Theorem symggrpi 10406

Description: The symmetry group on A is a group (inference version). (Contributed by Paul Chapman, 25-Feb-2008.)

Hypothesis

Ref	Expression
symggrpi.1	|- A e. V

Assertion

Ref	Expression
symggrpi	|- (SymGrp` A) e. Grp

Proof of Theorem symggrpi

Step	Hyp	Ref	Expression
1		symggrpi.1	. . 3 |- A e. V
2		eqid 1475	. . . . 5 |- {x | x:A-1-1-onto->A} = {x | x:A-1-1-onto->A}
3		equid 1126	. . . . . . 7 |- x = x
4	3	biantru 724	. . . . . 6 |- (x:A-1-1-onto->A <-> (x:A-1-1-onto->A /\ x = x))
5	4	abbii 1575	. . . . 5 |- {x | x:A-1-1-onto->A} = {x | (x:A-1-1-onto->A /\ x = x)}
6	2, 5	eqtr 1495	. . . 4 |- {x | x:A-1-1-onto->A} = {x | (x:A-1-1-onto->A /\ x = x)}
7	6	f1oabexg 3700	. . 3 |- ((A e. V /\ A e. V) -> {x | x:A-1-1-onto->A} e. V)
8	1, 1, 7	mp2an 697	. 2 |- {x | x:A-1-1-onto->A} e. V
9	1, 2	symgf 10405	. 2 |- (SymGrp` A):({x | x:A-1-1-onto->A} X. {x | x:A-1-1-onto->A})-->{x | x:A-1-1-onto->A}
10		coass 3512	. . 3 |- ((f o. g) o. h) = (f o. (g o. h))
11	1, 2	symgoprval 10404	. . . . . 6 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A}) -> (f(SymGrp` A)g) = (f o. g))
12	11	3adant3 799	. . . . 5 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> (f(SymGrp` A)g) = (f o. g))
13	12	opreq1d 3975	. . . 4 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> ((f(SymGrp` A)g)(SymGrp` A)h) = ((f o. g)(SymGrp` A)h))
14	1, 2	symgoprval 10404	. . . . . 6 |- (((f o. g) e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> ((f o. g)(SymGrp` A)h) = ((f o. g) o. h))
15		f1oco 3707	. . . . . . 7 |- ((f:A-1-1-onto->A /\ g:A-1-1-onto->A) -> (f o. g):A-1-1-onto->A)
16	1, 2	elsymgrn 10401	. . . . . . . 8 |- (f e. {x | x:A-1-1-onto->A} <-> f:A-1-1-onto->A)
17	1, 2	elsymgrn 10401	. . . . . . . 8 |- (g e. {x | x:A-1-1-onto->A} <-> g:A-1-1-onto->A)
18	16, 17	anbi12i 482	. . . . . . 7 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A}) <-> (f:A-1-1-onto->A /\ g:A-1-1-onto->A))
19	1, 2	elsymgrn 10401	. . . . . . 7 |- ((f o. g) e. {x | x:A-1-1-onto->A} <-> (f o. g):A-1-1-onto->A)
20	15, 18, 19	3imtr4 219	. . . . . 6 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A}) -> (f o. g) e. {x | x:A-1-1-onto->A})
21	14, 20	sylan 448	. . . . 5 |- (((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A}) /\ h e. {x | x:A-1-1-onto->A}) -> ((f o. g)(SymGrp` A)h) = ((f o. g) o. h))
22	21	3impa 828	. . . 4 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> ((f o. g)(SymGrp` A)h) = ((f o. g) o. h))
23	13, 22	eqtrd 1507	. . 3 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> ((f(SymGrp` A)g)(SymGrp` A)h) = ((f o. g) o. h))
24	1, 2	symgoprval 10404	. . . . . 6 |- ((g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> (g(SymGrp` A)h) = (g o. h))
25	24	3adant1 797	. . . . 5 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> (g(SymGrp` A)h) = (g o. h))
26	25	opreq2d 3976	. . . 4 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> (f(SymGrp` A)(g(SymGrp` A)h)) = (f(SymGrp` A)(g o. h)))
27	1, 2	symgoprval 10404	. . . . . 6 |- ((f e. {x | x:A-1-1-onto->A} /\ (g o. h) e. {x | x:A-1-1-onto->A}) -> (f(SymGrp` A)(g o. h)) = (f o. (g o. h)))
28		f1oco 3707	. . . . . . 7 |- ((g:A-1-1-onto->A /\ h:A-1-1-onto->A) -> (g o. h):A-1-1-onto->A)
29	1, 2	elsymgrn 10401	. . . . . . . 8 |- (h e. {x | x:A-1-1-onto->A} <-> h:A-1-1-onto->A)
30	17, 29	anbi12i 482	. . . . . . 7 |- ((g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) <-> (g:A-1-1-onto->A /\ h:A-1-1-onto->A))
31	1, 2	elsymgrn 10401	. . . . . . 7 |- ((g o. h) e. {x | x:A-1-1-onto->A} <-> (g o. h):A-1-1-onto->A)
32	28, 30, 31	3imtr4 219	. . . . . 6 |- ((g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> (g o. h) e. {x | x:A-1-1-onto->A})
33	27, 32	sylan2 451	. . . . 5 |- ((f e. {x | x:A-1-1-onto->A} /\ (g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A})) -> (f(SymGrp` A)(g o. h)) = (f o. (g o. h)))
34	33	3impb 829	. . . 4 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> (f(SymGrp` A)(g o. h)) = (f o. (g o. h)))
35	26, 34	eqtrd 1507	. . 3 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> (f(SymGrp` A)(g(SymGrp` A)h)) = (f o. (g o. h)))
36	10, 23, 35	3eqtr4a 1532	. 2 |- ((f e. {x | x:A-1-1-onto->A} /\ g e. {x | x:A-1-1-onto->A} /\ h e. {x | x:A-1-1-onto->A}) -> ((f(SymGrp` A)g)(SymGrp` A)h) = (f(SymGrp` A)(g(SymGrp` A)h)))
37		f1oi 3717	. . 3 |- (I |` A):A-1-1-onto->A
38	1, 2	elsymgrn 10401	. . 3 |- ((I |` A) e. {x | x:A-1-1-onto->A} <-> (I |` A):A-1-1-onto->A)
39	37, 38	mpbir 190	. 2 |- (I |` A) e. {x | x:A-1-1-onto->A}
40	1, 2	symgoprval 10404	. . . 4 |- (((I |` A) e. {x | x:A-1-1-onto->A} /\ f e. {x | x:A-1-1-onto->A}) -> ((I |` A)(SymGrp` A)f) = ((I |` A) o. f))
41	39, 40	mpan 695	. . 3 |- (f e. {x | x:A-1-1-onto->A} -> ((I |` A)(SymGrp` A)f) = ((I |` A) o. f))
42		f1of 3689	. . . . 5 |- (f:A-1-1-onto->A -> f:A-->A)
43		fcoi2 3646	. . . . 5 |- (f:A-->A -> ((I |` A) o. f) = f)
44	42, 43	syl 10	. . . 4 |- (f:A-1-1-onto->A -> ((I |` A) o. f) = f)
45	16, 44	sylbi 199	. . 3 |- (f e. {x | x:A-1-1-onto->A} -> ((I |` A) o. f) = f)
46	41, 45	eqtrd 1507	. 2 |- (f e. {x | x:A-1-1-onto->A} -> ((I |` A)(SymGrp` A)f) = f)
47		f1ocnv 3701	. . 3 |- (f:A-1-1-onto->A -> `'f:A-1-1-onto->A)
48	1, 2	elsymgrn 10401	. . 3 |- (`'f e. {x | x:A-1-1-onto->A} <-> `'f:A-1-1-onto->A)
49	47, 16, 48	3imtr4 219	. 2 |- (f e. {x | x:A-1-1-onto->A} -> `'f e. {x | x:A-1-1-onto->A})
50	1, 2	symgoprval 10404	. . . 4 |- ((`'f e. {x | x:A-1-1-onto->A} /\ f e. {x | x:A-1-1-onto->A}) -> (`'f(SymGrp` A)f) = (`'f o. f))
51	49, 50	mpancom 705	. . 3 |- (f e. {x | x:A-1-1-onto->A} -> (`'f(SymGrp` A)f) = (`'f o. f))
52		f1ococnv1 3709	. . . 4 |- (f:A-1-1-onto->A -> (`'f o. f) = (I |` A))
53	16, 52	sylbi 199	. . 3 |- (f e. {x | x:A-1-1-onto->A} -> (`'f o. f) = (I |` A))
54	51, 53	eqtrd 1507	. 2 |- (f e. {x | x:A-1-1-onto->A} -> (`'f(SymGrp` A)f) = (I |` A))
55	8, 9, 36, 39, 46, 49, 54	isgrpi 8042	1 |- (SymGrp` A) e. Grp

Syntax hints:   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811  Icid 2831  `'ccnv 3169   |` cres 3172   o. ccom 3174  -->wf 3178  -1-1-onto->wf1o 3181  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  SymGrpcsymgrp 10399

This theorem is referenced by:  symgidi 10407  symggrp 10408  cayleylem2 10410  cayleylem3 10411

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-