Theorem xpriindim 4589

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 4586	. 2 |- ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i ( C X. |^|_ x e. A B ) )
2		xpiindim 4588	. . 3 |- ( E. y y e. A -> ( C X. |^|_ x e. A B ) = |^|_ x e. A ( C X. B ) )
3	2	ineq2d 3204	. 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	syl5eq 2133	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 1290   E.wex 1427   e. wcel 1439   i^i cin 3001   |^|_ciin 3739   X. cxp 4452

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-iin 3741  df-opab 3908  df-xp 4460  df-rel 4461

This theorem is referenced by: (None)