wex 1506   e. wcel 2167   i^i cin 3156   |^|_ciin 3918   X. cxp 4662

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-iin 3920  df-opab 4096  df-xp 4670  df-rel 4671

This theorem is referenced by: (None)