Theorem evpmodpmf1o 19942

Description: The function for performing an even permutation after a fixed odd permutation is one to one onto all odd permutations. (Contributed by SO, 9-Jul-2018.)

Hypotheses

Ref	Expression
evpmodpmf1o.s	|- S = ( SymGrp ` D )
evpmodpmf1o.p	|- P = ( Base ` S )

Assertion

Ref	Expression
evpmodpmf1o	|- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) : (pmEven ` D ) -1-1-onto-> ( P \ (pmEven ` D ) ) )

Distinct variable groups:   S, f   D, f   P, f   f, F

Proof of Theorem evpmodpmf1o

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 790	. . . 4 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> D e. Fin )
2		evpmodpmf1o.s	. . . . . . 7 |- S = ( SymGrp ` D )
3	2	symggrp 17820	. . . . . 6 |- ( D e. Fin -> S e. Grp )
4	3	ad2antrr 762	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> S e. Grp )
5		eldifi 3732	. . . . . 6 |- ( F e. ( P \ (pmEven ` D ) ) -> F e. P )
6	5	ad2antlr 763	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> F e. P )
7		evpmodpmf1o.p	. . . . . . . 8 |- P = ( Base ` S )
8	2, 7	evpmss 19932	. . . . . . 7 |- (pmEven ` D ) C_ P
9	8	sseli 3599	. . . . . 6 |- ( f e. (pmEven ` D ) -> f e. P )
10	9	adantl 482	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> f e. P )
11		eqid 2622	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
12	7, 11	grpcl 17430	. . . . 5 |- ( ( S e. Grp /\ F e. P /\ f e. P ) -> ( F ( +g ` S ) f ) e. P )
13	4, 6, 10, 12	syl3anc 1326	. . . 4 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( F ( +g ` S ) f ) e. P )
14		eqid 2622	. . . . . . . 8 |- (pmSgn ` D ) = (pmSgn ` D )
15		eqid 2622	. . . . . . . 8 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
16	2, 14, 15	psgnghm2 19927	. . . . . . 7 |- ( D e. Fin -> (pmSgn ` D ) e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
17	16	ad2antrr 762	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> (pmSgn ` D ) e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
18		prex 4909	. . . . . . . 8 |- { 1 , -u 1 } e. _V
19		eqid 2622	. . . . . . . . . 10 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
20		cnfldmul 19752	. . . . . . . . . 10 |- x. = ( .r ` ℂfld )
21	19, 20	mgpplusg 18493	. . . . . . . . 9 |- x. = ( +g ` (mulGrp ` ℂfld ) )
22	15, 21	ressplusg 15993	. . . . . . . 8 |- ( { 1 , -u 1 } e. _V -> x. = ( +g ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
23	18, 22	ax-mp 5	. . . . . . 7 |- x. = ( +g ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
24	7, 11, 23	ghmlin 17665	. . . . . 6 |- ( ( (pmSgn ` D ) e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) /\ F e. P /\ f e. P ) -> ( (pmSgn ` D ) ` ( F ( +g ` S ) f ) ) = ( ( (pmSgn ` D ) ` F ) x. ( (pmSgn ` D ) ` f ) ) )
25	17, 6, 10, 24	syl3anc 1326	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( (pmSgn ` D ) ` ( F ( +g ` S ) f ) ) = ( ( (pmSgn ` D ) ` F ) x. ( (pmSgn ` D ) ` f ) ) )
26	2, 7, 14	psgnodpm 19934	. . . . . . . 8 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( (pmSgn ` D ) ` F ) = -u 1 )
27	26	adantr 481	. . . . . . 7 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( (pmSgn ` D ) ` F ) = -u 1 )
28	2, 7, 14	psgnevpm 19935	. . . . . . . 8 |- ( ( D e. Fin /\ f e. (pmEven ` D ) ) -> ( (pmSgn ` D ) ` f ) = 1 )
29	28	adantlr 751	. . . . . . 7 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( (pmSgn ` D ) ` f ) = 1 )
30	27, 29	oveq12d 6668	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( ( (pmSgn ` D ) ` F ) x. ( (pmSgn ` D ) ` f ) ) = ( -u 1 x. 1 ) )
31		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
32	31	mulm1i 10475	. . . . . 6 |- ( -u 1 x. 1 ) = -u 1
33	30, 32	syl6eq 2672	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( ( (pmSgn ` D ) ` F ) x. ( (pmSgn ` D ) ` f ) ) = -u 1 )
34	25, 33	eqtrd 2656	. . . 4 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( (pmSgn ` D ) ` ( F ( +g ` S ) f ) ) = -u 1 )
35	2, 7, 14	psgnodpmr 19936	. . . 4 |- ( ( D e. Fin /\ ( F ( +g ` S ) f ) e. P /\ ( (pmSgn ` D ) ` ( F ( +g ` S ) f ) ) = -u 1 ) -> ( F ( +g ` S ) f ) e. ( P \ (pmEven ` D ) ) )
36	1, 13, 34, 35	syl3anc 1326	. . 3 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( F ( +g ` S ) f ) e. ( P \ (pmEven ` D ) ) )
37		eqid 2622	. . 3 |- ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) = ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) )
38	36, 37	fmptd 6385	. 2 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) : (pmEven ` D ) --> ( P \ (pmEven ` D ) ) )
39	3	ad2antrr 762	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> S e. Grp )
40		eqid 2622	. . . . . . . 8 |- ( invg ` S ) = ( invg ` S )
41	7, 40	grpinvcl 17467	. . . . . . 7 |- ( ( S e. Grp /\ F e. P ) -> ( ( invg ` S ) ` F ) e. P )
42	3, 5, 41	syl2an 494	. . . . . 6 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( ( invg ` S ) ` F ) e. P )
43	42	adantr 481	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( ( invg ` S ) ` F ) e. P )
44		eldifi 3732	. . . . . 6 |- ( g e. ( P \ (pmEven ` D ) ) -> g e. P )
45	44	adantl 482	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> g e. P )
46	7, 11	grpcl 17430	. . . . 5 |- ( ( S e. Grp /\ ( ( invg ` S ) ` F ) e. P /\ g e. P ) -> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) e. P )
47	39, 43, 45, 46	syl3anc 1326	. . . 4 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) e. P )
48	16	ad2antrr 762	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> (pmSgn ` D ) e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
49	7, 11, 23	ghmlin 17665	. . . . . 6 |- ( ( (pmSgn ` D ) e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) /\ ( ( invg ` S ) ` F ) e. P /\ g e. P ) -> ( (pmSgn ` D ) ` ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) = ( ( (pmSgn ` D ) ` ( ( invg ` S ) ` F ) ) x. ( (pmSgn ` D ) ` g ) ) )
50	48, 43, 45, 49	syl3anc 1326	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( (pmSgn ` D ) ` ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) = ( ( (pmSgn ` D ) ` ( ( invg ` S ) ` F ) ) x. ( (pmSgn ` D ) ` g ) ) )
51	2, 7, 40	symginv 17822	. . . . . . . . 9 |- ( F e. P -> ( ( invg ` S ) ` F ) = `' F )
52	5, 51	syl 17	. . . . . . . 8 |- ( F e. ( P \ (pmEven ` D ) ) -> ( ( invg ` S ) ` F ) = `' F )
53	52	ad2antlr 763	. . . . . . 7 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( ( invg ` S ) ` F ) = `' F )
54	53	fveq2d 6195	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( (pmSgn ` D ) ` ( ( invg ` S ) ` F ) ) = ( (pmSgn ` D ) ` `' F ) )
55	2, 7, 14	psgnodpm 19934	. . . . . . 7 |- ( ( D e. Fin /\ g e. ( P \ (pmEven ` D ) ) ) -> ( (pmSgn ` D ) ` g ) = -u 1 )
56	55	adantlr 751	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( (pmSgn ` D ) ` g ) = -u 1 )
57	54, 56	oveq12d 6668	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( ( (pmSgn ` D ) ` ( ( invg ` S ) ` F ) ) x. ( (pmSgn ` D ) ` g ) ) = ( ( (pmSgn ` D ) ` `' F ) x. -u 1 ) )
58		simpll 790	. . . . . . . . 9 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> D e. Fin )
59	5	ad2antlr 763	. . . . . . . . 9 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> F e. P )
60	2, 14, 7	psgninv 19928	. . . . . . . . 9 |- ( ( D e. Fin /\ F e. P ) -> ( (pmSgn ` D ) ` `' F ) = ( (pmSgn ` D ) ` F ) )
61	58, 59, 60	syl2anc 693	. . . . . . . 8 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( (pmSgn ` D ) ` `' F ) = ( (pmSgn ` D ) ` F ) )
62	26	adantr 481	. . . . . . . 8 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( (pmSgn ` D ) ` F ) = -u 1 )
63	61, 62	eqtrd 2656	. . . . . . 7 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( (pmSgn ` D ) ` `' F ) = -u 1 )
64	63	oveq1d 6665	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( ( (pmSgn ` D ) ` `' F ) x. -u 1 ) = ( -u 1 x. -u 1 ) )
65		neg1mulneg1e1 11245	. . . . . 6 |- ( -u 1 x. -u 1 ) = 1
66	64, 65	syl6eq 2672	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( ( (pmSgn ` D ) ` `' F ) x. -u 1 ) = 1 )
67	50, 57, 66	3eqtrd 2660	. . . 4 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( (pmSgn ` D ) ` ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) = 1 )
68	2, 7, 14	psgnevpmb 19933	. . . . 5 |- ( D e. Fin -> ( ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) e. (pmEven ` D ) <-> ( ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) e. P /\ ( (pmSgn ` D ) ` ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) = 1 ) ) )
69	68	ad2antrr 762	. . . 4 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) e. (pmEven ` D ) <-> ( ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) e. P /\ ( (pmSgn ` D ) ` ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) = 1 ) ) )
70	47, 67, 69	mpbir2and 957	. . 3 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) e. (pmEven ` D ) )
71		eqid 2622	. . 3 |- ( g e. ( P \ (pmEven ` D ) ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) = ( g e. ( P \ (pmEven ` D ) ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) )
72	70, 71	fmptd 6385	. 2 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( g e. ( P \ (pmEven ` D ) ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) : ( P \ (pmEven ` D ) ) --> (pmEven ` D ) )
73		eqidd 2623	. . . . 5 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) = ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) )
74		eqidd 2623	. . . . 5 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( g e. ( P \ (pmEven ` D ) ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) = ( g e. ( P \ (pmEven ` D ) ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) )
75		oveq2 6658	. . . . 5 |- ( g = ( F ( +g ` S ) f ) -> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) = ( ( ( invg ` S ) ` F ) ( +g ` S ) ( F ( +g ` S ) f ) ) )
76	36, 73, 74, 75	fmptco 6396	. . . 4 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( ( g e. ( P \ (pmEven ` D ) ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) o. ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) ) = ( f e. (pmEven ` D ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) ( F ( +g ` S ) f ) ) ) )
77		eqid 2622	. . . . . . . . 9 |- ( 0g ` S ) = ( 0g ` S )
78	7, 11, 77, 40	grplinv 17468	. . . . . . . 8 |- ( ( S e. Grp /\ F e. P ) -> ( ( ( invg ` S ) ` F ) ( +g ` S ) F ) = ( 0g ` S ) )
79	4, 6, 78	syl2anc 693	. . . . . . 7 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( ( ( invg ` S ) ` F ) ( +g ` S ) F ) = ( 0g ` S ) )
80	79	oveq1d 6665	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( ( ( ( invg ` S ) ` F ) ( +g ` S ) F ) ( +g ` S ) f ) = ( ( 0g ` S ) ( +g ` S ) f ) )
81	42	adantr 481	. . . . . . 7 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( ( invg ` S ) ` F ) e. P )
82	7, 11	grpass 17431	. . . . . . 7 |- ( ( S e. Grp /\ ( ( ( invg ` S ) ` F ) e. P /\ F e. P /\ f e. P ) ) -> ( ( ( ( invg ` S ) ` F ) ( +g ` S ) F ) ( +g ` S ) f ) = ( ( ( invg ` S ) ` F ) ( +g ` S ) ( F ( +g ` S ) f ) ) )
83	4, 81, 6, 10, 82	syl13anc 1328	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( ( ( ( invg ` S ) ` F ) ( +g ` S ) F ) ( +g ` S ) f ) = ( ( ( invg ` S ) ` F ) ( +g ` S ) ( F ( +g ` S ) f ) ) )
84	7, 11, 77	grplid 17452	. . . . . . 7 |- ( ( S e. Grp /\ f e. P ) -> ( ( 0g ` S ) ( +g ` S ) f ) = f )
85	4, 10, 84	syl2anc 693	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( ( 0g ` S ) ( +g ` S ) f ) = f )
86	80, 83, 85	3eqtr3d 2664	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ f e. (pmEven ` D ) ) -> ( ( ( invg ` S ) ` F ) ( +g ` S ) ( F ( +g ` S ) f ) ) = f )
87	86	mpteq2dva 4744	. . . 4 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( f e. (pmEven ` D ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) ( F ( +g ` S ) f ) ) ) = ( f e. (pmEven ` D ) |-> f ) )
88	76, 87	eqtrd 2656	. . 3 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( ( g e. ( P \ (pmEven ` D ) ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) o. ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) ) = ( f e. (pmEven ` D ) |-> f ) )
89		mptresid 5456	. . 3 |- ( f e. (pmEven ` D ) |-> f ) = ( _I |` (pmEven ` D ) )
90	88, 89	syl6eq 2672	. 2 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( ( g e. ( P \ (pmEven ` D ) ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) o. ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) ) = ( _I |` (pmEven ` D ) ) )
91		oveq2 6658	. . . . 5 |- ( f = ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) -> ( F ( +g ` S ) f ) = ( F ( +g ` S ) ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) )
92	70, 74, 73, 91	fmptco 6396	. . . 4 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) o. ( g e. ( P \ (pmEven ` D ) ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) ) = ( g e. ( P \ (pmEven ` D ) ) |-> ( F ( +g ` S ) ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) ) )
93	7, 11, 77, 40	grprinv 17469	. . . . . . . . 9 |- ( ( S e. Grp /\ F e. P ) -> ( F ( +g ` S ) ( ( invg ` S ) ` F ) ) = ( 0g ` S ) )
94	3, 5, 93	syl2an 494	. . . . . . . 8 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( F ( +g ` S ) ( ( invg ` S ) ` F ) ) = ( 0g ` S ) )
95	94	oveq1d 6665	. . . . . . 7 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( ( F ( +g ` S ) ( ( invg ` S ) ` F ) ) ( +g ` S ) g ) = ( ( 0g ` S ) ( +g ` S ) g ) )
96	95	adantr 481	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( ( F ( +g ` S ) ( ( invg ` S ) ` F ) ) ( +g ` S ) g ) = ( ( 0g ` S ) ( +g ` S ) g ) )
97	7, 11	grpass 17431	. . . . . . 7 |- ( ( S e. Grp /\ ( F e. P /\ ( ( invg ` S ) ` F ) e. P /\ g e. P ) ) -> ( ( F ( +g ` S ) ( ( invg ` S ) ` F ) ) ( +g ` S ) g ) = ( F ( +g ` S ) ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) )
98	39, 59, 43, 45, 97	syl13anc 1328	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( ( F ( +g ` S ) ( ( invg ` S ) ` F ) ) ( +g ` S ) g ) = ( F ( +g ` S ) ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) )
99	7, 11, 77	grplid 17452	. . . . . . 7 |- ( ( S e. Grp /\ g e. P ) -> ( ( 0g ` S ) ( +g ` S ) g ) = g )
100	39, 45, 99	syl2anc 693	. . . . . 6 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( ( 0g ` S ) ( +g ` S ) g ) = g )
101	96, 98, 100	3eqtr3d 2664	. . . . 5 |- ( ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) /\ g e. ( P \ (pmEven ` D ) ) ) -> ( F ( +g ` S ) ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) = g )
102	101	mpteq2dva 4744	. . . 4 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( g e. ( P \ (pmEven ` D ) ) |-> ( F ( +g ` S ) ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) ) = ( g e. ( P \ (pmEven ` D ) ) |-> g ) )
103	92, 102	eqtrd 2656	. . 3 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) o. ( g e. ( P \ (pmEven ` D ) ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) ) = ( g e. ( P \ (pmEven ` D ) ) |-> g ) )
104		mptresid 5456	. . 3 |- ( g e. ( P \ (pmEven ` D ) ) |-> g ) = ( _I |` ( P \ (pmEven ` D ) ) )
105	103, 104	syl6eq 2672	. 2 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) o. ( g e. ( P \ (pmEven ` D ) ) |-> ( ( ( invg ` S ) ` F ) ( +g ` S ) g ) ) ) = ( _I |` ( P \ (pmEven ` D ) ) ) )
106	38, 72, 90, 105	fcof1od 6549	1 |- ( ( D e. Fin /\ F e. ( P \ (pmEven ` D ) ) ) -> ( f e. (pmEven ` D ) |-> ( F ( +g ` S ) f ) ) : (pmEven ` D ) -1-1-onto-> ( P \ (pmEven ` D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   {cpr 4179   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Fincfn 7955   1c1 9937   x. cmul 9941   -ucneg 10267   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   GrpHom cghm 17657   SymGrpcsymg 17797  pmSgncpsgn 17909  pmEvencevpm 17910  mulGrpcmgp 18489  ℂ_fldccnfld 19746

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-ghm 17658  df-gim 17701  df-oppg 17776  df-symg 17798  df-pmtr 17862  df-psgn 17911  df-evpm 17912  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-cnfld 19747

This theorem is referenced by:  mdetralt  20414