Theorem ndmaovdistr 41287

Description: Any operation is distributive outside its domain. In contrast to ndmovdistr 6823 where it is required that the operation's domain doesn't contain the empty set ( -. (/) e. S), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypotheses

Ref	Expression
ndmaov.1	|- dom F = ( S X. S )
ndmaov.6	|- dom G = ( S X. S )

Assertion

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

Proof of Theorem ndmaovdistr

Step	Hyp	Ref	Expression
1		ndmaov.6	. . . . . . 7 |- dom G = ( S X. S )
2	1	eleq2i 2693	. . . . . 6 |- ( <. A , (( B F C)) >. e. dom G <-> <. A , (( B F C)) >. e. ( S X. S ) )
3		opelxp 5146	. . . . . 6 |- ( <. A , (( B F C)) >. e. ( S X. S ) <-> ( A e. S /\ (( B F C)) e. S ) )
4	2, 3	bitri 264	. . . . 5 |- ( <. A , (( B F C)) >. e. dom G <-> ( A e. S /\ (( B F C)) e. S ) )
5		aovvdm 41265	. . . . . . 7 |- ( (( B F C)) e. S -> <. B , C >. e. dom F )
6		ndmaov.1	. . . . . . . . . 10 |- dom F = ( S X. S )
7	6	eleq2i 2693	. . . . . . . . 9 |- ( <. B , C >. e. dom F <-> <. B , C >. e. ( S X. S ) )
8		opelxp 5146	. . . . . . . . 9 |- ( <. B , C >. e. ( S X. S ) <-> ( B e. S /\ C e. S ) )
9	7, 8	bitri 264	. . . . . . . 8 |- ( <. B , C >. e. dom F <-> ( B e. S /\ C e. S ) )
10		3anass 1042	. . . . . . . . 9 |- ( ( A e. S /\ B e. S /\ C e. S ) <-> ( A e. S /\ ( B e. S /\ C e. S ) ) )
11	10	simplbi2com 657	. . . . . . . 8 |- ( ( B e. S /\ C e. S ) -> ( A e. S -> ( A e. S /\ B e. S /\ C e. S ) ) )
12	9, 11	sylbi 207	. . . . . . 7 |- ( <. B , C >. e. dom F -> ( A e. S -> ( A e. S /\ B e. S /\ C e. S ) ) )
13	5, 12	syl 17	. . . . . 6 |- ( (( B F C)) e. S -> ( A e. S -> ( A e. S /\ B e. S /\ C e. S ) ) )
14	13	impcom 446	. . . . 5 |- ( ( A e. S /\ (( B F C)) e. S ) -> ( A e. S /\ B e. S /\ C e. S ) )
15	4, 14	sylbi 207	. . . 4 |- ( <. A , (( B F C)) >. e. dom G -> ( A e. S /\ B e. S /\ C e. S ) )
16	15	con3i 150	. . 3 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> -. <. A , (( B F C)) >. e. dom G )
17		ndmaov 41263	. . 3 |- ( -. <. A , (( B F C)) >. e. dom G -> (( A G (( B F C)) )) = _V )
18	16, 17	syl 17	. 2 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> (( A G (( B F C)) )) = _V )
19	6	eleq2i 2693	. . . . . 6 |- ( <. (( A G B)) , (( A G C)) >. e. dom F <-> <. (( A G B)) , (( A G C)) >. e. ( S X. S ) )
20		opelxp 5146	. . . . . 6 |- ( <. (( A G B)) , (( A G C)) >. e. ( S X. S ) <-> ( (( A G B)) e. S /\ (( A G C)) e. S ) )
21	19, 20	bitri 264	. . . . 5 |- ( <. (( A G B)) , (( A G C)) >. e. dom F <-> ( (( A G B)) e. S /\ (( A G C)) e. S ) )
22		aovvdm 41265	. . . . . . 7 |- ( (( A G B)) e. S -> <. A , B >. e. dom G )
23	1	eleq2i 2693	. . . . . . . . 9 |- ( <. A , B >. e. dom G <-> <. A , B >. e. ( S X. S ) )
24		opelxp 5146	. . . . . . . . 9 |- ( <. A , B >. e. ( S X. S ) <-> ( A e. S /\ B e. S ) )
25	23, 24	bitri 264	. . . . . . . 8 |- ( <. A , B >. e. dom G <-> ( A e. S /\ B e. S ) )
26	1	eleq2i 2693	. . . . . . . . . . 11 |- ( <. A , C >. e. dom G <-> <. A , C >. e. ( S X. S ) )
27		opelxp 5146	. . . . . . . . . . 11 |- ( <. A , C >. e. ( S X. S ) <-> ( A e. S /\ C e. S ) )
28	26, 27	bitri 264	. . . . . . . . . 10 |- ( <. A , C >. e. dom G <-> ( A e. S /\ C e. S ) )
29		simpll 790	. . . . . . . . . . . 12 |- ( ( ( A e. S /\ C e. S ) /\ ( A e. S /\ B e. S ) ) -> A e. S )
30		simprr 796	. . . . . . . . . . . 12 |- ( ( ( A e. S /\ C e. S ) /\ ( A e. S /\ B e. S ) ) -> B e. S )
31		simplr 792	. . . . . . . . . . . 12 |- ( ( ( A e. S /\ C e. S ) /\ ( A e. S /\ B e. S ) ) -> C e. S )
32	29, 30, 31	3jca 1242	. . . . . . . . . . 11 |- ( ( ( A e. S /\ C e. S ) /\ ( A e. S /\ B e. S ) ) -> ( A e. S /\ B e. S /\ C e. S ) )
33	32	ex 450	. . . . . . . . . 10 |- ( ( A e. S /\ C e. S ) -> ( ( A e. S /\ B e. S ) -> ( A e. S /\ B e. S /\ C e. S ) ) )
34	28, 33	sylbi 207	. . . . . . . . 9 |- ( <. A , C >. e. dom G -> ( ( A e. S /\ B e. S ) -> ( A e. S /\ B e. S /\ C e. S ) ) )
35		aovvdm 41265	. . . . . . . . 9 |- ( (( A G C)) e. S -> <. A , C >. e. dom G )
36	34, 35	syl11 33	. . . . . . . 8 |- ( ( A e. S /\ B e. S ) -> ( (( A G C)) e. S -> ( A e. S /\ B e. S /\ C e. S ) ) )
37	25, 36	sylbi 207	. . . . . . 7 |- ( <. A , B >. e. dom G -> ( (( A G C)) e. S -> ( A e. S /\ B e. S /\ C e. S ) ) )
38	22, 37	syl 17	. . . . . 6 |- ( (( A G B)) e. S -> ( (( A G C)) e. S -> ( A e. S /\ B e. S /\ C e. S ) ) )
39	38	imp 445	. . . . 5 |- ( ( (( A G B)) e. S /\ (( A G C)) e. S ) -> ( A e. S /\ B e. S /\ C e. S ) )
40	21, 39	sylbi 207	. . . 4 |- ( <. (( A G B)) , (( A G C)) >. e. dom F -> ( A e. S /\ B e. S /\ C e. S ) )
41	40	con3i 150	. . 3 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> -. <. (( A G B)) , (( A G C)) >. e. dom F )
42		ndmaov 41263	. . 3 |- ( -. <. (( A G B)) , (( A G C)) >. e. dom F -> (( (( A G B)) F (( A G C)) )) = _V )
43	41, 42	syl 17	. 2 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> (( (( A G B)) F (( A G C)) )) = _V )
44	18, 43	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   _Vcvv 3200   <.cop 4183   X. cxp 5112   dom cdm 5114   ((caov 41195

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-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-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-opab 4713  df-xp 5120  df-fv 5896  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)