Theorem mdetralt 19625

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 2423	. . . 4 |- ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` N ) )
6		eqid 2423	. . . 4 |- ( ZRHom ` R ) = ( ZRHom ` R )
7		eqid 2423	. . . 4 |- (pmSgn ` N ) = (pmSgn ` N )
8		eqid 2423	. . . 4 |- ( .r ` R ) = ( .r ` R )
9		eqid 2423	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
10	2, 3, 4, 5, 6, 7, 8, 9	mdetleib 19604	. . 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 17	. 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 2423	. . 3 |- ( Base ` R ) = ( Base ` R )
13		eqid 2423	. . 3 |- ( +g ` R ) = ( +g ` R )
14		mdetralt.r	. . . . 5 |- ( ph -> R e. CRing )
15		crngring 17784	. . . . 5 |- ( R e. CRing -> R e. Ring )
16	14, 15	syl 17	. . . 4 |- ( ph -> R e. Ring )
17		ringcmn 17804	. . . 4 |- ( R e. Ring -> R e. CMnd )
18	16, 17	syl 17	. . 3 |- ( ph -> R e. CMnd )
19	3, 4	matrcl 19429	. . . . . 6 |- ( X e. B -> ( N e. Fin /\ R e. _V ) )
20	1, 19	syl 17	. . . . 5 |- ( ph -> ( N e. Fin /\ R e. _V ) )
21	20	simpld 461	. . . 4 |- ( ph -> N e. Fin )
22		eqid 2423	. . . . 5 |- ( SymGrp ` N ) = ( SymGrp ` N )
23	22, 5	symgbasfi 17020	. . . 4 |- ( N e. Fin -> ( Base ` ( SymGrp ` N ) ) e. Fin )
24	21, 23	syl 17	. . 3 |- ( ph -> ( Base ` ( SymGrp ` N ) ) e. Fin )
25	16	adantr 467	. . . 4 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> R e. Ring )
26		zrhpsgnmhm 19144	. . . . . . 7 |- ( ( R e. Ring /\ N e. Fin ) -> ( ( ZRHom ` R ) o. (pmSgn ` N ) ) e. ( ( SymGrp ` N ) MndHom (mulGrp ` R ) ) )
27	16, 21, 26	syl2anc 666	. . . . . 6 |- ( ph -> ( ( ZRHom ` R ) o. (pmSgn ` N ) ) e. ( ( SymGrp ` N ) MndHom (mulGrp ` R ) ) )
28	9, 12	mgpbas 17722	. . . . . . 7 |- ( Base ` R ) = ( Base ` (mulGrp ` R ) )
29	5, 28	mhmf 16580	. . . . . 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 17	. . . . 5 |- ( ph -> ( ( ZRHom ` R ) o. (pmSgn ` N ) ) : ( Base ` ( SymGrp ` N ) ) --> ( Base ` R ) )
31	30	ffvelrnda 6035	. . . 4 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) e. ( Base ` R ) )
32	9	crngmgp 17781	. . . . . . 7 |- ( R e. CRing -> (mulGrp ` R ) e. CMnd )
33	14, 32	syl 17	. . . . . 6 |- ( ph -> (mulGrp ` R ) e. CMnd )
34	33	adantr 467	. . . . 5 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> (mulGrp ` R ) e. CMnd )
35	21	adantr 467	. . . . 5 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> N e. Fin )
36	3, 12, 4	matbas2i 19439	. . . . . . . . . 10 |- ( X e. B -> X e. ( ( Base ` R ) ^m ( N X. N ) ) )
37	1, 36	syl 17	. . . . . . . . 9 |- ( ph -> X e. ( ( Base ` R ) ^m ( N X. N ) ) )
38		elmapi 7499	. . . . . . . . 9 |- ( X e. ( ( Base ` R ) ^m ( N X. N ) ) -> X : ( N X. N ) --> ( Base ` R ) )
39	37, 38	syl 17	. . . . . . . 8 |- ( ph -> X : ( N X. N ) --> ( Base ` R ) )
40	39	ad2antrr 731	. . . . . . 7 |- ( ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) /\ c e. N ) -> X : ( N X. N ) --> ( Base ` R ) )
41	22, 5	symgbasf1o 17017	. . . . . . . . . 10 |- ( p e. ( Base ` ( SymGrp ` N ) ) -> p : N -1-1-onto-> N )
42	41	adantl 468	. . . . . . . . 9 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> p : N -1-1-onto-> N )
43		f1of 5829	. . . . . . . . 9 |- ( p : N -1-1-onto-> N -> p : N --> N )
44	42, 43	syl 17	. . . . . . . 8 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> p : N --> N )
45	44	ffvelrnda 6035	. . . . . . 7 |- ( ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) /\ c e. N ) -> ( p ` c ) e. N )
46		simpr 463	. . . . . . 7 |- ( ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) /\ c e. N ) -> c e. N )
47	40, 45, 46	fovrnd 6453	. . . . . 6 |- ( ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) /\ c e. N ) -> ( ( p ` c ) X c ) e. ( Base ` R ) )
48	47	ralrimiva 2840	. . . . 5 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> A. c e. N ( ( p ` c ) X c ) e. ( Base ` R ) )
49	28, 34, 35, 48	gsummptcl 17592	. . . 4 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) e. ( Base ` R ) )
50	12, 8	ringcl 17787	. . . 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 1265	. . 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 3868	. . . 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 19146	. . . . . 6 |- (pmEven ` N ) C_ ( Base ` ( SymGrp ` N ) )
55		undif 3877	. . . . . 6 |- ( (pmEven ` N ) C_ ( Base ` ( SymGrp ` N ) ) <-> ( (pmEven ` N ) u. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) = ( Base ` ( SymGrp ` N ) ) )
56	54, 55	mpbi 212	. . . . 5 |- ( (pmEven ` N ) u. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) = ( Base ` ( SymGrp ` N ) )
57	56	eqcomi 2436	. . . 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 2423	. . 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 17557	. 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 5171	. . . . . . 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 467	. . . . . . . . . 10 |- ( ( ph /\ p e. (pmEven ` N ) ) -> R e. Ring )
64	21	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ p e. (pmEven ` N ) ) -> N e. Fin )
65		simpr 463	. . . . . . . . . 10 |- ( ( ph /\ p e. (pmEven ` N ) ) -> p e. (pmEven ` N ) )
66		eqid 2423	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
67	6, 7, 66	zrhpsgnevpm 19151	. . . . . . . . . 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 1265	. . . . . . . . 9 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( 1r ` R ) )
69	68	oveq1d 6318	. . . . . . . 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 3461	. . . . . . . . . 10 |- ( p e. (pmEven ` N ) -> p e. ( Base ` ( SymGrp ` N ) ) )
71	70, 49	sylan2 477	. . . . . . . . 9 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) e. ( Base ` R ) )
72	12, 8, 66	ringlidm 17797	. . . . . . . . 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 666	. . . . . . . 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 2464	. . . . . . 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 4508	. . . . . 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 2476	. . . . 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 6319	. . . 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 3593	. . . . . . . 8 |- ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) C_ ( Base ` ( SymGrp ` N ) )
79		resmpt 5171	. . . . . . . 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 467	. . . . . . . . . . . 12 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> R e. Ring )
82	21	adantr 467	. . . . . . . . . . . 12 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> N e. Fin )
83		simpr 463	. . . . . . . . . . . 12 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
84		eqid 2423	. . . . . . . . . . . . 13 |- ( invg ` R ) = ( invg ` R )
85	6, 7, 66, 5, 84	zrhpsgnodpm 19152	. . . . . . . . . . . 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 1265	. . . . . . . . . . 11 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( ( invg ` R ) ` ( 1r ` R ) ) )
87	86	oveq1d 6318	. . . . . . . . . 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 3588	. . . . . . . . . . . 12 |- ( p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) -> p e. ( Base ` ( SymGrp ` N ) ) )
89	88, 49	sylan2 477	. . . . . . . . . . 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 17815	. . . . . . . . . 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 2464	. . . . . . . . 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 4508	. . . . . . . 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 2424	. . . . . . . . 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 17778	. . . . . . . . . . . 12 |- ( R e. Ring -> R e. Grp )
95	16, 94	syl 17	. . . . . . . . . . 11 |- ( ph -> R e. Grp )
96	12, 84	grpinvf 16703	. . . . . . . . . . 11 |- ( R e. Grp -> ( invg ` R ) : ( Base ` R ) --> ( Base ` R ) )
97	95, 96	syl 17	. . . . . . . . . 10 |- ( ph -> ( invg ` R ) : ( Base ` R ) --> ( Base ` R ) )
98	97	feqmptd 5932	. . . . . . . . 9 |- ( ph -> ( invg ` R ) = ( q e. ( Base ` R ) |-> ( ( invg ` R ) ` q ) ) )
99		fveq2 5879	. . . . . . . . 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 6069	. . . . . . . 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 2467	. . . . . . 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 2476	. . . . . 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 6319	. . . . 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 17803	. . . . . . 7 |- ( R e. Ring -> R e. Abel )
106	16, 105	syl 17	. . . . . 6 |- ( ph -> R e. Abel )
107		difssd 3594	. . . . . . 7 |- ( ph -> ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) C_ ( Base ` ( SymGrp ` N ) ) )
108		ssfi 7796	. . . . . . 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 666	. . . . . 6 |- ( ph -> ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) e. Fin )
110		eqid 2423	. . . . . 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 17573	. . . . 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 2840	. . . . . . . 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 4154	. . . . . . . . . . . 12 |- ( ( I e. N /\ J e. N ) -> { I , J } C_ N )
116	113, 114, 115	syl2anc 666	. . . . . . . . . . 11 |- ( ph -> { I , J } C_ N )
117		mdetralt.ij	. . . . . . . . . . . 12 |- ( ph -> I =/= J )
118		pr2nelem 8438	. . . . . . . . . . . 12 |- ( ( I e. N /\ J e. N /\ I =/= J ) -> { I , J } ~~ 2o )
119	113, 114, 117, 118	syl3anc 1265	. . . . . . . . . . 11 |- ( ph -> { I , J } ~~ 2o )
120		eqid 2423	. . . . . . . . . . . 12 |- (pmTrsp ` N ) = (pmTrsp ` N )
121		eqid 2423	. . . . . . . . . . . 12 |- ran (pmTrsp ` N ) = ran (pmTrsp ` N )
122	120, 121	pmtrrn 17091	. . . . . . . . . . 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 1265	. . . . . . . . . 10 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) e. ran (pmTrsp ` N ) )
124	22, 5, 121	pmtrodpm 19157	. . . . . . . . . 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 666	. . . . . . . . 9 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
126	22, 5	evpmodpmf1o 19156	. . . . . . . . 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 666	. . . . . . . 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 17593	. . . . . . 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 2495	. . . . . . . . . . . . 13 |- ( p = q -> ( p e. (pmEven ` N ) <-> q e. (pmEven ` N ) ) )
130	129	anbi2d 709	. . . . . . . . . . . 12 |- ( p = q -> ( ( ph /\ p e. (pmEven ` N ) ) <-> ( ph /\ q e. (pmEven ` N ) ) ) )
131		oveq2 6311	. . . . . . . . . . . . 13 |- ( p = q -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) )
132	131	eleq1d 2492	. . . . . . . . . . . 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 322	. . . . . . . . . . 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 17034	. . . . . . . . . . . . . . 15 |- ( N e. Fin -> ( SymGrp ` N ) e. Grp )
135	21, 134	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( SymGrp ` N ) e. Grp )
136	135	adantr 467	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( SymGrp ` N ) e. Grp )
137	121, 22, 5	symgtrf 17103	. . . . . . . . . . . . . 14 |- ran (pmTrsp ` N ) C_ ( Base ` ( SymGrp ` N ) )
138	123	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmTrsp ` N ) ` { I , J } ) e. ran (pmTrsp ` N ) )
139	137, 138	sseldi 3463	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmTrsp ` N ) ` { I , J } ) e. ( Base ` ( SymGrp ` N ) ) )
140	70	adantl 468	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. (pmEven ` N ) ) -> p e. ( Base ` ( SymGrp ` N ) ) )
141		eqid 2423	. . . . . . . . . . . . . 14 |- ( +g ` ( SymGrp ` N ) ) = ( +g ` ( SymGrp ` N ) )
142	5, 141	grpcl 16672	. . . . . . . . . . . . 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 1265	. . . . . . . . . . . 12 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( Base ` ( SymGrp ` N ) ) )
144		eqid 2423	. . . . . . . . . . . . . . . . 17 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
145	22, 7, 144	psgnghm2 19141	. . . . . . . . . . . . . . . 16 |- ( N e. Fin -> (pmSgn ` N ) e. ( ( SymGrp ` N ) GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
146	21, 145	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> (pmSgn ` N ) e. ( ( SymGrp ` N ) GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
147	146	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ph /\ p e. (pmEven ` N ) ) -> (pmSgn ` N ) e. ( ( SymGrp ` N ) GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
148		prex 4661	. . . . . . . . . . . . . . . 16 |- { 1 , -u 1 } e. _V
149		eqid 2423	. . . . . . . . . . . . . . . . . 18 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
150		cnfldmul 18969	. . . . . . . . . . . . . . . . . 18 |- x. = ( .r ` ℂfld )
151	149, 150	mgpplusg 17720	. . . . . . . . . . . . . . . . 17 |- x. = ( +g ` (mulGrp ` ℂfld ) )
152	144, 151	ressplusg 15232	. . . . . . . . . . . . . . . 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 16881	. . . . . . . . . . . . . 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 1265	. . . . . . . . . . . . 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 17144	. . . . . . . . . . . . . . . 16 |- ( ( (pmTrsp ` N ) ` { I , J } ) e. ran (pmTrsp ` N ) -> ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) = -u 1 )
157	138, 156	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) = -u 1 )
158	22, 5, 7	psgnevpm 19149	. . . . . . . . . . . . . . . 16 |- ( ( N e. Fin /\ p e. (pmEven ` N ) ) -> ( (pmSgn ` N ) ` p ) = 1 )
159	21, 158	sylan 474	. . . . . . . . . . . . . . 15 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmSgn ` N ) ` p ) = 1 )
160	157, 159	oveq12d 6321	. . . . . . . . . . . . . 14 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) x. ( (pmSgn ` N ) ` p ) ) = ( -u 1 x. 1 ) )
161		neg1cn 10715	. . . . . . . . . . . . . . 15 |- -u 1 e. CC
162	161	mulid1i 9647	. . . . . . . . . . . . . 14 |- ( -u 1 x. 1 ) = -u 1
163	160, 162	syl6eq 2480	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) x. ( (pmSgn ` N ) ` p ) ) = -u 1 )
164	155, 163	eqtrd 2464	. . . . . . . . . . . 12 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmSgn ` N ) ` ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ) = -u 1 )
165	22, 5, 7	psgnodpmr 19150	. . . . . . . . . . . 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 1265	. . . . . . . . . . 11 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
167	133, 166	chvarv 2069	. . . . . . . . . 10 |- ( ( ph /\ q e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
168		eqidd 2424	. . . . . . . . . 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 5878	. . . . . . . . . . . . 13 |- ( p = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) -> ( p ` c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) )
170	169	oveq1d 6318	. . . . . . . . . . . 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 4509	. . . . . . . . . . 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 6319	. . . . . . . . . 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 6069	. . . . . . . . 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 6311	. . . . . . . . . . . . . . 15 |- ( q = p -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) )
175	174	fveq1d 5881	. . . . . . . . . . . . . 14 |- ( q = p -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) )
176	175	oveq1d 6318	. . . . . . . . . . . . 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 4509	. . . . . . . . . . . 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 6319	. . . . . . . . . . 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 4514	. . . . . . . . . 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 467	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( (pmTrsp ` N ) ` { I , J } ) e. ( Base ` ( SymGrp ` N ) ) )
182	140	adantr 467	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> p e. ( Base ` ( SymGrp ` N ) ) )
183	22, 5, 141	symgov 17024	. . . . . . . . . . . . . . . . 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 666	. . . . . . . . . . . . . . . 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 5881	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. (pmEven ` N ) ) -> p : N --> N )
187		fvco3 5956	. . . . . . . . . . . . . . . 16 |- ( ( p : N --> N /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) o. p ) ` c ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) )
188	186, 187	sylan 474	. . . . . . . . . . . . . . 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 2464	. . . . . . . . . . . . . 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 6318	. . . . . . . . . . . . 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 17087	. . . . . . . . . . . . . . . . . . 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 1267	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) = J )
193	192	ad2antrr 731	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) = J )
194	193	oveq1d 6318	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) X c ) = ( J X c ) )
195		simpr 463	. . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> A. a e. N ( I X a ) = ( J X a ) )
198		oveq2 6311	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> ( I X a ) = ( I X c ) )
199		oveq2 6311	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> ( J X a ) = ( J X c ) )
200	198, 199	eqeq12d 2445	. . . . . . . . . . . . . . . . . 18 |- ( a = c -> ( ( I X a ) = ( J X a ) <-> ( I X c ) = ( J X c ) ) )
201	200	rspcv 3179	. . . . . . . . . . . . . . . . 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 63	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( I X c ) = ( J X c ) )
203	194, 202	eqtr4d 2467	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) X c ) = ( I X c ) )
204		fveq2 5879	. . . . . . . . . . . . . . . . 17 |- ( ( p ` c ) = I -> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) )
205	204	oveq1d 6318	. . . . . . . . . . . . . . . 16 |- ( ( p ` c ) = I -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) X c ) )
206		oveq1 6310	. . . . . . . . . . . . . . . 16 |- ( ( p ` c ) = I -> ( ( p ` c ) X c ) = ( I X c ) )
207	205, 206	eqeq12d 2445	. . . . . . . . . . . . . . 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 226	. . . . . . . . . . . . . 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 4076	. . . . . . . . . . . . . . . . . . . . . . 23 |- { I , J } = { J , I }
210	209	fveq2i 5882	. . . . . . . . . . . . . . . . . . . . . 22 |- ( (pmTrsp ` N ) ` { I , J } ) = ( (pmTrsp ` N ) ` { J , I } )
211	210	fveq1i 5880	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) = ( ( (pmTrsp ` N ) ` { J , I } ) ` J )
212	117	necomd 2696	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> J =/= I )
213	120	pmtrprfv 17087	. . . . . . . . . . . . . . . . . . . . . 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 1267	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> ( ( (pmTrsp ` N ) ` { J , I } ) ` J ) = I )
215	211, 214	syl5eq 2476	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) = I )
216	215	oveq1d 6318	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) = ( I X c ) )
217	216	ad2antrr 731	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) = ( I X c ) )
218	217, 202	eqtrd 2464	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) = ( J X c ) )
219		fveq2 5879	. . . . . . . . . . . . . . . . . . 19 |- ( ( p ` c ) = J -> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) )
220	219	oveq1d 6318	. . . . . . . . . . . . . . . . . 18 |- ( ( p ` c ) = J -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) )
221		oveq1 6310	. . . . . . . . . . . . . . . . . 18 |- ( ( p ` c ) = J -> ( ( p ` c ) X c ) = ( J X c ) )
222	220, 221	eqeq12d 2445	. . . . . . . . . . . . . . . . 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 226	. . . . . . . . . . . . . . . 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 48	. . . . . . . . . . . . . . 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 2750	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( p ` c ) =/= J /\ ( p ` c ) =/= I ) <-> -. ( ( p ` c ) = J \/ ( p ` c ) = I ) )
226		elpri 4014	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( p ` c ) e. { I , J } -> ( ( p ` c ) = I \/ ( p ` c ) = J ) )
227	226	orcomd 390	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( p ` c ) e. { I , J } -> ( ( p ` c ) = J \/ ( p ` c ) = I ) )
228	227	con3i 141	. . . . . . . . . . . . . . . . . . . . 21 |- ( -. ( ( p ` c ) = J \/ ( p ` c ) = I ) -> -. ( p ` c ) e. { I , J } )
229	225, 228	sylbi 199	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> -. ( p ` c ) e. { I , J } )
230	229	3adant1 1024	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) /\ ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> -. ( p ` c ) e. { I , J } )
231	120	pmtrmvd 17090	. . . . . . . . . . . . . . . . . . . . . 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 1265	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> dom ( ( (pmTrsp ` N ) ` { I , J } ) \ _I ) = { I , J } )
233	232	ad2antrr 731	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> dom ( ( (pmTrsp ` N ) ` { I , J } ) \ _I ) = { I , J } )
234	233	3ad2ant1 1027	. . . . . . . . . . . . . . . . . . 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 2536	. . . . . . . . . . . . . . . . . 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 17089	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N e. Fin /\ { I , J } C_ N /\ { I , J } ~~ 2o ) -> ( (pmTrsp ` N ) ` { I , J } ) : N --> N )
237	21, 116, 119, 236	syl3anc 1265	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) : N --> N )
238		ffn 5744	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( (pmTrsp ` N ) ` { I , J } ) : N --> N -> ( (pmTrsp ` N ) ` { I , J } ) Fn N )
239	237, 238	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) Fn N )
240	239	ad2antrr 731	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( (pmTrsp ` N ) ` { I , J } ) Fn N )
241	186	ffvelrnda 6035	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( p ` c ) e. N )
242		fnelnfp 6107	. . . . . . . . . . . . . . . . . . . . 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 666	. . . . . . . . . . . . . . . . . . . 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 1027	. . . . . . . . . . . . . . . . . . 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 2681	. . . . . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . . . . 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 6318	. . . . . . . . . . . . . . . 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 1205	. . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . 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 2464	. . . . . . . . . . . 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 4508	. . . . . . . . . . 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 6319	. . . . . . . . . 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 4508	. . . . . . . . 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 2468	. . . . . . . 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 6319	. . . . . . 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 2464	. . . . . 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 5883	. . . . 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 2468	. . . 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 6321	. . 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 7796	. . . . . 6 |- ( ( ( Base ` ( SymGrp ` N ) ) e. Fin /\ (pmEven ` N ) C_ ( Base ` ( SymGrp ` N ) ) ) -> (pmEven ` N ) e. Fin )
263	24, 261, 262	syl2anc 666	. . . . 5 |- ( ph -> (pmEven ` N ) e. Fin )
264	71	ralrimiva 2840	. . . . 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 17592	. . . 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 16706	. . . 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 666	. . 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 2464	. 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 2468	1 |- ( ph -> ( D ` X ) = .0. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   /\ w3a 983   = wceq 1438   e. wcel 1869   =/= wne 2619   A.wral 2776   _Vcvv 3082   \ cdif 3434   u. cun 3435   i^i cin 3436   C_ wss 3437   (/)c0 3762   {cpr 3999   class class class wbr 4421   |-> cmpt 4480   _I cid 4761   X. cxp 4849   dom cdm 4851   ran crn 4852   |` cres 4853   o. ccom 4855   Fn wfn 5594   -->wf 5595   -1-1-onto->wf1o 5598   ` cfv 5599  (class class class)co 6303   2oc2o 7182   ^m cmap 7478   ~~ cen 7572   Fincfn 7575   1c1 9542   x. cmul 9546   -ucneg 9863   Basecbs 15114   ↾_s cress 15115   +g cplusg 15183   .rcmulr 15184   0gc0g 15331   gsum_g cgsu 15332   MndHom cmhm 16573   Grpcgrp 16662   invgcminusg 16663   GrpHom cghm 16873   SymGrpcsymg 17011  pmTrspcpmtr 17075  pmSgncpsgn 17123  pmEvencevpm 17124  CMndccmn 17423   Abelcabl 17424  mulGrpcmgp 17716   1rcur 17728   Ringcrg 17773   CRingccrg 17774  ℂ_fldccnfld 18963   ZRHomczrh 19063   Mat cmat 19424   maDet cmdat 19601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-addf 9620  ax-mulf 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-xor 1402  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-ot 4006  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-tpos 6979  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-sup 7960  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-rp 11305  df-fz 11787  df-fzo 11918  df-seq 12215  df-exp 12274  df-hash 12517  df-word 12662  df-lsw 12663  df-concat 12664  df-s1 12665  df-substr 12666  df-splice 12667  df-reverse 12668  df-s2 12940  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-0g 15333  df-gsum 15334  df-prds 15339  df-pws 15341  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-submnd 16576  df-grp 16666  df-minusg 16667  df-mulg 16669  df-subg 16807  df-ghm 16874  df-gim 16916  df-cntz 16964  df-oppg 16990  df-symg 17012  df-pmtr 17076  df-psgn 17125  df-evpm 17126  df-cmn 17425  df-abl 17426  df-mgp 17717  df-ur 17729  df-ring 17775  df-cring 17776  df-oppr 17844  df-dvdsr 17862  df-unit 17863  df-invr 17893  df-dvr 17904  df-rnghom 17936  df-drng 17970  df-subrg 17999  df-sra 18388  df-rgmod 18389  df-cnfld 18964  df-zring 19032  df-zrh 19067  df-dsmm 19287  df-frlm 19302  df-mat 19425  df-mdet 19602

This theorem is referenced by:  mdetralt2  19626  mdetuni0  19638  mdetmul  19640