Theorem ndmaovass 41286

Description: Any operation is associative outside its domain. In contrast to ndmovass 6822 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.)

Hypothesis

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

Assertion

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

Proof of Theorem ndmaovass

Step	Hyp	Ref	Expression
1		ndmaov.1	. . . . . . 7 |- dom F = ( S X. S )
2	1	eleq2i 2693	. . . . . 6 |- ( <. (( A F B)) , C >. e. dom F <-> <. (( A F B)) , C >. e. ( S X. S ) )
3		opelxp 5146	. . . . . 6 |- ( <. (( A F B)) , C >. e. ( S X. S ) <-> ( (( A F B)) e. S /\ C e. S ) )
4	2, 3	bitri 264	. . . . 5 |- ( <. (( A F B)) , C >. e. dom F <-> ( (( A F B)) e. S /\ C e. S ) )
5		aovvdm 41265	. . . . . . 7 |- ( (( A F B)) e. S -> <. A , B >. e. dom F )
6	1	eleq2i 2693	. . . . . . . . 9 |- ( <. A , B >. e. dom F <-> <. A , B >. e. ( S X. S ) )
7		opelxp 5146	. . . . . . . . 9 |- ( <. A , B >. e. ( S X. S ) <-> ( A e. S /\ B e. S ) )
8	6, 7	bitri 264	. . . . . . . 8 |- ( <. A , B >. e. dom F <-> ( A e. S /\ B e. S ) )
9		df-3an 1039	. . . . . . . . 9 |- ( ( A e. S /\ B e. S /\ C e. S ) <-> ( ( A e. S /\ B e. S ) /\ C e. S ) )
10	9	simplbi2 655	. . . . . . . 8 |- ( ( A e. S /\ B e. S ) -> ( C e. S -> ( A e. S /\ B e. S /\ C e. S ) ) )
11	8, 10	sylbi 207	. . . . . . 7 |- ( <. A , B >. e. dom F -> ( C e. S -> ( A e. S /\ B e. S /\ C e. S ) ) )
12	5, 11	syl 17	. . . . . 6 |- ( (( A F B)) e. S -> ( C e. S -> ( A e. S /\ B e. S /\ C e. S ) ) )
13	12	imp 445	. . . . 5 |- ( ( (( A F B)) e. S /\ C e. S ) -> ( A e. S /\ B e. S /\ C e. S ) )
14	4, 13	sylbi 207	. . . 4 |- ( <. (( A F B)) , C >. e. dom F -> ( A e. S /\ B e. S /\ C e. S ) )
15	14	con3i 150	. . 3 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> -. <. (( A F B)) , C >. e. dom F )
16		ndmaov 41263	. . 3 |- ( -. <. (( A F B)) , C >. e. dom F -> (( (( A F B)) F C)) = _V )
17	15, 16	syl 17	. 2 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> (( (( A F B)) F C)) = _V )
18	1	eleq2i 2693	. . . . . . 7 |- ( <. A , (( B F C)) >. e. dom F <-> <. A , (( B F C)) >. e. ( S X. S ) )
19		opelxp 5146	. . . . . . 7 |- ( <. A , (( B F C)) >. e. ( S X. S ) <-> ( A e. S /\ (( B F C)) e. S ) )
20	18, 19	bitri 264	. . . . . 6 |- ( <. A , (( B F C)) >. e. dom F <-> ( A e. S /\ (( B F C)) e. S ) )
21		aovvdm 41265	. . . . . . . 8 |- ( (( B F C)) e. S -> <. B , C >. e. dom F )
22	1	eleq2i 2693	. . . . . . . . . 10 |- ( <. B , C >. e. dom F <-> <. B , C >. e. ( S X. S ) )
23		opelxp 5146	. . . . . . . . . 10 |- ( <. B , C >. e. ( S X. S ) <-> ( B e. S /\ C e. S ) )
24	22, 23	bitri 264	. . . . . . . . 9 |- ( <. B , C >. e. dom F <-> ( B e. S /\ C e. S ) )
25		3anass 1042	. . . . . . . . . . . 12 |- ( ( A e. S /\ B e. S /\ C e. S ) <-> ( A e. S /\ ( B e. S /\ C e. S ) ) )
26	25	biimpri 218	. . . . . . . . . . 11 |- ( ( A e. S /\ ( B e. S /\ C e. S ) ) -> ( A e. S /\ B e. S /\ C e. S ) )
27	26	a1d 25	. . . . . . . . . 10 |- ( ( A e. S /\ ( B e. S /\ C e. S ) ) -> ( <. A , (( B F C)) >. e. dom F -> ( A e. S /\ B e. S /\ C e. S ) ) )
28	27	expcom 451	. . . . . . . . 9 |- ( ( B e. S /\ C e. S ) -> ( A e. S -> ( <. A , (( B F C)) >. e. dom F -> ( A e. S /\ B e. S /\ C e. S ) ) ) )
29	24, 28	sylbi 207	. . . . . . . 8 |- ( <. B , C >. e. dom F -> ( A e. S -> ( <. A , (( B F C)) >. e. dom F -> ( A e. S /\ B e. S /\ C e. S ) ) ) )
30	21, 29	syl 17	. . . . . . 7 |- ( (( B F C)) e. S -> ( A e. S -> ( <. A , (( B F C)) >. e. dom F -> ( A e. S /\ B e. S /\ C e. S ) ) ) )
31	30	impcom 446	. . . . . 6 |- ( ( A e. S /\ (( B F C)) e. S ) -> ( <. A , (( B F C)) >. e. dom F -> ( A e. S /\ B e. S /\ C e. S ) ) )
32	20, 31	sylbi 207	. . . . 5 |- ( <. A , (( B F C)) >. e. dom F -> ( <. A , (( B F C)) >. e. dom F -> ( A e. S /\ B e. S /\ C e. S ) ) )
33	32	pm2.43i 52	. . . 4 |- ( <. A , (( B F C)) >. e. dom F -> ( A e. S /\ B e. S /\ C e. S ) )
34	33	con3i 150	. . 3 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> -. <. A , (( B F C)) >. e. dom F )
35		ndmaov 41263	. . 3 |- ( -. <. A , (( B F C)) >. e. dom F -> (( A F (( B F C)) )) = _V )
36	34, 35	syl 17	. 2 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> (( A F (( B F C)) )) = _V )
37	17, 36	eqtr4d 2659	1 |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> (( (( A F B)) F C)) = (( A F (( B F 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)