Theorem xpriindi 5258

Description: Distributive law for Cartesian product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
xpriindi	|- ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) )

Distinct variable groups:   x, A   x, C

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

Proof of Theorem xpriindi

Step	Hyp	Ref	Expression
1		iineq1 4535	. . . . . . 7 |- ( A = (/) -> |^|_ x e. A B = |^|_ x e. (/) B )
2		0iin 4578	. . . . . . 7 |- |^|_ x e. (/) B = _V
3	1, 2	syl6eq 2672	. . . . . 6 |- ( A = (/) -> |^|_ x e. A B = _V )
4	3	ineq2d 3814	. . . . 5 |- ( A = (/) -> ( D i^i |^|_ x e. A B ) = ( D i^i _V ) )
5		inv1 3970	. . . . 5 |- ( D i^i _V ) = D
6	4, 5	syl6eq 2672	. . . 4 |- ( A = (/) -> ( D i^i |^|_ x e. A B ) = D )
7	6	xpeq2d 5139	. . 3 |- ( A = (/) -> ( C X. ( D i^i |^|_ x e. A B ) ) = ( C X. D ) )
8		iineq1 4535	. . . . . 6 |- ( A = (/) -> |^|_ x e. A ( C X. B ) = |^|_ x e. (/) ( C X. B ) )
9		0iin 4578	. . . . . 6 |- |^|_ x e. (/) ( C X. B ) = _V
10	8, 9	syl6eq 2672	. . . . 5 |- ( A = (/) -> |^|_ x e. A ( C X. B ) = _V )
11	10	ineq2d 3814	. . . 4 |- ( A = (/) -> ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) = ( ( C X. D ) i^i _V ) )
12		inv1 3970	. . . 4 |- ( ( C X. D ) i^i _V ) = ( C X. D )
13	11, 12	syl6eq 2672	. . 3 |- ( A = (/) -> ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) = ( C X. D ) )
14	7, 13	eqtr4d 2659	. 2 |- ( A = (/) -> ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) )
15		xpindi 5255	. . 3 |- ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i ( C X. |^|_ x e. A B ) )
16		xpiindi 5257	. . . 4 |- ( A =/= (/) -> ( C X. |^|_ x e. A B ) = |^|_ x e. A ( C X. B ) )
17	16	ineq2d 3814	. . 3 |- ( A =/= (/) -> ( ( C X. D ) i^i ( C X. |^|_ x e. A B ) ) = ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) )
18	15, 17	syl5eq 2668	. 2 |- ( A =/= (/) -> ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) )
19	14, 18	pm2.61ine 2877	1 |- ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) )

Syntax hints:   = wceq 1483   =/= wne 2794   _Vcvv 3200   i^i cin 3573   (/)c0 3915   |^|_ciin 4521   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iin 4523  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by: (None)