Theorem mdetralt 18980

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 2441	. . . 4 |- ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` N ) )
6		eqid 2441	. . . 4 |- ( ZRHom ` R ) = ( ZRHom ` R )
7		eqid 2441	. . . 4 |- (pmSgn ` N ) = (pmSgn ` N )
8		eqid 2441	. . . 4 |- ( .r ` R ) = ( .r ` R )
9		eqid 2441	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
10	2, 3, 4, 5, 6, 7, 8, 9	mdetleib 18959	. . 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 2441	. . 3 |- ( Base ` R ) = ( Base ` R )
13		eqid 2441	. . 3 |- ( +g ` R ) = ( +g ` R )
14		mdetralt.r	. . . . 5 |- ( ph -> R e. CRing )
15		crngring 17080	. . . . 5 |- ( R e. CRing -> R e. Ring )
16	14, 15	syl 16	. . . 4 |- ( ph -> R e. Ring )
17		ringcmn 17100	. . . 4 |- ( R e. Ring -> R e. CMnd )
18	16, 17	syl 16	. . 3 |- ( ph -> R e. CMnd )
19	3, 4	matrcl 18784	. . . . . 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 2441	. . . . 5 |- ( SymGrp ` N ) = ( SymGrp ` N )
23	22, 5	symgbasfi 16282	. . . 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 18490	. . . . . . 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 17018	. . . . . . 7 |- ( Base ` R ) = ( Base ` (mulGrp ` R ) )
29	5, 28	mhmf 15842	. . . . . 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 6013	. . . 4 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) e. ( Base ` R ) )
32	9	crngmgp 17077	. . . . . . 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 18794	. . . . . . . . . 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 7439	. . . . . . . . 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 16279	. . . . . . . . . 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 5803	. . . . . . . . 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 6013	. . . . . . 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 6429	. . . . . 6 |- ( ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) /\ c e. N ) -> ( ( p ` c ) X c ) e. ( Base ` R ) )
48	47	ralrimiva 2855	. . . . 5 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> A. c e. N ( ( p ` c ) X c ) e. ( Base ` R ) )
49	28, 34, 35, 48	gsummptcl 16865	. . . 4 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) e. ( Base ` R ) )
50	12, 8	ringcl 17083	. . . 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 1227	. . 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 3883	. . . 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 18492	. . . . . 6 |- (pmEven ` N ) C_ ( Base ` ( SymGrp ` N ) )
55		undif 3891	. . . . . 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 2454	. . . 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 2441	. . 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 16822	. 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 5310	. . . . . . 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 2441	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
67	6, 7, 66	zrhpsgnevpm 18497	. . . . . . . . . 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 1227	. . . . . . . . 9 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( 1r ` R ) )
69	68	oveq1d 6293	. . . . . . . 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 3483	. . . . . . . . . 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 17093	. . . . . . . . 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 2482	. . . . . . 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 4520	. . . . . 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 2494	. . . . 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 6294	. . . 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 3614	. . . . . . . 8 |- ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) C_ ( Base ` ( SymGrp ` N ) )
79		resmpt 5310	. . . . . . . 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 2441	. . . . . . . . . . . . 13 |- ( invg ` R ) = ( invg ` R )
85	6, 7, 66, 5, 84	zrhpsgnodpm 18498	. . . . . . . . . . . 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 1227	. . . . . . . . . . 11 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( ( invg ` R ) ` ( 1r ` R ) ) )
87	86	oveq1d 6293	. . . . . . . . . 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 3609	. . . . . . . . . . . 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 17111	. . . . . . . . . 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 2482	. . . . . . . . 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 4520	. . . . . . . 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 2442	. . . . . . . . 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 17074	. . . . . . . . . . . 12 |- ( R e. Ring -> R e. Grp )
95	16, 94	syl 16	. . . . . . . . . . 11 |- ( ph -> R e. Grp )
96	12, 84	grpinvf 15965	. . . . . . . . . . 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 5908	. . . . . . . . 9 |- ( ph -> ( invg ` R ) = ( q e. ( Base ` R ) |-> ( ( invg ` R ) ` q ) ) )
99		fveq2 5853	. . . . . . . . 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 6046	. . . . . . . 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 2485	. . . . . . 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 2494	. . . . . 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 6294	. . . . 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 17099	. . . . . . 7 |- ( R e. Ring -> R e. Abel )
106	16, 105	syl 16	. . . . . 6 |- ( ph -> R e. Abel )
107		difssd 3615	. . . . . . 7 |- ( ph -> ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) C_ ( Base ` ( SymGrp ` N ) ) )
108		ssfi 7739	. . . . . . 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 2441	. . . . . 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 16843	. . . . 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 2855	. . . . . . . 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 4168	. . . . . . . . . . . 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 8382	. . . . . . . . . . . 12 |- ( ( I e. N /\ J e. N /\ I =/= J ) -> { I , J } ~~ 2o )
119	113, 114, 117, 118	syl3anc 1227	. . . . . . . . . . 11 |- ( ph -> { I , J } ~~ 2o )
120		eqid 2441	. . . . . . . . . . . 12 |- (pmTrsp ` N ) = (pmTrsp ` N )
121		eqid 2441	. . . . . . . . . . . 12 |- ran (pmTrsp ` N ) = ran (pmTrsp ` N )
122	120, 121	pmtrrn 16353	. . . . . . . . . . 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 1227	. . . . . . . . . 10 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) e. ran (pmTrsp ` N ) )
124	22, 5, 121	pmtrodpm 18503	. . . . . . . . . 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 18502	. . . . . . . . 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 16866	. . . . . . 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 2513	. . . . . . . . . . . . 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 6286	. . . . . . . . . . . . 13 |- ( p = q -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) )
132	131	eleq1d 2510	. . . . . . . . . . . 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 16296	. . . . . . . . . . . . . . 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 16365	. . . . . . . . . . . . . 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 3485	. . . . . . . . . . . . 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 2441	. . . . . . . . . . . . . 14 |- ( +g ` ( SymGrp ` N ) ) = ( +g ` ( SymGrp ` N ) )
142	5, 141	grpcl 15934	. . . . . . . . . . . . 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 1227	. . . . . . . . . . . 12 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( Base ` ( SymGrp ` N ) ) )
144		eqid 2441	. . . . . . . . . . . . . . . . 17 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
145	22, 7, 144	psgnghm2 18487	. . . . . . . . . . . . . . . 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 4676	. . . . . . . . . . . . . . . 16 |- { 1 , -u 1 } e. _V
149		eqid 2441	. . . . . . . . . . . . . . . . . 18 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
150		cnfldmul 18297	. . . . . . . . . . . . . . . . . 18 |- x. = ( .r ` ℂfld )
151	149, 150	mgpplusg 17016	. . . . . . . . . . . . . . . . 17 |- x. = ( +g ` (mulGrp ` ℂfld ) )
152	144, 151	ressplusg 14613	. . . . . . . . . . . . . . . 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 16143	. . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . 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 16406	. . . . . . . . . . . . . . . 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 18495	. . . . . . . . . . . . . . . 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 6296	. . . . . . . . . . . . . 14 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) x. ( (pmSgn ` N ) ` p ) ) = ( -u 1 x. 1 ) )
161		neg1cn 10642	. . . . . . . . . . . . . . 15 |- -u 1 e. CC
162	161	mulid1i 9598	. . . . . . . . . . . . . 14 |- ( -u 1 x. 1 ) = -u 1
163	160, 162	syl6eq 2498	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) x. ( (pmSgn ` N ) ` p ) ) = -u 1 )
164	155, 163	eqtrd 2482	. . . . . . . . . . . 12 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmSgn ` N ) ` ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ) = -u 1 )
165	22, 5, 7	psgnodpmr 18496	. . . . . . . . . . . 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 1227	. . . . . . . . . . 11 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
167	133, 166	chvarv 1998	. . . . . . . . . 10 |- ( ( ph /\ q e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
168		eqidd 2442	. . . . . . . . . 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 5852	. . . . . . . . . . . . 13 |- ( p = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) -> ( p ` c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) )
170	169	oveq1d 6293	. . . . . . . . . . . 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 4521	. . . . . . . . . . 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 6294	. . . . . . . . . 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 6046	. . . . . . . . 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 6286	. . . . . . . . . . . . . . 15 |- ( q = p -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) )
175	174	fveq1d 5855	. . . . . . . . . . . . . 14 |- ( q = p -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) )
176	175	oveq1d 6293	. . . . . . . . . . . . 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 4521	. . . . . . . . . . . 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 6294	. . . . . . . . . . 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 4525	. . . . . . . . . 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 16286	. . . . . . . . . . . . . . . . 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 5855	. . . . . . . . . . . . . . 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 5932	. . . . . . . . . . . . . . . 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 2482	. . . . . . . . . . . . . 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 6293	. . . . . . . . . . . . 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 16349	. . . . . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . . . . 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 6293	. . . . . . . . . . . . . . . 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 6286	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> ( I X a ) = ( I X c ) )
199		oveq2 6286	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> ( J X a ) = ( J X c ) )
200	198, 199	eqeq12d 2463	. . . . . . . . . . . . . . . . . 18 |- ( a = c -> ( ( I X a ) = ( J X a ) <-> ( I X c ) = ( J X c ) ) )
201	200	rspcv 3190	. . . . . . . . . . . . . . . . 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 2485	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) X c ) = ( I X c ) )
204		fveq2 5853	. . . . . . . . . . . . . . . . 17 |- ( ( p ` c ) = I -> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) )
205	204	oveq1d 6293	. . . . . . . . . . . . . . . 16 |- ( ( p ` c ) = I -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) X c ) )
206		oveq1 6285	. . . . . . . . . . . . . . . 16 |- ( ( p ` c ) = I -> ( ( p ` c ) X c ) = ( I X c ) )
207	205, 206	eqeq12d 2463	. . . . . . . . . . . . . . 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 4090	. . . . . . . . . . . . . . . . . . . . . . 23 |- { I , J } = { J , I }
210	209	fveq2i 5856	. . . . . . . . . . . . . . . . . . . . . 22 |- ( (pmTrsp ` N ) ` { I , J } ) = ( (pmTrsp ` N ) ` { J , I } )
211	210	fveq1i 5854	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) = ( ( (pmTrsp ` N ) ` { J , I } ) ` J )
212	117	necomd 2712	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> J =/= I )
213	120	pmtrprfv 16349	. . . . . . . . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> ( ( (pmTrsp ` N ) ` { J , I } ) ` J ) = I )
215	211, 214	syl5eq 2494	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) = I )
216	215	oveq1d 6293	. . . . . . . . . . . . . . . . . . 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 2482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) = ( J X c ) )
219		fveq2 5853	. . . . . . . . . . . . . . . . . . 19 |- ( ( p ` c ) = J -> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) )
220	219	oveq1d 6293	. . . . . . . . . . . . . . . . . 18 |- ( ( p ` c ) = J -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) )
221		oveq1 6285	. . . . . . . . . . . . . . . . . 18 |- ( ( p ` c ) = J -> ( ( p ` c ) X c ) = ( J X c ) )
222	220, 221	eqeq12d 2463	. . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( p ` c ) =/= J /\ ( p ` c ) =/= I ) <-> -. ( ( p ` c ) = J \/ ( p ` c ) = I ) )
226		elpri 4031	. . . . . . . . . . . . . . . . . . . . . . 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 1013	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) /\ ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> -. ( p ` c ) e. { I , J } )
231	120	pmtrmvd 16352	. . . . . . . . . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . . . . . . . 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 1016	. . . . . . . . . . . . . . . . . . 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 2554	. . . . . . . . . . . . . . . . . 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 16351	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N e. Fin /\ { I , J } C_ N /\ { I , J } ~~ 2o ) -> ( (pmTrsp ` N ) ` { I , J } ) : N --> N )
237	21, 116, 119, 236	syl3anc 1227	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) : N --> N )
238		ffn 5718	. . . . . . . . . . . . . . . . . . . . . . 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 6013	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( p ` c ) e. N )
242		fnelnfp 6083	. . . . . . . . . . . . . . . . . . . . 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 1016	. . . . . . . . . . . . . . . . . . 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 2697	. . . . . . . . . . . . . . . . . 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 6293	. . . . . . . . . . . . . . . 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 1194	. . . . . . . . . . . . . . 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 2758	. . . . . . . . . . . . . 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 2758	. . . . . . . . . . . . 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 2482	. . . . . . . . . . . 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 4520	. . . . . . . . . . 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 6294	. . . . . . . . . 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 4520	. . . . . . . . 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 2486	. . . . . . . 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 6294	. . . . . . 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 2482	. . . . . 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 5857	. . . . 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 2486	. . . 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 6296	. . 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 7739	. . . . . 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 2855	. . . . 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 16865	. . . 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 15968	. . . 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 2482	. 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 2486	1 |- ( ph -> ( D ` X ) = .0. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 972   = wceq 1381   e. wcel 1802   =/= wne 2636   A.wral 2791   _Vcvv 3093   \ cdif 3456   u. cun 3457   i^i cin 3458   C_ wss 3459   (/)c0 3768   {cpr 4013   class class class wbr 4434   |-> cmpt 4492   _I cid 4777   X. cxp 4984   dom cdm 4986   ran crn 4987   |` cres 4988   o. ccom 4990   Fn wfn 5570   -->wf 5571   -1-1-onto->wf1o 5574   ` cfv 5575  (class class class)co 6278   2oc2o 7123   ^m cmap 7419   ~~ cen 7512   Fincfn 7515   1c1 9493   x. cmul 9497   -ucneg 9808   Basecbs 14506   ↾_s cress 14507   +g cplusg 14571   .rcmulr 14572   0gc0g 14711   gsum_g cgsu 14712   MndHom cmhm 15835   Grpcgrp 15924   invgcminusg 15925   GrpHom cghm 16135   SymGrpcsymg 16273  pmTrspcpmtr 16337  pmSgncpsgn 16385  pmEvencevpm 16386  CMndccmn 16669   Abelcabl 16670  mulGrpcmgp 17012   1rcur 17024   Ringcrg 17069   CRingccrg 17070  ℂ_fldccnfld 18291   ZRHomczrh 18407   Mat cmat 18779   maDet cmdat 18956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-addf 9571  ax-mulf 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-xor 1363  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-ot 4020  df-uni 4232  df-int 4269  df-iun 4314  df-iin 4315  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-se 4826  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-isom 5584  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6522  df-om 6683  df-1st 6782  df-2nd 6783  df-supp 6901  df-tpos 6954  df-recs 7041  df-rdg 7075  df-1o 7129  df-2o 7130  df-oadd 7133  df-er 7310  df-map 7421  df-pm 7422  df-ixp 7469  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-fsupp 7829  df-sup 7900  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-div 10210  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-7 10602  df-8 10603  df-9 10604  df-10 10605  df-n0 10799  df-z 10868  df-dec 10982  df-uz 11088  df-rp 11227  df-fz 11679  df-fzo 11801  df-seq 12084  df-exp 12143  df-hash 12382  df-word 12518  df-concat 12520  df-s1 12521  df-substr 12522  df-splice 12523  df-reverse 12524  df-s2 12789  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-hom 14595  df-cco 14596  df-0g 14713  df-gsum 14714  df-prds 14719  df-pws 14721  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-mhm 15837  df-submnd 15838  df-grp 15928  df-minusg 15929  df-mulg 15931  df-subg 16069  df-ghm 16136  df-gim 16178  df-cntz 16226  df-oppg 16252  df-symg 16274  df-pmtr 16338  df-psgn 16387  df-evpm 16388  df-cmn 16671  df-abl 16672  df-mgp 17013  df-ur 17025  df-ring 17071  df-cring 17072  df-oppr 17143  df-dvdsr 17161  df-unit 17162  df-invr 17192  df-dvr 17203  df-rnghom 17235  df-drng 17269  df-subrg 17298  df-sra 17689  df-rgmod 17690  df-cnfld 18292  df-zring 18360  df-zrh 18411  df-dsmm 18633  df-frlm 18648  df-mat 18780  df-mdet 18957

This theorem is referenced by:  mdetralt2  18981  mdetuni0  18993  mdetmul  18995