Theorem mdetralt 18979

Description: The determinant function is alternating regarding rows: if a matrix has two identical rows, its determinant is 0. Corollary 4.9 in [Lang] p. 515. (Contributed by SO, 10-Jul-2018.) (Proof shortened by AV, 23-Jul-2018.)

Hypotheses

Ref	Expression
mdetralt.d	|- D = ( N maDet R )
mdetralt.a	|- A = ( N Mat R )
mdetralt.b	|- B = ( Base ` A )
mdetralt.z	|- .0. = ( 0g ` R )
mdetralt.r	|- ( ph -> R e. CRing )
mdetralt.x	|- ( ph -> X e. B )
mdetralt.i	|- ( ph -> I e. N )
mdetralt.j	|- ( ph -> J e. N )
mdetralt.ij	|- ( ph -> I =/= J )
mdetralt.eq	|- ( ph -> A. a e. N ( I X a ) = ( J X a ) )

Assertion

Ref	Expression
mdetralt	|- ( ph -> ( D ` X ) = .0. )

Distinct variable groups:   I, a   J, a   N, a   X, a

Allowed substitution hints:   ph( a)   A( a)   B( a)   D( a)   R( a)   .0. ( a)

Proof of Theorem mdetralt

Dummy variables c p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdetralt.x	. . 3 |- ( ph -> X e. B )
2		mdetralt.d	. . . 4 |- D = ( N maDet R )
3		mdetralt.a	. . . 4 |- A = ( N Mat R )
4		mdetralt.b	. . . 4 |- B = ( Base ` A )
5		eqid 2467	. . . 4 |- ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` N ) )
6		eqid 2467	. . . 4 |- ( ZRHom ` R ) = ( ZRHom ` R )
7		eqid 2467	. . . 4 |- (pmSgn ` N ) = (pmSgn ` N )
8		eqid 2467	. . . 4 |- ( .r ` R ) = ( .r ` R )
9		eqid 2467	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
10	2, 3, 4, 5, 6, 7, 8, 9	mdetleib 18958	. . 3 |- ( X e. B -> ( D ` X ) = ( R gsumg ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) )
11	1, 10	syl 16	. 2 |- ( ph -> ( D ` X ) = ( R gsumg ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) )
12		eqid 2467	. . 3 |- ( Base ` R ) = ( Base ` R )
13		eqid 2467	. . 3 |- ( +g ` R ) = ( +g ` R )
14		mdetralt.r	. . . . 5 |- ( ph -> R e. CRing )
15		crngring 17081	. . . . 5 |- ( R e. CRing -> R e. Ring )
16	14, 15	syl 16	. . . 4 |- ( ph -> R e. Ring )
17		ringcmn 17101	. . . 4 |- ( R e. Ring -> R e. CMnd )
18	16, 17	syl 16	. . 3 |- ( ph -> R e. CMnd )
19	3, 4	matrcl 18783	. . . . . 6 |- ( X e. B -> ( N e. Fin /\ R e. _V ) )
20	1, 19	syl 16	. . . . 5 |- ( ph -> ( N e. Fin /\ R e. _V ) )
21	20	simpld 459	. . . 4 |- ( ph -> N e. Fin )
22		eqid 2467	. . . . 5 |- ( SymGrp ` N ) = ( SymGrp ` N )
23	22, 5	symgbasfi 16283	. . . 4 |- ( N e. Fin -> ( Base ` ( SymGrp ` N ) ) e. Fin )
24	21, 23	syl 16	. . 3 |- ( ph -> ( Base ` ( SymGrp ` N ) ) e. Fin )
25	16	adantr 465	. . . 4 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> R e. Ring )
26		zrhpsgnmhm 18489	. . . . . . 7 |- ( ( R e. Ring /\ N e. Fin ) -> ( ( ZRHom ` R ) o. (pmSgn ` N ) ) e. ( ( SymGrp ` N ) MndHom (mulGrp ` R ) ) )
27	16, 21, 26	syl2anc 661	. . . . . 6 |- ( ph -> ( ( ZRHom ` R ) o. (pmSgn ` N ) ) e. ( ( SymGrp ` N ) MndHom (mulGrp ` R ) ) )
28	9, 12	mgpbas 17019	. . . . . . 7 |- ( Base ` R ) = ( Base ` (mulGrp ` R ) )
29	5, 28	mhmf 15844	. . . . . 6 |- ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) e. ( ( SymGrp ` N ) MndHom (mulGrp ` R ) ) -> ( ( ZRHom ` R ) o. (pmSgn ` N ) ) : ( Base ` ( SymGrp ` N ) ) --> ( Base ` R ) )
30	27, 29	syl 16	. . . . 5 |- ( ph -> ( ( ZRHom ` R ) o. (pmSgn ` N ) ) : ( Base ` ( SymGrp ` N ) ) --> ( Base ` R ) )
31	30	ffvelrnda 6032	. . . 4 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) e. ( Base ` R ) )
32	9	crngmgp 17078	. . . . . . 7 |- ( R e. CRing -> (mulGrp ` R ) e. CMnd )
33	14, 32	syl 16	. . . . . 6 |- ( ph -> (mulGrp ` R ) e. CMnd )
34	33	adantr 465	. . . . 5 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> (mulGrp ` R ) e. CMnd )
35	21	adantr 465	. . . . 5 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> N e. Fin )
36	3, 12, 4	matbas2i 18793	. . . . . . . . . 10 |- ( X e. B -> X e. ( ( Base ` R ) ^m ( N X. N ) ) )
37	1, 36	syl 16	. . . . . . . . 9 |- ( ph -> X e. ( ( Base ` R ) ^m ( N X. N ) ) )
38		elmapi 7452	. . . . . . . . 9 |- ( X e. ( ( Base ` R ) ^m ( N X. N ) ) -> X : ( N X. N ) --> ( Base ` R ) )
39	37, 38	syl 16	. . . . . . . 8 |- ( ph -> X : ( N X. N ) --> ( Base ` R ) )
40	39	ad2antrr 725	. . . . . . 7 |- ( ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) /\ c e. N ) -> X : ( N X. N ) --> ( Base ` R ) )
41	22, 5	symgbasf1o 16280	. . . . . . . . . 10 |- ( p e. ( Base ` ( SymGrp ` N ) ) -> p : N -1-1-onto-> N )
42	41	adantl 466	. . . . . . . . 9 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> p : N -1-1-onto-> N )
43		f1of 5822	. . . . . . . . 9 |- ( p : N -1-1-onto-> N -> p : N --> N )
44	42, 43	syl 16	. . . . . . . 8 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> p : N --> N )
45	44	ffvelrnda 6032	. . . . . . 7 |- ( ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) /\ c e. N ) -> ( p ` c ) e. N )
46		simpr 461	. . . . . . 7 |- ( ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) /\ c e. N ) -> c e. N )
47	40, 45, 46	fovrnd 6442	. . . . . 6 |- ( ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) /\ c e. N ) -> ( ( p ` c ) X c ) e. ( Base ` R ) )
48	47	ralrimiva 2881	. . . . 5 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> A. c e. N ( ( p ` c ) X c ) e. ( Base ` R ) )
49	28, 34, 35, 48	gsummptcl 16867	. . . 4 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) e. ( Base ` R ) )
50	12, 8	ringcl 17084	. . . 4 |- ( ( R e. Ring /\ ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) e. ( Base ` R ) /\ ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) e. ( Base ` R ) ) -> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) e. ( Base ` R ) )
51	25, 31, 49, 50	syl3anc 1228	. . 3 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) e. ( Base ` R ) )
52		disjdif 3905	. . . 4 |- ( (pmEven ` N ) i^i ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) = (/)
53	52	a1i 11	. . 3 |- ( ph -> ( (pmEven ` N ) i^i ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) = (/) )
54	22, 5	evpmss 18491	. . . . . 6 |- (pmEven ` N ) C_ ( Base ` ( SymGrp ` N ) )
55		undif 3913	. . . . . 6 |- ( (pmEven ` N ) C_ ( Base ` ( SymGrp ` N ) ) <-> ( (pmEven ` N ) u. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) = ( Base ` ( SymGrp ` N ) ) )
56	54, 55	mpbi 208	. . . . 5 |- ( (pmEven ` N ) u. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) = ( Base ` ( SymGrp ` N ) )
57	56	eqcomi 2480	. . . 4 |- ( Base ` ( SymGrp ` N ) ) = ( (pmEven ` N ) u. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
58	57	a1i 11	. . 3 |- ( ph -> ( Base ` ( SymGrp ` N ) ) = ( (pmEven ` N ) u. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) )
59		eqid 2467	. . 3 |- ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) = ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
60	12, 13, 18, 24, 51, 53, 58, 59	gsummptfidmsplitres 16824	. 2 |- ( ph -> ( R gsumg ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) = ( ( R gsumg ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` (pmEven ` N ) ) ) ( +g ` R ) ( R gsumg ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) ) ) )
61		resmpt 5329	. . . . . . 7 |- ( (pmEven ` N ) C_ ( Base ` ( SymGrp ` N ) ) -> ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` (pmEven ` N ) ) = ( p e. (pmEven ` N ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) )
62	54, 61	ax-mp 5	. . . . . 6 |- ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` (pmEven ` N ) ) = ( p e. (pmEven ` N ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
63	16	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ p e. (pmEven ` N ) ) -> R e. Ring )
64	21	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ p e. (pmEven ` N ) ) -> N e. Fin )
65		simpr 461	. . . . . . . . . 10 |- ( ( ph /\ p e. (pmEven ` N ) ) -> p e. (pmEven ` N ) )
66		eqid 2467	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
67	6, 7, 66	zrhpsgnevpm 18496	. . . . . . . . . 10 |- ( ( R e. Ring /\ N e. Fin /\ p e. (pmEven ` N ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( 1r ` R ) )
68	63, 64, 65, 67	syl3anc 1228	. . . . . . . . 9 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( 1r ` R ) )
69	68	oveq1d 6310	. . . . . . . 8 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) = ( ( 1r ` R ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
70	54	sseli 3505	. . . . . . . . . 10 |- ( p e. (pmEven ` N ) -> p e. ( Base ` ( SymGrp ` N ) ) )
71	70, 49	sylan2 474	. . . . . . . . 9 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) e. ( Base ` R ) )
72	12, 8, 66	ringlidm 17094	. . . . . . . . 9 |- ( ( R e. Ring /\ ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) = ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) )
73	63, 71, 72	syl2anc 661	. . . . . . . 8 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( 1r ` R ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) = ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) )
74	69, 73	eqtrd 2508	. . . . . . 7 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) = ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) )
75	74	mpteq2dva 4539	. . . . . 6 |- ( ph -> ( p e. (pmEven ` N ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) = ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
76	62, 75	syl5eq 2520	. . . . 5 |- ( ph -> ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` (pmEven ` N ) ) = ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
77	76	oveq2d 6311	. . . 4 |- ( ph -> ( R gsumg ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` (pmEven ` N ) ) ) = ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) )
78		difss 3636	. . . . . . . 8 |- ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) C_ ( Base ` ( SymGrp ` N ) )
79		resmpt 5329	. . . . . . . 8 |- ( ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) C_ ( Base ` ( SymGrp ` N ) ) -> ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) = ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) )
80	78, 79	ax-mp 5	. . . . . . 7 |- ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) = ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
81	16	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> R e. Ring )
82	21	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> N e. Fin )
83		simpr 461	. . . . . . . . . . . 12 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
84		eqid 2467	. . . . . . . . . . . . 13 |- ( invg ` R ) = ( invg ` R )
85	6, 7, 66, 5, 84	zrhpsgnodpm 18497	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ N e. Fin /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( ( invg ` R ) ` ( 1r ` R ) ) )
86	81, 82, 83, 85	syl3anc 1228	. . . . . . . . . . 11 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( ( invg ` R ) ` ( 1r ` R ) ) )
87	86	oveq1d 6310	. . . . . . . . . 10 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) = ( ( ( invg ` R ) ` ( 1r ` R ) ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
88		eldifi 3631	. . . . . . . . . . . 12 |- ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) -> p e. ( Base ` ( SymGrp ` N ) ) )
89	88, 49	sylan2 474	. . . . . . . . . . 11 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) e. ( Base ` R ) )
90	12, 8, 66, 84, 81, 89	ringnegl 17112	. . . . . . . . . 10 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> ( ( ( invg ` R ) ` ( 1r ` R ) ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) = ( ( invg ` R ) ` ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
91	87, 90	eqtrd 2508	. . . . . . . . 9 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) = ( ( invg ` R ) ` ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
92	91	mpteq2dva 4539	. . . . . . . 8 |- ( ph -> ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) = ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( ( invg ` R ) ` ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) )
93		eqidd 2468	. . . . . . . . 9 |- ( ph -> ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) = ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
94		ringgrp 17075	. . . . . . . . . . . 12 |- ( R e. Ring -> R e. Grp )
95	16, 94	syl 16	. . . . . . . . . . 11 |- ( ph -> R e. Grp )
96	12, 84	grpinvf 15966	. . . . . . . . . . 11 |- ( R e. Grp -> ( invg ` R ) : ( Base ` R ) --> ( Base ` R ) )
97	95, 96	syl 16	. . . . . . . . . 10 |- ( ph -> ( invg ` R ) : ( Base ` R ) --> ( Base ` R ) )
98	97	feqmptd 5927	. . . . . . . . 9 |- ( ph -> ( invg ` R ) = ( q e. ( Base ` R ) |-> ( ( invg ` R ) ` q ) ) )
99		fveq2 5872	. . . . . . . . 9 |- ( q = ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) -> ( ( invg ` R ) ` q ) = ( ( invg ` R ) ` ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
100	89, 93, 98, 99	fmptco 6065	. . . . . . . 8 |- ( ph -> ( ( invg ` R ) o. ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) = ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( ( invg ` R ) ` ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) )
101	92, 100	eqtr4d 2511	. . . . . . 7 |- ( ph -> ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) = ( ( invg ` R ) o. ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) )
102	80, 101	syl5eq 2520	. . . . . 6 |- ( ph -> ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) = ( ( invg ` R ) o. ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) )
103	102	oveq2d 6311	. . . . 5 |- ( ph -> ( R gsumg ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) ) = ( R gsumg ( ( invg ` R ) o. ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) )
104		mdetralt.z	. . . . . 6 |- .0. = ( 0g ` R )
105		ringabl 17100	. . . . . . 7 |- ( R e. Ring -> R e. Abel )
106	16, 105	syl 16	. . . . . 6 |- ( ph -> R e. Abel )
107		difssd 3637	. . . . . . 7 |- ( ph -> ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) C_ ( Base ` ( SymGrp ` N ) ) )
108		ssfi 7752	. . . . . . 7 |- ( ( ( Base ` ( SymGrp ` N ) ) e. Fin /\ ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) C_ ( Base ` ( SymGrp ` N ) ) ) -> ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) e. Fin )
109	24, 107, 108	syl2anc 661	. . . . . 6 |- ( ph -> ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) e. Fin )
110		eqid 2467	. . . . . 6 |- ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) = ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) )
111	12, 104, 84, 106, 109, 89, 110	gsummptfidminv 16845	. . . . 5 |- ( ph -> ( R gsumg ( ( invg ` R ) o. ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) = ( ( invg ` R ) ` ( R gsumg ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) )
112	89	ralrimiva 2881	. . . . . . . 8 |- ( ph -> A. p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) e. ( Base ` R ) )
113		mdetralt.i	. . . . . . . . . . . 12 |- ( ph -> I e. N )
114		mdetralt.j	. . . . . . . . . . . 12 |- ( ph -> J e. N )
115		prssi 4189	. . . . . . . . . . . 12 |- ( ( I e. N /\ J e. N ) -> { I , J } C_ N )
116	113, 114, 115	syl2anc 661	. . . . . . . . . . 11 |- ( ph -> { I , J } C_ N )
117		mdetralt.ij	. . . . . . . . . . . 12 |- ( ph -> I =/= J )
118		pr2nelem 8394	. . . . . . . . . . . 12 |- ( ( I e. N /\ J e. N /\ I =/= J ) -> { I , J } ~~ 2o )
119	113, 114, 117, 118	syl3anc 1228	. . . . . . . . . . 11 |- ( ph -> { I , J } ~~ 2o )
120		eqid 2467	. . . . . . . . . . . 12 |- (pmTrsp ` N ) = (pmTrsp ` N )
121		eqid 2467	. . . . . . . . . . . 12 |- ran (pmTrsp ` N ) = ran (pmTrsp ` N )
122	120, 121	pmtrrn 16355	. . . . . . . . . . 11 |- ( ( N e. Fin /\ { I , J } C_ N /\ { I , J } ~~ 2o ) -> ( (pmTrsp ` N ) ` { I , J } ) e. ran (pmTrsp ` N ) )
123	21, 116, 119, 122	syl3anc 1228	. . . . . . . . . 10 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) e. ran (pmTrsp ` N ) )
124	22, 5, 121	pmtrodpm 18502	. . . . . . . . . 10 |- ( ( N e. Fin /\ ( (pmTrsp ` N ) ` { I , J } ) e. ran (pmTrsp ` N ) ) -> ( (pmTrsp ` N ) ` { I , J } ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
125	21, 123, 124	syl2anc 661	. . . . . . . . 9 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
126	22, 5	evpmodpmf1o 18501	. . . . . . . . 9 |- ( ( N e. Fin /\ ( (pmTrsp ` N ) ` { I , J } ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> ( q e. (pmEven ` N ) |-> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ) : (pmEven ` N ) -1-1-onto-> ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
127	21, 125, 126	syl2anc 661	. . . . . . . 8 |- ( ph -> ( q e. (pmEven ` N ) |-> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ) : (pmEven ` N ) -1-1-onto-> ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
128	12, 18, 109, 112, 110, 127	gsummptfif1o 16868	. . . . . . 7 |- ( ph -> ( R gsumg ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) = ( R gsumg ( ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) o. ( q e. (pmEven ` N ) |-> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ) ) ) )
129		eleq1 2539	. . . . . . . . . . . . 13 |- ( p = q -> ( p e. (pmEven ` N ) <-> q e. (pmEven ` N ) ) )
130	129	anbi2d 703	. . . . . . . . . . . 12 |- ( p = q -> ( ( ph /\ p e. (pmEven ` N ) ) <-> ( ph /\ q e. (pmEven ` N ) ) ) )
131		oveq2 6303	. . . . . . . . . . . . 13 |- ( p = q -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) )
132	131	eleq1d 2536	. . . . . . . . . . . 12 |- ( p = q -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) <-> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) )
133	130, 132	imbi12d 320	. . . . . . . . . . 11 |- ( p = q -> ( ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) <-> ( ( ph /\ q e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) ) )
134	22	symggrp 16297	. . . . . . . . . . . . . . 15 |- ( N e. Fin -> ( SymGrp ` N ) e. Grp )
135	21, 134	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> ( SymGrp ` N ) e. Grp )
136	135	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( SymGrp ` N ) e. Grp )
137	121, 22, 5	symgtrf 16367	. . . . . . . . . . . . . 14 |- ran (pmTrsp ` N ) C_ ( Base ` ( SymGrp ` N ) )
138	123	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmTrsp ` N ) ` { I , J } ) e. ran (pmTrsp ` N ) )
139	137, 138	sseldi 3507	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmTrsp ` N ) ` { I , J } ) e. ( Base ` ( SymGrp ` N ) ) )
140	70	adantl 466	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. (pmEven ` N ) ) -> p e. ( Base ` ( SymGrp ` N ) ) )
141		eqid 2467	. . . . . . . . . . . . . 14 |- ( +g ` ( SymGrp ` N ) ) = ( +g ` ( SymGrp ` N ) )
142	5, 141	grpcl 15935	. . . . . . . . . . . . 13 |- ( ( ( SymGrp ` N ) e. Grp /\ ( (pmTrsp ` N ) ` { I , J } ) e. ( Base ` ( SymGrp ` N ) ) /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( Base ` ( SymGrp ` N ) ) )
143	136, 139, 140, 142	syl3anc 1228	. . . . . . . . . . . 12 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( Base ` ( SymGrp ` N ) ) )
144		eqid 2467	. . . . . . . . . . . . . . . . 17 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
145	22, 7, 144	psgnghm2 18486	. . . . . . . . . . . . . . . 16 |- ( N e. Fin -> (pmSgn ` N ) e. ( ( SymGrp ` N ) GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
146	21, 145	syl 16	. . . . . . . . . . . . . . 15 |- ( ph -> (pmSgn ` N ) e. ( ( SymGrp ` N ) GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
147	146	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ p e. (pmEven ` N ) ) -> (pmSgn ` N ) e. ( ( SymGrp ` N ) GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
148		prex 4695	. . . . . . . . . . . . . . . 16 |- { 1 , -u 1 } e. _V
149		eqid 2467	. . . . . . . . . . . . . . . . . 18 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
150		cnfldmul 18296	. . . . . . . . . . . . . . . . . 18 |- x. = ( .r ` ℂfld )
151	149, 150	mgpplusg 17017	. . . . . . . . . . . . . . . . 17 |- x. = ( +g ` (mulGrp ` ℂfld ) )
152	144, 151	ressplusg 14614	. . . . . . . . . . . . . . . 16 |- ( { 1 , -u 1 } e. _V -> x. = ( +g ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
153	148, 152	ax-mp 5	. . . . . . . . . . . . . . 15 |- x. = ( +g ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
154	5, 141, 153	ghmlin 16144	. . . . . . . . . . . . . 14 |- ( ( (pmSgn ` N ) e. ( ( SymGrp ` N ) GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) /\ ( (pmTrsp ` N ) ` { I , J } ) e. ( Base ` ( SymGrp ` N ) ) /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( (pmSgn ` N ) ` ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ) = ( ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) x. ( (pmSgn ` N ) ` p ) ) )
155	147, 139, 140, 154	syl3anc 1228	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmSgn ` N ) ` ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ) = ( ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) x. ( (pmSgn ` N ) ` p ) ) )
156	22, 121, 7	psgnpmtr 16408	. . . . . . . . . . . . . . . 16 |- ( ( (pmTrsp ` N ) ` { I , J } ) e. ran (pmTrsp ` N ) -> ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) = -u 1 )
157	138, 156	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) = -u 1 )
158	22, 5, 7	psgnevpm 18494	. . . . . . . . . . . . . . . 16 |- ( ( N e. Fin /\ p e. (pmEven ` N ) ) -> ( (pmSgn ` N ) ` p ) = 1 )
159	21, 158	sylan 471	. . . . . . . . . . . . . . 15 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmSgn ` N ) ` p ) = 1 )
160	157, 159	oveq12d 6313	. . . . . . . . . . . . . 14 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) x. ( (pmSgn ` N ) ` p ) ) = ( -u 1 x. 1 ) )
161		neg1cn 10651	. . . . . . . . . . . . . . 15 |- -u 1 e. CC
162	161	mulid1i 9610	. . . . . . . . . . . . . 14 |- ( -u 1 x. 1 ) = -u 1
163	160, 162	syl6eq 2524	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) x. ( (pmSgn ` N ) ` p ) ) = -u 1 )
164	155, 163	eqtrd 2508	. . . . . . . . . . . 12 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmSgn ` N ) ` ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ) = -u 1 )
165	22, 5, 7	psgnodpmr 18495	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( Base ` ( SymGrp ` N ) ) /\ ( (pmSgn ` N ) ` ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ) = -u 1 ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
166	64, 143, 164, 165	syl3anc 1228	. . . . . . . . . . 11 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
167	133, 166	chvarv 1983	. . . . . . . . . 10 |- ( ( ph /\ q e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
168		eqidd 2468	. . . . . . . . . 10 |- ( ph -> ( q e. (pmEven ` N ) |-> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ) = ( q e. (pmEven ` N ) |-> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ) )
169		fveq1 5871	. . . . . . . . . . . . 13 |- ( p = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) -> ( p ` c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) )
170	169	oveq1d 6310	. . . . . . . . . . . 12 |- ( p = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) -> ( ( p ` c ) X c ) = ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) X c ) )
171	170	mpteq2dv 4540	. . . . . . . . . . 11 |- ( p = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) -> ( c e. N |-> ( ( p ` c ) X c ) ) = ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) X c ) ) )
172	171	oveq2d 6311	. . . . . . . . . 10 |- ( p = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) -> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) = ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) X c ) ) ) )
173	167, 168, 93, 172	fmptco 6065	. . . . . . . . 9 |- ( ph -> ( ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) o. ( q e. (pmEven ` N ) |-> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ) ) = ( q e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) X c ) ) ) ) )
174		oveq2 6303	. . . . . . . . . . . . . . 15 |- ( q = p -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) )
175	174	fveq1d 5874	. . . . . . . . . . . . . 14 |- ( q = p -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) )
176	175	oveq1d 6310	. . . . . . . . . . . . 13 |- ( q = p -> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) X c ) = ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) X c ) )
177	176	mpteq2dv 4540	. . . . . . . . . . . 12 |- ( q = p -> ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) X c ) ) = ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) X c ) ) )
178	177	oveq2d 6311	. . . . . . . . . . 11 |- ( q = p -> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) X c ) ) ) = ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) X c ) ) ) )
179	178	cbvmptv 4544	. . . . . . . . . 10 |- ( q e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) X c ) ) ) ) = ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) X c ) ) ) )
180	179	a1i 11	. . . . . . . . 9 |- ( ph -> ( q e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) X c ) ) ) ) = ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) X c ) ) ) ) )
181	139	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( (pmTrsp ` N ) ` { I , J } ) e. ( Base ` ( SymGrp ` N ) ) )
182	140	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> p e. ( Base ` ( SymGrp ` N ) ) )
183	22, 5, 141	symgov 16287	. . . . . . . . . . . . . . . . 17 |- ( ( ( (pmTrsp ` N ) ` { I , J } ) e. ( Base ` ( SymGrp ` N ) ) /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) = ( ( (pmTrsp ` N ) ` { I , J } ) o. p ) )
184	181, 182, 183	syl2anc 661	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) = ( ( (pmTrsp ` N ) ` { I , J } ) o. p ) )
185	184	fveq1d 5874	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) o. p ) ` c ) )
186	70, 44	sylan2 474	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. (pmEven ` N ) ) -> p : N --> N )
187		fvco3 5951	. . . . . . . . . . . . . . . 16 |- ( ( p : N --> N /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) o. p ) ` c ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) )
188	186, 187	sylan 471	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) o. p ) ` c ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) )
189	185, 188	eqtrd 2508	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) )
190	189	oveq1d 6310	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) X c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) )
191	120	pmtrprfv 16351	. . . . . . . . . . . . . . . . . . 19 |- ( ( N e. Fin /\ ( I e. N /\ J e. N /\ I =/= J ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) = J )
192	21, 113, 114, 117, 191	syl13anc 1230	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) = J )
193	192	ad2antrr 725	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) = J )
194	193	oveq1d 6310	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) X c ) = ( J X c ) )
195		simpr 461	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> c e. N )
196		mdetralt.eq	. . . . . . . . . . . . . . . . . 18 |- ( ph -> A. a e. N ( I X a ) = ( J X a ) )
197	196	ad2antrr 725	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> A. a e. N ( I X a ) = ( J X a ) )
198		oveq2 6303	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> ( I X a ) = ( I X c ) )
199		oveq2 6303	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> ( J X a ) = ( J X c ) )
200	198, 199	eqeq12d 2489	. . . . . . . . . . . . . . . . . 18 |- ( a = c -> ( ( I X a ) = ( J X a ) <-> ( I X c ) = ( J X c ) ) )
201	200	rspcv 3215	. . . . . . . . . . . . . . . . 17 |- ( c e. N -> ( A. a e. N ( I X a ) = ( J X a ) -> ( I X c ) = ( J X c ) ) )
202	195, 197, 201	sylc 60	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( I X c ) = ( J X c ) )
203	194, 202	eqtr4d 2511	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) X c ) = ( I X c ) )
204		fveq2 5872	. . . . . . . . . . . . . . . . 17 |- ( ( p ` c ) = I -> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) )
205	204	oveq1d 6310	. . . . . . . . . . . . . . . 16 |- ( ( p ` c ) = I -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) X c ) )
206		oveq1 6302	. . . . . . . . . . . . . . . 16 |- ( ( p ` c ) = I -> ( ( p ` c ) X c ) = ( I X c ) )
207	205, 206	eqeq12d 2489	. . . . . . . . . . . . . . 15 |- ( ( p ` c ) = I -> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( p ` c ) X c ) <-> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) X c ) = ( I X c ) ) )
208	203, 207	syl5ibrcom 222	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( p ` c ) = I -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( p ` c ) X c ) ) )
209		prcom 4111	. . . . . . . . . . . . . . . . . . . . . . 23 |- { I , J } = { J , I }
210	209	fveq2i 5875	. . . . . . . . . . . . . . . . . . . . . 22 |- ( (pmTrsp ` N ) ` { I , J } ) = ( (pmTrsp ` N ) ` { J , I } )
211	210	fveq1i 5873	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) = ( ( (pmTrsp ` N ) ` { J , I } ) ` J )
212	117	necomd 2738	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> J =/= I )
213	120	pmtrprfv 16351	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( N e. Fin /\ ( J e. N /\ I e. N /\ J =/= I ) ) -> ( ( (pmTrsp ` N ) ` { J , I } ) ` J ) = I )
214	21, 114, 113, 212, 213	syl13anc 1230	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> ( ( (pmTrsp ` N ) ` { J , I } ) ` J ) = I )
215	211, 214	syl5eq 2520	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) = I )
216	215	oveq1d 6310	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) = ( I X c ) )
217	216	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) = ( I X c ) )
218	217, 202	eqtrd 2508	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) = ( J X c ) )
219		fveq2 5872	. . . . . . . . . . . . . . . . . . 19 |- ( ( p ` c ) = J -> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) )
220	219	oveq1d 6310	. . . . . . . . . . . . . . . . . 18 |- ( ( p ` c ) = J -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) )
221		oveq1 6302	. . . . . . . . . . . . . . . . . 18 |- ( ( p ` c ) = J -> ( ( p ` c ) X c ) = ( J X c ) )
222	220, 221	eqeq12d 2489	. . . . . . . . . . . . . . . . 17 |- ( ( p ` c ) = J -> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( p ` c ) X c ) <-> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) = ( J X c ) ) )
223	218, 222	syl5ibrcom 222	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( p ` c ) = J -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( p ` c ) X c ) ) )
224	223	a1dd 46	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( p ` c ) = J -> ( ( p ` c ) =/= I -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( p ` c ) X c ) ) ) )
225		neanior 2792	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( p ` c ) =/= J /\ ( p ` c ) =/= I ) <-> -. ( ( p ` c ) = J \/ ( p ` c ) = I ) )
226		elpri 4053	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( p ` c ) e. { I , J } -> ( ( p ` c ) = I \/ ( p ` c ) = J ) )
227	226	orcomd 388	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( p ` c ) e. { I , J } -> ( ( p ` c ) = J \/ ( p ` c ) = I ) )
228	227	con3i 135	. . . . . . . . . . . . . . . . . . . . 21 |- ( -. ( ( p ` c ) = J \/ ( p ` c ) = I ) -> -. ( p ` c ) e. { I , J } )
229	225, 228	sylbi 195	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> -. ( p ` c ) e. { I , J } )
230	229	3adant1 1014	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) /\ ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> -. ( p ` c ) e. { I , J } )
231	120	pmtrmvd 16354	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( N e. Fin /\ { I , J } C_ N /\ { I , J } ~~ 2o ) -> dom ( ( (pmTrsp ` N ) ` { I , J } ) \ _I ) = { I , J } )
232	21, 116, 119, 231	syl3anc 1228	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> dom ( ( (pmTrsp ` N ) ` { I , J } ) \ _I ) = { I , J } )
233	232	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> dom ( ( (pmTrsp ` N ) ` { I , J } ) \ _I ) = { I , J } )
234	233	3ad2ant1 1017	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) /\ ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> dom ( ( (pmTrsp ` N ) ` { I , J } ) \ _I ) = { I , J } )
235	230, 234	neleqtrrd 2580	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) /\ ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> -. ( p ` c ) e. dom ( ( (pmTrsp ` N ) ` { I , J } ) \ _I ) )
236	120	pmtrf 16353	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N e. Fin /\ { I , J } C_ N /\ { I , J } ~~ 2o ) -> ( (pmTrsp ` N ) ` { I , J } ) : N --> N )
237	21, 116, 119, 236	syl3anc 1228	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) : N --> N )
238		ffn 5737	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( (pmTrsp ` N ) ` { I , J } ) : N --> N -> ( (pmTrsp ` N ) ` { I , J } ) Fn N )
239	237, 238	syl 16	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) Fn N )
240	239	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( (pmTrsp ` N ) ` { I , J } ) Fn N )
241	186	ffvelrnda 6032	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( p ` c ) e. N )
242		fnelnfp 6102	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( (pmTrsp ` N ) ` { I , J } ) Fn N /\ ( p ` c ) e. N ) -> ( ( p ` c ) e. dom ( ( (pmTrsp ` N ) ` { I , J } ) \ _I ) <-> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) =/= ( p ` c ) ) )
243	240, 241, 242	syl2anc 661	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( p ` c ) e. dom ( ( (pmTrsp ` N ) ` { I , J } ) \ _I ) <-> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) =/= ( p ` c ) ) )
244	243	3ad2ant1 1017	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) /\ ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> ( ( p ` c ) e. dom ( ( (pmTrsp ` N ) ` { I , J } ) \ _I ) <-> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) =/= ( p ` c ) ) )
245	244	necon2bbid 2723	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) /\ ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) = ( p ` c ) <-> -. ( p ` c ) e. dom ( ( (pmTrsp ` N ) ` { I , J } ) \ _I ) ) )
246	235, 245	mpbird 232	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) /\ ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) = ( p ` c ) )
247	246	oveq1d 6310	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) /\ ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( p ` c ) X c ) )
248	247	3exp 1195	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( p ` c ) =/= J -> ( ( p ` c ) =/= I -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( p ` c ) X c ) ) ) )
249	224, 248	pm2.61dne 2784	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( p ` c ) =/= I -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( p ` c ) X c ) ) )
250	208, 249	pm2.61dne 2784	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( p ` c ) X c ) )
251	190, 250	eqtrd 2508	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) X c ) = ( ( p ` c ) X c ) )
252	251	mpteq2dva 4539	. . . . . . . . . . 11 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) X c ) ) = ( c e. N |-> ( ( p ` c ) X c ) ) )
253	252	oveq2d 6311	. . . . . . . . . 10 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) X c ) ) ) = ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) )
254	253	mpteq2dva 4539	. . . . . . . . 9 |- ( ph -> ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) X c ) ) ) ) = ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
255	173, 180, 254	3eqtrd 2512	. . . . . . . 8 |- ( ph -> ( ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) o. ( q e. (pmEven ` N ) |-> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ) ) = ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) )
256	255	oveq2d 6311	. . . . . . 7 |- ( ph -> ( R gsumg ( ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) o. ( q e. (pmEven ` N ) |-> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ) ) ) = ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) )
257	128, 256	eqtrd 2508	. . . . . 6 |- ( ph -> ( R gsumg ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) = ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) )
258	257	fveq2d 5876	. . . . 5 |- ( ph -> ( ( invg ` R ) ` ( R gsumg ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) = ( ( invg ` R ) ` ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) )
259	103, 111, 258	3eqtrd 2512	. . . 4 |- ( ph -> ( R gsumg ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) ) = ( ( invg ` R ) ` ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) )
260	77, 259	oveq12d 6313	. . 3 |- ( ph -> ( ( R gsumg ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` (pmEven ` N ) ) ) ( +g ` R ) ( R gsumg ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) ) ) = ( ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ( +g ` R ) ( ( invg ` R ) ` ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) ) )
261	54	a1i 11	. . . . . 6 |- ( ph -> (pmEven ` N ) C_ ( Base ` ( SymGrp ` N ) ) )
262		ssfi 7752	. . . . . 6 |- ( ( ( Base ` ( SymGrp ` N ) ) e. Fin /\ (pmEven ` N ) C_ ( Base ` ( SymGrp ` N ) ) ) -> (pmEven ` N ) e. Fin )
263	24, 261, 262	syl2anc 661	. . . . 5 |- ( ph -> (pmEven ` N ) e. Fin )
264	71	ralrimiva 2881	. . . . 5 |- ( ph -> A. p e. (pmEven ` N ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) e. ( Base ` R ) )
265	12, 18, 263, 264	gsummptcl 16867	. . . 4 |- ( ph -> ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) e. ( Base ` R ) )
266	12, 13, 104, 84	grprinv 15969	. . . 4 |- ( ( R e. Grp /\ ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) e. ( Base ` R ) ) -> ( ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ( +g ` R ) ( ( invg ` R ) ` ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) ) = .0. )
267	95, 265, 266	syl2anc 661	. . 3 |- ( ph -> ( ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ( +g ` R ) ( ( invg ` R ) ` ( R gsumg ( p e. (pmEven ` N ) |-> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) ) ) = .0. )
268	260, 267	eqtrd 2508	. 2 |- ( ph -> ( ( R gsumg ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` (pmEven ` N ) ) ) ( +g ` R ) ( R gsumg ( ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) ) ) |` ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) ) ) = .0. )
269	11, 60, 268	3eqtrd 2512	1 |- ( ph -> ( D ` X ) = .0. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2817   _Vcvv 3118   \ cdif 3478   u. cun 3479   i^i cin 3480   C_ wss 3481   (/)c0 3790   {cpr 4035   class class class wbr 4453   |-> cmpt 4511   _I cid 4796   X. cxp 5003   dom cdm 5005   ran crn 5006   |` cres 5007   o. ccom 5009   Fn wfn 5589   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   2oc2o 7136   ^m cmap 7432   ~~ cen 7525   Fincfn 7528   1c1 9505   x. cmul 9509   -ucneg 9818   Basecbs 14507   ↾_s cress 14508   +g cplusg 14572   .rcmulr 14573   0gc0g 14712   gsum_g cgsu 14713   MndHom cmhm 15837   Grpcgrp 15925   invgcminusg 15926   GrpHom cghm 16136   SymGrpcsymg 16274  pmTrspcpmtr 16339  pmSgncpsgn 16387  pmEvencevpm 16388  CMndccmn 16671   Abelcabl 16672  mulGrpcmgp 17013   1rcur 17025   Ringcrg 17070   CRingccrg 17071  ℂ_fldccnfld 18290   ZRHomczrh 18406   Mat cmat 18778   maDet cmdat 18955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-addf 9583  ax-mulf 9584

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-word 12523  df-concat 12525  df-s1 12526  df-substr 12527  df-splice 12528  df-reverse 12529  df-s2 12793  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-0g 14714  df-gsum 14715  df-prds 14720  df-pws 14722  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-mulg 15932  df-subg 16070  df-ghm 16137  df-gim 16179  df-cntz 16227  df-oppg 16253  df-symg 16275  df-pmtr 16340  df-psgn 16389  df-evpm 16390  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-cring 17073  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-rnghom 17236  df-drng 17269  df-subrg 17298  df-sra 17689  df-rgmod 17690  df-cnfld 18291  df-zring 18359  df-zrh 18410  df-dsmm 18632  df-frlm 18647  df-mat 18779  df-mdet 18956

This theorem is referenced by:  mdetralt2  18980  mdetuni0  18992  mdetmul  18994