Theorem ndmovdistr 6823

Description: Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Hypotheses

Ref	Expression
ndmov.1	|- dom F = ( S X. S )
ndmov.5	|- -. (/) e. S
ndmov.6	|- dom G = ( S X. S )

Assertion

Ref	Expression
ndmovdistr	|- ( -. ( A e. S /\ B e. S /\ C e. S ) -> ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) ) )

Proof of Theorem ndmovdistr

Step	Hyp	Ref	Expression
1		ndmov.1	. . . . . . 7 |- dom F = ( S X. S )
2		ndmov.5	. . . . . . 7 |- -. (/) e. S
3	1, 2	ndmovrcl 6820	. . . . . 6 |- ( ( B F C ) e. S -> ( B e. S /\ C e. S ) )
4	3	anim2i 593	. . . . 5 |- ( ( A e. S /\ ( B F C ) e. S ) -> ( A e. S /\ ( B e. S /\ C e. S ) ) )
5		3anass 1042	. . . . 5 |- ( ( A e. S /\ B e. S /\ C e. S ) <-> ( A e. S /\ ( B e. S /\ C e. S ) ) )
6	4, 5	sylibr 224	. . . 4 |- ( ( A e. S /\ ( B F C ) e. S ) -> ( A e. S /\ B e. S /\ C e. S ) )
7	6	con3i 150	. . 3 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> -. ( A e. S /\ ( B F C ) e. S ) )
8		ndmov.6	. . . 4 |- dom G = ( S X. S )
9	8	ndmov 6818	. . 3 |- ( -. ( A e. S /\ ( B F C ) e. S ) -> ( A G ( B F C ) ) = (/) )
10	7, 9	syl 17	. 2 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> ( A G ( B F C ) ) = (/) )
11	8, 2	ndmovrcl 6820	. . . . . 6 |- ( ( A G B ) e. S -> ( A e. S /\ B e. S ) )
12	8, 2	ndmovrcl 6820	. . . . . 6 |- ( ( A G C ) e. S -> ( A e. S /\ C e. S ) )
13	11, 12	anim12i 590	. . . . 5 |- ( ( ( A G B ) e. S /\ ( A G C ) e. S ) -> ( ( A e. S /\ B e. S ) /\ ( A e. S /\ C e. S ) ) )
14		anandi3 1052	. . . . 5 |- ( ( A e. S /\ B e. S /\ C e. S ) <-> ( ( A e. S /\ B e. S ) /\ ( A e. S /\ C e. S ) ) )
15	13, 14	sylibr 224	. . . 4 |- ( ( ( A G B ) e. S /\ ( A G C ) e. S ) -> ( A e. S /\ B e. S /\ C e. S ) )
16	15	con3i 150	. . 3 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> -. ( ( A G B ) e. S /\ ( A G C ) e. S ) )
17	1	ndmov 6818	. . 3 |- ( -. ( ( A G B ) e. S /\ ( A G C ) e. S ) -> ( ( A G B ) F ( A G C ) ) = (/) )
18	16, 17	syl 17	. 2 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> ( ( A G B ) F ( A G C ) ) = (/) )
19	10, 18	eqtr4d 2659	1 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   (/)c0 3915   X. cxp 5112   dom cdm 5114  (class class class)co 6650

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-8 1992  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-pow 4843  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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  distrpi  9720  distrnq  9783  distrpr  9850  distrsr  9912