Theorem madjusmdetlem4 29896

Description: Lemma for madjusmdet 29897. (Contributed by Thierry Arnoux, 22-Aug-2020.)

Hypotheses

Ref	Expression
madjusmdet.b	|- B = ( Base ` A )
madjusmdet.a	|- A = ( ( 1 ... N ) Mat R )
madjusmdet.d	|- D = ( ( 1 ... N ) maDet R )
madjusmdet.k	|- K = ( ( 1 ... N ) maAdju R )
madjusmdet.t	|- .x. = ( .r ` R )
madjusmdet.z	|- Z = ( ZRHom ` R )
madjusmdet.e	|- E = ( ( 1 ... ( N - 1 ) ) maDet R )
madjusmdet.n	|- ( ph -> N e. NN )
madjusmdet.r	|- ( ph -> R e. CRing )
madjusmdet.i	|- ( ph -> I e. ( 1 ... N ) )
madjusmdet.j	|- ( ph -> J e. ( 1 ... N ) )
madjusmdet.m	|- ( ph -> M e. B )
madjusmdetlem2.p	|- P = ( i e. ( 1 ... N ) |-> if ( i = 1 , I , if ( i <_ I , ( i - 1 ) , i ) ) )
madjusmdetlem2.s	|- S = ( i e. ( 1 ... N ) |-> if ( i = 1 , N , if ( i <_ N , ( i - 1 ) , i ) ) )
madjusmdetlem4.q	|- Q = ( j e. ( 1 ... N ) |-> if ( j = 1 , J , if ( j <_ J , ( j - 1 ) , j ) ) )
madjusmdetlem4.t	|- T = ( j e. ( 1 ... N ) |-> if ( j = 1 , N , if ( j <_ N , ( j - 1 ) , j ) ) )

Assertion

Ref	Expression
madjusmdetlem4	|- ( ph -> ( J ( K ` M ) I ) = ( ( Z ` ( -u 1 ^ ( I + J ) ) ) .x. ( E ` ( I (subMat1 ` M ) J ) ) ) )

Distinct variable groups:   B, i, j   i, I, j   i, J, j   i, M, j   i, N, j   P, i, j   Q, i, j   R, i, j   ph, i, j   S, i, j   T, i, j

Allowed substitution hints:   A( i, j)   D( i, j)   .x. ( i, j)   E( i, j)   K( i, j)   Z( i, j)

Proof of Theorem madjusmdetlem4

Dummy variables k l are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		madjusmdet.b	. . 3 |- B = ( Base ` A )
2		madjusmdet.a	. . 3 |- A = ( ( 1 ... N ) Mat R )
3		madjusmdet.d	. . 3 |- D = ( ( 1 ... N ) maDet R )
4		madjusmdet.k	. . 3 |- K = ( ( 1 ... N ) maAdju R )
5		madjusmdet.t	. . 3 |- .x. = ( .r ` R )
6		madjusmdet.z	. . 3 |- Z = ( ZRHom ` R )
7		madjusmdet.e	. . 3 |- E = ( ( 1 ... ( N - 1 ) ) maDet R )
8		madjusmdet.n	. . 3 |- ( ph -> N e. NN )
9		madjusmdet.r	. . 3 |- ( ph -> R e. CRing )
10		madjusmdet.i	. . 3 |- ( ph -> I e. ( 1 ... N ) )
11		madjusmdet.j	. . 3 |- ( ph -> J e. ( 1 ... N ) )
12		madjusmdet.m	. . 3 |- ( ph -> M e. B )
13		eqid 2622	. . 3 |- ( Base ` ( SymGrp ` ( 1 ... N ) ) ) = ( Base ` ( SymGrp ` ( 1 ... N ) ) )
14		eqid 2622	. . 3 |- (pmSgn ` ( 1 ... N ) ) = (pmSgn ` ( 1 ... N ) )
15		eqid 2622	. . 3 |- ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) = ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J )
16		fveq2 6191	. . . . 5 |- ( k = i -> ( ( P o. `' S ) ` k ) = ( ( P o. `' S ) ` i ) )
17	16	oveq1d 6665	. . . 4 |- ( k = i -> ( ( ( P o. `' S ) ` k ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( ( Q o. `' T ) ` l ) ) = ( ( ( P o. `' S ) ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( ( Q o. `' T ) ` l ) ) )
18		fveq2 6191	. . . . 5 |- ( l = j -> ( ( Q o. `' T ) ` l ) = ( ( Q o. `' T ) ` j ) )
19	18	oveq2d 6666	. . . 4 |- ( l = j -> ( ( ( P o. `' S ) ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( ( Q o. `' T ) ` l ) ) = ( ( ( P o. `' S ) ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( ( Q o. `' T ) ` j ) ) )
20	17, 19	cbvmpt2v 6735	. . 3 |- ( k e. ( 1 ... N ) , l e. ( 1 ... N ) |-> ( ( ( P o. `' S ) ` k ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( ( Q o. `' T ) ` l ) ) ) = ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( ( P o. `' S ) ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( ( Q o. `' T ) ` j ) ) )
21		eqid 2622	. . . . . 6 |- ( 1 ... N ) = ( 1 ... N )
22		madjusmdetlem2.p	. . . . . 6 |- P = ( i e. ( 1 ... N ) |-> if ( i = 1 , I , if ( i <_ I , ( i - 1 ) , i ) ) )
23		eqid 2622	. . . . . 6 |- ( SymGrp ` ( 1 ... N ) ) = ( SymGrp ` ( 1 ... N ) )
24	21, 22, 23, 13	fzto1st 29853	. . . . 5 |- ( I e. ( 1 ... N ) -> P e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
25	10, 24	syl 17	. . . 4 |- ( ph -> P e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
26		nnuz 11723	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
27	8, 26	syl6eleq 2711	. . . . . . . 8 |- ( ph -> N e. ( ZZ>= ` 1 ) )
28		eluzfz2 12349	. . . . . . . 8 |- ( N e. ( ZZ>= ` 1 ) -> N e. ( 1 ... N ) )
29	27, 28	syl 17	. . . . . . 7 |- ( ph -> N e. ( 1 ... N ) )
30		madjusmdetlem2.s	. . . . . . . 8 |- S = ( i e. ( 1 ... N ) |-> if ( i = 1 , N , if ( i <_ N , ( i - 1 ) , i ) ) )
31	21, 30, 23, 13	fzto1st 29853	. . . . . . 7 |- ( N e. ( 1 ... N ) -> S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
32	29, 31	syl 17	. . . . . 6 |- ( ph -> S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
33		eqid 2622	. . . . . . 7 |- ( invg ` ( SymGrp ` ( 1 ... N ) ) ) = ( invg ` ( SymGrp ` ( 1 ... N ) ) )
34	23, 13, 33	symginv 17822	. . . . . 6 |- ( S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` S ) = `' S )
35	32, 34	syl 17	. . . . 5 |- ( ph -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` S ) = `' S )
36		fzfid 12772	. . . . . . 7 |- ( ph -> ( 1 ... N ) e. Fin )
37	23	symggrp 17820	. . . . . . 7 |- ( ( 1 ... N ) e. Fin -> ( SymGrp ` ( 1 ... N ) ) e. Grp )
38	36, 37	syl 17	. . . . . 6 |- ( ph -> ( SymGrp ` ( 1 ... N ) ) e. Grp )
39	13, 33	grpinvcl 17467	. . . . . 6 |- ( ( ( SymGrp ` ( 1 ... N ) ) e. Grp /\ S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
40	38, 32, 39	syl2anc 693	. . . . 5 |- ( ph -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
41	35, 40	eqeltrrd 2702	. . . 4 |- ( ph -> `' S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
42		eqid 2622	. . . . . 6 |- ( +g ` ( SymGrp ` ( 1 ... N ) ) ) = ( +g ` ( SymGrp ` ( 1 ... N ) ) )
43	23, 13, 42	symgov 17810	. . . . 5 |- ( ( P e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( P ( +g ` ( SymGrp ` ( 1 ... N ) ) ) `' S ) = ( P o. `' S ) )
44	23, 13, 42	symgcl 17811	. . . . 5 |- ( ( P e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( P ( +g ` ( SymGrp ` ( 1 ... N ) ) ) `' S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
45	43, 44	eqeltrrd 2702	. . . 4 |- ( ( P e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( P o. `' S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
46	25, 41, 45	syl2anc 693	. . 3 |- ( ph -> ( P o. `' S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
47		madjusmdetlem4.q	. . . . . 6 |- Q = ( j e. ( 1 ... N ) |-> if ( j = 1 , J , if ( j <_ J , ( j - 1 ) , j ) ) )
48	21, 47, 23, 13	fzto1st 29853	. . . . 5 |- ( J e. ( 1 ... N ) -> Q e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
49	11, 48	syl 17	. . . 4 |- ( ph -> Q e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
50		madjusmdetlem4.t	. . . . . . . 8 |- T = ( j e. ( 1 ... N ) |-> if ( j = 1 , N , if ( j <_ N , ( j - 1 ) , j ) ) )
51	21, 50, 23, 13	fzto1st 29853	. . . . . . 7 |- ( N e. ( 1 ... N ) -> T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
52	29, 51	syl 17	. . . . . 6 |- ( ph -> T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
53	23, 13, 33	symginv 17822	. . . . . 6 |- ( T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` T ) = `' T )
54	52, 53	syl 17	. . . . 5 |- ( ph -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` T ) = `' T )
55	13, 33	grpinvcl 17467	. . . . . 6 |- ( ( ( SymGrp ` ( 1 ... N ) ) e. Grp /\ T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
56	38, 52, 55	syl2anc 693	. . . . 5 |- ( ph -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
57	54, 56	eqeltrrd 2702	. . . 4 |- ( ph -> `' T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
58	23, 13, 42	symgov 17810	. . . . 5 |- ( ( Q e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( Q ( +g ` ( SymGrp ` ( 1 ... N ) ) ) `' T ) = ( Q o. `' T ) )
59	23, 13, 42	symgcl 17811	. . . . 5 |- ( ( Q e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( Q ( +g ` ( SymGrp ` ( 1 ... N ) ) ) `' T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
60	58, 59	eqeltrrd 2702	. . . 4 |- ( ( Q e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( Q o. `' T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
61	49, 57, 60	syl2anc 693	. . 3 |- ( ph -> ( Q o. `' T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
62	23, 13	symgbasf1o 17803	. . . . . . 7 |- ( S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) -> S : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
63	32, 62	syl 17	. . . . . 6 |- ( ph -> S : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
64		f1of1 6136	. . . . . 6 |- ( S : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> S : ( 1 ... N ) -1-1-> ( 1 ... N ) )
65		df-f1 5893	. . . . . . 7 |- ( S : ( 1 ... N ) -1-1-> ( 1 ... N ) <-> ( S : ( 1 ... N ) --> ( 1 ... N ) /\ Fun `' S ) )
66	65	simprbi 480	. . . . . 6 |- ( S : ( 1 ... N ) -1-1-> ( 1 ... N ) -> Fun `' S )
67	63, 64, 66	3syl 18	. . . . 5 |- ( ph -> Fun `' S )
68		f1ocnv 6149	. . . . . . 7 |- ( S : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> `' S : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
69		f1odm 6141	. . . . . . 7 |- ( `' S : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> dom `' S = ( 1 ... N ) )
70	63, 68, 69	3syl 18	. . . . . 6 |- ( ph -> dom `' S = ( 1 ... N ) )
71	29, 70	eleqtrrd 2704	. . . . 5 |- ( ph -> N e. dom `' S )
72		fvco 6274	. . . . 5 |- ( ( Fun `' S /\ N e. dom `' S ) -> ( ( P o. `' S ) ` N ) = ( P ` ( `' S ` N ) ) )
73	67, 71, 72	syl2anc 693	. . . 4 |- ( ph -> ( ( P o. `' S ) ` N ) = ( P ` ( `' S ` N ) ) )
74	21, 30, 23, 13	fzto1stinvn 29854	. . . . . 6 |- ( N e. ( 1 ... N ) -> ( `' S ` N ) = 1 )
75	29, 74	syl 17	. . . . 5 |- ( ph -> ( `' S ` N ) = 1 )
76	75	fveq2d 6195	. . . 4 |- ( ph -> ( P ` ( `' S ` N ) ) = ( P ` 1 ) )
77	21, 22	fzto1stfv1 29851	. . . . 5 |- ( I e. ( 1 ... N ) -> ( P ` 1 ) = I )
78	10, 77	syl 17	. . . 4 |- ( ph -> ( P ` 1 ) = I )
79	73, 76, 78	3eqtrd 2660	. . 3 |- ( ph -> ( ( P o. `' S ) ` N ) = I )
80	23, 13	symgbasf1o 17803	. . . . . . 7 |- ( T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) -> T : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
81	52, 80	syl 17	. . . . . 6 |- ( ph -> T : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
82		f1of1 6136	. . . . . 6 |- ( T : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> T : ( 1 ... N ) -1-1-> ( 1 ... N ) )
83		df-f1 5893	. . . . . . 7 |- ( T : ( 1 ... N ) -1-1-> ( 1 ... N ) <-> ( T : ( 1 ... N ) --> ( 1 ... N ) /\ Fun `' T ) )
84	83	simprbi 480	. . . . . 6 |- ( T : ( 1 ... N ) -1-1-> ( 1 ... N ) -> Fun `' T )
85	81, 82, 84	3syl 18	. . . . 5 |- ( ph -> Fun `' T )
86		f1ocnv 6149	. . . . . . 7 |- ( T : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> `' T : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
87		f1odm 6141	. . . . . . 7 |- ( `' T : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> dom `' T = ( 1 ... N ) )
88	81, 86, 87	3syl 18	. . . . . 6 |- ( ph -> dom `' T = ( 1 ... N ) )
89	29, 88	eleqtrrd 2704	. . . . 5 |- ( ph -> N e. dom `' T )
90		fvco 6274	. . . . 5 |- ( ( Fun `' T /\ N e. dom `' T ) -> ( ( Q o. `' T ) ` N ) = ( Q ` ( `' T ` N ) ) )
91	85, 89, 90	syl2anc 693	. . . 4 |- ( ph -> ( ( Q o. `' T ) ` N ) = ( Q ` ( `' T ` N ) ) )
92	21, 50, 23, 13	fzto1stinvn 29854	. . . . . 6 |- ( N e. ( 1 ... N ) -> ( `' T ` N ) = 1 )
93	29, 92	syl 17	. . . . 5 |- ( ph -> ( `' T ` N ) = 1 )
94	93	fveq2d 6195	. . . 4 |- ( ph -> ( Q ` ( `' T ` N ) ) = ( Q ` 1 ) )
95	21, 47	fzto1stfv1 29851	. . . . 5 |- ( J e. ( 1 ... N ) -> ( Q ` 1 ) = J )
96	11, 95	syl 17	. . . 4 |- ( ph -> ( Q ` 1 ) = J )
97	91, 94, 96	3eqtrd 2660	. . 3 |- ( ph -> ( ( Q o. `' T ) ` N ) = J )
98		crngring 18558	. . . . . 6 |- ( R e. CRing -> R e. Ring )
99	9, 98	syl 17	. . . . 5 |- ( ph -> R e. Ring )
100	2, 1	minmar1cl 20457	. . . . 5 |- ( ( ( R e. Ring /\ M e. B ) /\ ( I e. ( 1 ... N ) /\ J e. ( 1 ... N ) ) ) -> ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) e. B )
101	99, 12, 10, 11, 100	syl22anc 1327	. . . 4 |- ( ph -> ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) e. B )
102	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 22, 30, 47, 50, 20, 101	madjusmdetlem3 29895	. . 3 |- ( ph -> ( I (subMat1 ` ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ) J ) = ( N (subMat1 ` ( k e. ( 1 ... N ) , l e. ( 1 ... N ) |-> ( ( ( P o. `' S ) ` k ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( ( Q o. `' T ) ` l ) ) ) ) N ) )
103	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 46, 61, 79, 97, 102	madjusmdetlem1 29893	. 2 |- ( ph -> ( J ( K ` M ) I ) = ( ( Z ` ( ( (pmSgn ` ( 1 ... N ) ) ` ( P o. `' S ) ) x. ( (pmSgn ` ( 1 ... N ) ) ` ( Q o. `' T ) ) ) ) .x. ( E ` ( I (subMat1 ` M ) J ) ) ) )
104	23, 14, 13	psgnco 19929	. . . . . . . 8 |- ( ( ( 1 ... N ) e. Fin /\ P e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( (pmSgn ` ( 1 ... N ) ) ` ( P o. `' S ) ) = ( ( (pmSgn ` ( 1 ... N ) ) ` P ) x. ( (pmSgn ` ( 1 ... N ) ) ` `' S ) ) )
105	36, 25, 41, 104	syl3anc 1326	. . . . . . 7 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` ( P o. `' S ) ) = ( ( (pmSgn ` ( 1 ... N ) ) ` P ) x. ( (pmSgn ` ( 1 ... N ) ) ` `' S ) ) )
106	21, 22, 23, 13, 14	psgnfzto1st 29855	. . . . . . . . 9 |- ( I e. ( 1 ... N ) -> ( (pmSgn ` ( 1 ... N ) ) ` P ) = ( -u 1 ^ ( I + 1 ) ) )
107	10, 106	syl 17	. . . . . . . 8 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` P ) = ( -u 1 ^ ( I + 1 ) ) )
108	23, 14, 13	psgninv 19928	. . . . . . . . . 10 |- ( ( ( 1 ... N ) e. Fin /\ S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( (pmSgn ` ( 1 ... N ) ) ` `' S ) = ( (pmSgn ` ( 1 ... N ) ) ` S ) )
109	36, 32, 108	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` `' S ) = ( (pmSgn ` ( 1 ... N ) ) ` S ) )
110	21, 30, 23, 13, 14	psgnfzto1st 29855	. . . . . . . . . 10 |- ( N e. ( 1 ... N ) -> ( (pmSgn ` ( 1 ... N ) ) ` S ) = ( -u 1 ^ ( N + 1 ) ) )
111	29, 110	syl 17	. . . . . . . . 9 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` S ) = ( -u 1 ^ ( N + 1 ) ) )
112	109, 111	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` `' S ) = ( -u 1 ^ ( N + 1 ) ) )
113	107, 112	oveq12d 6668	. . . . . . 7 |- ( ph -> ( ( (pmSgn ` ( 1 ... N ) ) ` P ) x. ( (pmSgn ` ( 1 ... N ) ) ` `' S ) ) = ( ( -u 1 ^ ( I + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) )
114	105, 113	eqtrd 2656	. . . . . 6 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` ( P o. `' S ) ) = ( ( -u 1 ^ ( I + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) )
115	23, 14, 13	psgnco 19929	. . . . . . . 8 |- ( ( ( 1 ... N ) e. Fin /\ Q e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( (pmSgn ` ( 1 ... N ) ) ` ( Q o. `' T ) ) = ( ( (pmSgn ` ( 1 ... N ) ) ` Q ) x. ( (pmSgn ` ( 1 ... N ) ) ` `' T ) ) )
116	36, 49, 57, 115	syl3anc 1326	. . . . . . 7 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` ( Q o. `' T ) ) = ( ( (pmSgn ` ( 1 ... N ) ) ` Q ) x. ( (pmSgn ` ( 1 ... N ) ) ` `' T ) ) )
117	21, 47, 23, 13, 14	psgnfzto1st 29855	. . . . . . . . 9 |- ( J e. ( 1 ... N ) -> ( (pmSgn ` ( 1 ... N ) ) ` Q ) = ( -u 1 ^ ( J + 1 ) ) )
118	11, 117	syl 17	. . . . . . . 8 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` Q ) = ( -u 1 ^ ( J + 1 ) ) )
119	23, 14, 13	psgninv 19928	. . . . . . . . . 10 |- ( ( ( 1 ... N ) e. Fin /\ T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( (pmSgn ` ( 1 ... N ) ) ` `' T ) = ( (pmSgn ` ( 1 ... N ) ) ` T ) )
120	36, 52, 119	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` `' T ) = ( (pmSgn ` ( 1 ... N ) ) ` T ) )
121	21, 50, 23, 13, 14	psgnfzto1st 29855	. . . . . . . . . 10 |- ( N e. ( 1 ... N ) -> ( (pmSgn ` ( 1 ... N ) ) ` T ) = ( -u 1 ^ ( N + 1 ) ) )
122	29, 121	syl 17	. . . . . . . . 9 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` T ) = ( -u 1 ^ ( N + 1 ) ) )
123	120, 122	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` `' T ) = ( -u 1 ^ ( N + 1 ) ) )
124	118, 123	oveq12d 6668	. . . . . . 7 |- ( ph -> ( ( (pmSgn ` ( 1 ... N ) ) ` Q ) x. ( (pmSgn ` ( 1 ... N ) ) ` `' T ) ) = ( ( -u 1 ^ ( J + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) )
125	116, 124	eqtrd 2656	. . . . . 6 |- ( ph -> ( (pmSgn ` ( 1 ... N ) ) ` ( Q o. `' T ) ) = ( ( -u 1 ^ ( J + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) )
126	114, 125	oveq12d 6668	. . . . 5 |- ( ph -> ( ( (pmSgn ` ( 1 ... N ) ) ` ( P o. `' S ) ) x. ( (pmSgn ` ( 1 ... N ) ) ` ( Q o. `' T ) ) ) = ( ( ( -u 1 ^ ( I + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) x. ( ( -u 1 ^ ( J + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) ) )
127		1cnd 10056	. . . . . . . . 9 |- ( ph -> 1 e. CC )
128	127	negcld 10379	. . . . . . . 8 |- ( ph -> -u 1 e. CC )
129		fz1ssnn 12372	. . . . . . . . . . 11 |- ( 1 ... N ) C_ NN
130	129, 10	sseldi 3601	. . . . . . . . . 10 |- ( ph -> I e. NN )
131	130	nnnn0d 11351	. . . . . . . . 9 |- ( ph -> I e. NN0 )
132		1nn0 11308	. . . . . . . . . 10 |- 1 e. NN0
133	132	a1i 11	. . . . . . . . 9 |- ( ph -> 1 e. NN0 )
134	131, 133	nn0addcld 11355	. . . . . . . 8 |- ( ph -> ( I + 1 ) e. NN0 )
135	128, 134	expcld 13008	. . . . . . 7 |- ( ph -> ( -u 1 ^ ( I + 1 ) ) e. CC )
136	8	nnnn0d 11351	. . . . . . . . 9 |- ( ph -> N e. NN0 )
137	136, 133	nn0addcld 11355	. . . . . . . 8 |- ( ph -> ( N + 1 ) e. NN0 )
138	128, 137	expcld 13008	. . . . . . 7 |- ( ph -> ( -u 1 ^ ( N + 1 ) ) e. CC )
139	129, 11	sseldi 3601	. . . . . . . . . 10 |- ( ph -> J e. NN )
140	139	nnnn0d 11351	. . . . . . . . 9 |- ( ph -> J e. NN0 )
141	140, 133	nn0addcld 11355	. . . . . . . 8 |- ( ph -> ( J + 1 ) e. NN0 )
142	128, 141	expcld 13008	. . . . . . 7 |- ( ph -> ( -u 1 ^ ( J + 1 ) ) e. CC )
143	135, 138, 142, 138	mul4d 10248	. . . . . 6 |- ( ph -> ( ( ( -u 1 ^ ( I + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) x. ( ( -u 1 ^ ( J + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) ) = ( ( ( -u 1 ^ ( I + 1 ) ) x. ( -u 1 ^ ( J + 1 ) ) ) x. ( ( -u 1 ^ ( N + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) ) )
144	128, 141, 134	expaddd 13010	. . . . . . . 8 |- ( ph -> ( -u 1 ^ ( ( I + 1 ) + ( J + 1 ) ) ) = ( ( -u 1 ^ ( I + 1 ) ) x. ( -u 1 ^ ( J + 1 ) ) ) )
145	130	nncnd 11036	. . . . . . . . . . . 12 |- ( ph -> I e. CC )
146	139	nncnd 11036	. . . . . . . . . . . 12 |- ( ph -> J e. CC )
147	145, 127, 146, 127	add4d 10264	. . . . . . . . . . 11 |- ( ph -> ( ( I + 1 ) + ( J + 1 ) ) = ( ( I + J ) + ( 1 + 1 ) ) )
148		1p1e2 11134	. . . . . . . . . . . 12 |- ( 1 + 1 ) = 2
149	148	oveq2i 6661	. . . . . . . . . . 11 |- ( ( I + J ) + ( 1 + 1 ) ) = ( ( I + J ) + 2 )
150	147, 149	syl6eq 2672	. . . . . . . . . 10 |- ( ph -> ( ( I + 1 ) + ( J + 1 ) ) = ( ( I + J ) + 2 ) )
151	150	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( -u 1 ^ ( ( I + 1 ) + ( J + 1 ) ) ) = ( -u 1 ^ ( ( I + J ) + 2 ) ) )
152		2nn0 11309	. . . . . . . . . . . 12 |- 2 e. NN0
153	152	a1i 11	. . . . . . . . . . 11 |- ( ph -> 2 e. NN0 )
154	131, 140	nn0addcld 11355	. . . . . . . . . . 11 |- ( ph -> ( I + J ) e. NN0 )
155	128, 153, 154	expaddd 13010	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ ( ( I + J ) + 2 ) ) = ( ( -u 1 ^ ( I + J ) ) x. ( -u 1 ^ 2 ) ) )
156		neg1sqe1 12959	. . . . . . . . . . 11 |- ( -u 1 ^ 2 ) = 1
157	156	oveq2i 6661	. . . . . . . . . 10 |- ( ( -u 1 ^ ( I + J ) ) x. ( -u 1 ^ 2 ) ) = ( ( -u 1 ^ ( I + J ) ) x. 1 )
158	155, 157	syl6eq 2672	. . . . . . . . 9 |- ( ph -> ( -u 1 ^ ( ( I + J ) + 2 ) ) = ( ( -u 1 ^ ( I + J ) ) x. 1 ) )
159	128, 154	expcld 13008	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ ( I + J ) ) e. CC )
160	159	mulid1d 10057	. . . . . . . . 9 |- ( ph -> ( ( -u 1 ^ ( I + J ) ) x. 1 ) = ( -u 1 ^ ( I + J ) ) )
161	151, 158, 160	3eqtrd 2660	. . . . . . . 8 |- ( ph -> ( -u 1 ^ ( ( I + 1 ) + ( J + 1 ) ) ) = ( -u 1 ^ ( I + J ) ) )
162	144, 161	eqtr3d 2658	. . . . . . 7 |- ( ph -> ( ( -u 1 ^ ( I + 1 ) ) x. ( -u 1 ^ ( J + 1 ) ) ) = ( -u 1 ^ ( I + J ) ) )
163	137	nn0zd 11480	. . . . . . . 8 |- ( ph -> ( N + 1 ) e. ZZ )
164		m1expcl2 12882	. . . . . . . 8 |- ( ( N + 1 ) e. ZZ -> ( -u 1 ^ ( N + 1 ) ) e. { -u 1 , 1 } )
165		1neg1t1neg1 29514	. . . . . . . 8 |- ( ( -u 1 ^ ( N + 1 ) ) e. { -u 1 , 1 } -> ( ( -u 1 ^ ( N + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) = 1 )
166	163, 164, 165	3syl 18	. . . . . . 7 |- ( ph -> ( ( -u 1 ^ ( N + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) = 1 )
167	162, 166	oveq12d 6668	. . . . . 6 |- ( ph -> ( ( ( -u 1 ^ ( I + 1 ) ) x. ( -u 1 ^ ( J + 1 ) ) ) x. ( ( -u 1 ^ ( N + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) ) = ( ( -u 1 ^ ( I + J ) ) x. 1 ) )
168	143, 167, 160	3eqtrd 2660	. . . . 5 |- ( ph -> ( ( ( -u 1 ^ ( I + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) x. ( ( -u 1 ^ ( J + 1 ) ) x. ( -u 1 ^ ( N + 1 ) ) ) ) = ( -u 1 ^ ( I + J ) ) )
169	126, 168	eqtrd 2656	. . . 4 |- ( ph -> ( ( (pmSgn ` ( 1 ... N ) ) ` ( P o. `' S ) ) x. ( (pmSgn ` ( 1 ... N ) ) ` ( Q o. `' T ) ) ) = ( -u 1 ^ ( I + J ) ) )
170	169	fveq2d 6195	. . 3 |- ( ph -> ( Z ` ( ( (pmSgn ` ( 1 ... N ) ) ` ( P o. `' S ) ) x. ( (pmSgn ` ( 1 ... N ) ) ` ( Q o. `' T ) ) ) ) = ( Z ` ( -u 1 ^ ( I + J ) ) ) )
171	170	oveq1d 6665	. 2 |- ( ph -> ( ( Z ` ( ( (pmSgn ` ( 1 ... N ) ) ` ( P o. `' S ) ) x. ( (pmSgn ` ( 1 ... N ) ) ` ( Q o. `' T ) ) ) ) .x. ( E ` ( I (subMat1 ` M ) J ) ) ) = ( ( Z ` ( -u 1 ^ ( I + J ) ) ) .x. ( E ` ( I (subMat1 ` M ) J ) ) ) )
172	103, 171	eqtrd 2656	1 |- ( ph -> ( J ( K ` M ) I ) = ( ( Z ` ( -u 1 ^ ( I + J ) ) ) .x. ( E ` ( I (subMat1 ` M ) J ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   o. ccom 5118   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   -ucneg 10267   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   Grpcgrp 17422   invgcminusg 17423   SymGrpcsymg 17797  pmSgncpsgn 17909   Ringcrg 18547   CRingccrg 18548   ZRHomczrh 19848   Mat cmat 20213   maDet cmdat 20390   maAdju cmadu 20438   minMatR1 cminmar1 20439  subMat1csmat 29859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-ghm 17658  df-gim 17701  df-cntz 17750  df-oppg 17776  df-symg 17798  df-pmtr 17862  df-psgn 17911  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-sra 19172  df-rgmod 19173  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-dsmm 20076  df-frlm 20091  df-mat 20214  df-marrep 20364  df-subma 20383  df-mdet 20391  df-madu 20440  df-minmar1 20441  df-smat 29860

This theorem is referenced by:  madjusmdet  29897