Theorem mdetralt 18414

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 2443	. . . 4 |- ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` N ) )
6		eqid 2443	. . . 4 |- ( ZRHom ` R ) = ( ZRHom ` R )
7		eqid 2443	. . . 4 |- (pmSgn ` N ) = (pmSgn ` N )
8		eqid 2443	. . . 4 |- ( .r ` R ) = ( .r ` R )
9		eqid 2443	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
10	2, 3, 4, 5, 6, 7, 8, 9	mdetleib 18398	. . 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 2443	. . 3 |- ( Base ` R ) = ( Base ` R )
13		eqid 2443	. . 3 |- ( +g ` R ) = ( +g ` R )
14		mdetralt.r	. . . . 5 |- ( ph -> R e. CRing )
15		crngrng 16655	. . . . 5 |- ( R e. CRing -> R e. Ring )
16	14, 15	syl 16	. . . 4 |- ( ph -> R e. Ring )
17		rngcmn 16675	. . . 4 |- ( R e. Ring -> R e. CMnd )
18	16, 17	syl 16	. . 3 |- ( ph -> R e. CMnd )
19	3, 4	matrcl 18312	. . . . . 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 2443	. . . . 5 |- ( SymGrp ` N ) = ( SymGrp ` N )
23	22, 5	symgbasfi 15891	. . . 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 18014	. . . . . . 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 16597	. . . . . . 7 |- ( Base ` R ) = ( Base ` (mulGrp ` R ) )
29	5, 28	mhmf 15469	. . . . . 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 5843	. . . 4 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) e. ( Base ` R ) )
32	9	crngmgp 16653	. . . . . . 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 18323	. . . . . . . . . 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 7234	. . . . . . . . 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 15888	. . . . . . . . . 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 5641	. . . . . . . . 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 5843	. . . . . . 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 6235	. . . . . 6 |- ( ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) /\ c e. N ) -> ( ( p ` c ) X c ) e. ( Base ` R ) )
48	47	ralrimiva 2799	. . . . 5 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> A. c e. N ( ( p ` c ) X c ) e. ( Base ` R ) )
49	28, 34, 35, 48	gsummptcl 16458	. . . 4 |- ( ( ph /\ p e. ( Base ` ( SymGrp ` N ) ) ) -> ( (mulGrp ` R ) gsumg ( c e. N |-> ( ( p ` c ) X c ) ) ) e. ( Base ` R ) )
50	12, 8	rngcl 16658	. . . 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 1218	. . 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 3751	. . . 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 18016	. . . . . 6 |- (pmEven ` N ) C_ ( Base ` ( SymGrp ` N ) )
55		undif 3759	. . . . . 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 2447	. . . 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 2443	. . 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 16425	. 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 5156	. . . . . . 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 2443	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
67	6, 7, 66	zrhpsgnevpm 18021	. . . . . . . . . 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 1218	. . . . . . . . 9 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( 1r ` R ) )
69	68	oveq1d 6106	. . . . . . . 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 3352	. . . . . . . . . 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	rnglidm 16668	. . . . . . . . 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 2475	. . . . . . 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 4378	. . . . . 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 2487	. . . . 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 6107	. . . 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 3483	. . . . . . . 8 |- ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) C_ ( Base ` ( SymGrp ` N ) )
79		resmpt 5156	. . . . . . . 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 2443	. . . . . . . . . . . . 13 |- ( invg ` R ) = ( invg ` R )
85	6, 7, 66, 5, 84	zrhpsgnodpm 18022	. . . . . . . . . . . 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 1218	. . . . . . . . . . 11 |- ( ( ph /\ p e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( ( invg ` R ) ` ( 1r ` R ) ) )
87	86	oveq1d 6106	. . . . . . . . . 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 3478	. . . . . . . . . . . 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	rngnegl 16685	. . . . . . . . . 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 2475	. . . . . . . . 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 4378	. . . . . . . 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 2444	. . . . . . . . 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		rnggrp 16650	. . . . . . . . . . . 12 |- ( R e. Ring -> R e. Grp )
95	16, 94	syl 16	. . . . . . . . . . 11 |- ( ph -> R e. Grp )
96	12, 84	grpinvf 15582	. . . . . . . . . . 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 5744	. . . . . . . . 9 |- ( ph -> ( invg ` R ) = ( q e. ( Base ` R ) |-> ( ( invg ` R ) ` q ) ) )
99		fveq2 5691	. . . . . . . . 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 5876	. . . . . . . 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 2478	. . . . . . 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 2487	. . . . . 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 6107	. . . . 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		rngabl 16674	. . . . . . 7 |- ( R e. Ring -> R e. Abel )
106	16, 105	syl 16	. . . . . 6 |- ( ph -> R e. Abel )
107		difssd 3484	. . . . . . 7 |- ( ph -> ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) C_ ( Base ` ( SymGrp ` N ) ) )
108		ssfi 7533	. . . . . . 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 2443	. . . . . 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 16445	. . . . 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 2799	. . . . . . . 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 4029	. . . . . . . . . . . 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 8171	. . . . . . . . . . . 12 |- ( ( I e. N /\ J e. N /\ I =/= J ) -> { I , J } ~~ 2o )
119	113, 114, 117, 118	syl3anc 1218	. . . . . . . . . . 11 |- ( ph -> { I , J } ~~ 2o )
120		eqid 2443	. . . . . . . . . . . 12 |- (pmTrsp ` N ) = (pmTrsp ` N )
121		eqid 2443	. . . . . . . . . . . 12 |- ran (pmTrsp ` N ) = ran (pmTrsp ` N )
122	120, 121	pmtrrn 15963	. . . . . . . . . . 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 1218	. . . . . . . . . 10 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) e. ran (pmTrsp ` N ) )
124	22, 5, 121	pmtrodpm 18027	. . . . . . . . . 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 18026	. . . . . . . . 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 16459	. . . . . . 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 2503	. . . . . . . . . . . . 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 6099	. . . . . . . . . . . . 13 |- ( p = q -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) )
132	131	eleq1d 2509	. . . . . . . . . . . 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 15905	. . . . . . . . . . . . . . 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 15975	. . . . . . . . . . . . . 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 3354	. . . . . . . . . . . . 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 2443	. . . . . . . . . . . . . 14 |- ( +g ` ( SymGrp ` N ) ) = ( +g ` ( SymGrp ` N ) )
142	5, 141	grpcl 15551	. . . . . . . . . . . . 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 1218	. . . . . . . . . . . 12 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( Base ` ( SymGrp ` N ) ) )
144		eqid 2443	. . . . . . . . . . . . . . . . 17 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
145	22, 7, 144	psgnghm2 18011	. . . . . . . . . . . . . . . 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 4534	. . . . . . . . . . . . . . . 16 |- { 1 , -u 1 } e. _V
149		eqid 2443	. . . . . . . . . . . . . . . . . 18 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
150		cnfldmul 17824	. . . . . . . . . . . . . . . . . 18 |- x. = ( .r ` ℂfld )
151	149, 150	mgpplusg 16595	. . . . . . . . . . . . . . . . 17 |- x. = ( +g ` (mulGrp ` ℂfld ) )
152	144, 151	ressplusg 14280	. . . . . . . . . . . . . . . 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 15752	. . . . . . . . . . . . . 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 1218	. . . . . . . . . . . . 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 16016	. . . . . . . . . . . . . . . 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 18019	. . . . . . . . . . . . . . . 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 6109	. . . . . . . . . . . . . 14 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) x. ( (pmSgn ` N ) ` p ) ) = ( -u 1 x. 1 ) )
161		neg1cn 10425	. . . . . . . . . . . . . . 15 |- -u 1 e. CC
162	161	mulid1i 9388	. . . . . . . . . . . . . 14 |- ( -u 1 x. 1 ) = -u 1
163	160, 162	syl6eq 2491	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmSgn ` N ) ` ( (pmTrsp ` N ) ` { I , J } ) ) x. ( (pmSgn ` N ) ` p ) ) = -u 1 )
164	155, 163	eqtrd 2475	. . . . . . . . . . . 12 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( (pmSgn ` N ) ` ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ) = -u 1 )
165	22, 5, 7	psgnodpmr 18020	. . . . . . . . . . . 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 1218	. . . . . . . . . . 11 |- ( ( ph /\ p e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
167	133, 166	chvarv 1958	. . . . . . . . . 10 |- ( ( ph /\ q e. (pmEven ` N ) ) -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) e. ( ( Base ` ( SymGrp ` N ) ) \ (pmEven ` N ) ) )
168		eqidd 2444	. . . . . . . . . 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 5690	. . . . . . . . . . . . 13 |- ( p = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) -> ( p ` c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) )
170	169	oveq1d 6106	. . . . . . . . . . . 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 4379	. . . . . . . . . . 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 6107	. . . . . . . . . 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 5876	. . . . . . . . 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 6099	. . . . . . . . . . . . . . 15 |- ( q = p -> ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) = ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) )
175	174	fveq1d 5693	. . . . . . . . . . . . . 14 |- ( q = p -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) q ) ` c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ( +g ` ( SymGrp ` N ) ) p ) ` c ) )
176	175	oveq1d 6106	. . . . . . . . . . . . 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 4379	. . . . . . . . . . . 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 6107	. . . . . . . . . . 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 4383	. . . . . . . . . 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 15895	. . . . . . . . . . . . . . . . 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 5693	. . . . . . . . . . . . . . 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 5768	. . . . . . . . . . . . . . . 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 2475	. . . . . . . . . . . . . 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 6106	. . . . . . . . . . . . 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 15959	. . . . . . . . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . . . . . 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 6106	. . . . . . . . . . . . . . . 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 6099	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> ( I X a ) = ( I X c ) )
199		oveq2 6099	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> ( J X a ) = ( J X c ) )
200	198, 199	eqeq12d 2457	. . . . . . . . . . . . . . . . . 18 |- ( a = c -> ( ( I X a ) = ( J X a ) <-> ( I X c ) = ( J X c ) ) )
201	200	rspcv 3069	. . . . . . . . . . . . . . . . 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 2478	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) X c ) = ( I X c ) )
204		fveq2 5691	. . . . . . . . . . . . . . . . 17 |- ( ( p ` c ) = I -> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) )
205	204	oveq1d 6106	. . . . . . . . . . . . . . . 16 |- ( ( p ` c ) = I -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ` I ) X c ) )
206		oveq1 6098	. . . . . . . . . . . . . . . 16 |- ( ( p ` c ) = I -> ( ( p ` c ) X c ) = ( I X c ) )
207	205, 206	eqeq12d 2457	. . . . . . . . . . . . . . 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 3953	. . . . . . . . . . . . . . . . . . . . . . 23 |- { I , J } = { J , I }
210	209	fveq2i 5694	. . . . . . . . . . . . . . . . . . . . . 22 |- ( (pmTrsp ` N ) ` { I , J } ) = ( (pmTrsp ` N ) ` { J , I } )
211	210	fveq1i 5692	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) = ( ( (pmTrsp ` N ) ` { J , I } ) ` J )
212	117	necomd 2695	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> J =/= I )
213	120	pmtrprfv 15959	. . . . . . . . . . . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> ( ( (pmTrsp ` N ) ` { J , I } ) ` J ) = I )
215	211, 214	syl5eq 2487	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) = I )
216	215	oveq1d 6106	. . . . . . . . . . . . . . . . . . 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 2475	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) = ( J X c ) )
219		fveq2 5691	. . . . . . . . . . . . . . . . . . 19 |- ( ( p ` c ) = J -> ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) = ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) )
220	219	oveq1d 6106	. . . . . . . . . . . . . . . . . 18 |- ( ( p ` c ) = J -> ( ( ( (pmTrsp ` N ) ` { I , J } ) ` ( p ` c ) ) X c ) = ( ( ( (pmTrsp ` N ) ` { I , J } ) ` J ) X c ) )
221		oveq1 6098	. . . . . . . . . . . . . . . . . 18 |- ( ( p ` c ) = J -> ( ( p ` c ) X c ) = ( J X c ) )
222	220, 221	eqeq12d 2457	. . . . . . . . . . . . . . . . 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 2697	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( p ` c ) =/= J /\ ( p ` c ) =/= I ) <-> -. ( ( p ` c ) = J \/ ( p ` c ) = I ) )
226		elpri 3897	. . . . . . . . . . . . . . . . . . . . . . 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 1006	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) /\ ( p ` c ) =/= J /\ ( p ` c ) =/= I ) -> -. ( p ` c ) e. { I , J } )
231	120	pmtrmvd 15962	. . . . . . . . . . . . . . . . . . . . . 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 1218	. . . . . . . . . . . . . . . . . . . . 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 1009	. . . . . . . . . . . . . . . . . . 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 2539	. . . . . . . . . . . . . . . . . 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 15961	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N e. Fin /\ { I , J } C_ N /\ { I , J } ~~ 2o ) -> ( (pmTrsp ` N ) ` { I , J } ) : N --> N )
237	21, 116, 119, 236	syl3anc 1218	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> ( (pmTrsp ` N ) ` { I , J } ) : N --> N )
238		ffn 5559	. . . . . . . . . . . . . . . . . . . . . . 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 5843	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ p e. (pmEven ` N ) ) /\ c e. N ) -> ( p ` c ) e. N )
242		fnelnfp 5908	. . . . . . . . . . . . . . . . . . . . 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 1009	. . . . . . . . . . . . . . . . . . 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 2669	. . . . . . . . . . . . . . . . . 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 6106	. . . . . . . . . . . . . . . 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 1186	. . . . . . . . . . . . . . 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 2688	. . . . . . . . . . . . . 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 2688	. . . . . . . . . . . . 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 2475	. . . . . . . . . . . 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 4378	. . . . . . . . . . 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 6107	. . . . . . . . . 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 4378	. . . . . . . . 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 2479	. . . . . . . 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 6107	. . . . . . 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 2475	. . . . . 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 5695	. . . . 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 2479	. . . 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 6109	. . 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 7533	. . . . . 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 2799	. . . . 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 16458	. . . 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 15585	. . . 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 2475	. 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 2479	1 |- ( ph -> ( D ` X ) = .0. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2606   A.wral 2715   _Vcvv 2972   \ cdif 3325   u. cun 3326   i^i cin 3327   C_ wss 3328   (/)c0 3637   {cpr 3879   class class class wbr 4292   e. cmpt 4350   _I cid 4631   X. cxp 4838   dom cdm 4840   ran crn 4841   |` cres 4842   o. ccom 4844   Fn wfn 5413   -->wf 5414   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091   2oc2o 6914   ^m cmap 7214   ~~ cen 7307   Fincfn 7310   1c1 9283   x. cmul 9287   -ucneg 9596   Basecbs 14174   ↾_s cress 14175   +g cplusg 14238   .rcmulr 14239   0gc0g 14378   gsum_g cgsu 14379   Grpcgrp 15410   invgcminusg 15411   MndHom cmhm 15462   GrpHom cghm 15744   SymGrpcsymg 15882  pmTrspcpmtr 15947  pmSgncpsgn 15995  pmEvencevpm 15996  CMndccmn 16277   Abelcabel 16278  mulGrpcmgp 16591   1rcur 16603   Ringcrg 16645   CRingccrg 16646  ℂ_fldccnfld 17818   ZRHomczrh 17931   Mat cmat 18280   maDet cmdat 18395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-addf 9361  ax-mulf 9362

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-ot 3886  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-substr 12233  df-splice 12234  df-reverse 12235  df-s2 12475  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-0g 14380  df-gsum 14381  df-prds 14386  df-pws 14388  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-mulg 15548  df-subg 15678  df-ghm 15745  df-gim 15787  df-cntz 15835  df-oppg 15861  df-symg 15883  df-pmtr 15948  df-psgn 15997  df-evpm 15998  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-rnghom 16806  df-drng 16834  df-subrg 16863  df-sra 17253  df-rgmod 17254  df-cnfld 17819  df-zring 17884  df-zrh 17935  df-dsmm 18157  df-frlm 18172  df-mat 18282  df-mdet 18396

This theorem is referenced by:  mdetralt2  18415  mdetuni0  18427  mdetmul  18429