Theorem madjusmdetlem3 29895

Description: Lemma for madjusmdet 29897. (Contributed by Thierry Arnoux, 27-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 ) ) )
madjusmdetlem3.w	|- W = ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( ( P o. `' S ) ` i ) U ( ( Q o. `' T ) ` j ) ) )
madjusmdetlem3.u	|- ( ph -> U e. B )

Assertion

Ref	Expression
madjusmdetlem3	|- ( ph -> ( I (subMat1 ` U ) J ) = ( N (subMat1 ` W ) N ) )

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   U, i, j   i, W, 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 madjusmdetlem3

Step	Hyp	Ref	Expression
1		madjusmdet.n	. . . . . . . . . . 11 |- ( ph -> N e. NN )
2		nnuz 11723	. . . . . . . . . . 11 |- NN = ( ZZ>= ` 1 )
3	1, 2	syl6eleq 2711	. . . . . . . . . 10 |- ( ph -> N e. ( ZZ>= ` 1 ) )
4		fzdif2 29551	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 1 ) -> ( ( 1 ... N ) \ { N } ) = ( 1 ... ( N - 1 ) ) )
5	3, 4	syl 17	. . . . . . . . 9 |- ( ph -> ( ( 1 ... N ) \ { N } ) = ( 1 ... ( N - 1 ) ) )
6		difss 3737	. . . . . . . . 9 |- ( ( 1 ... N ) \ { N } ) C_ ( 1 ... N )
7	5, 6	syl6eqssr 3656	. . . . . . . 8 |- ( ph -> ( 1 ... ( N - 1 ) ) C_ ( 1 ... N ) )
8	7	adantr 481	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> ( 1 ... ( N - 1 ) ) C_ ( 1 ... N ) )
9		simprl 794	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> i e. ( 1 ... ( N - 1 ) ) )
10	8, 9	sseldd 3604	. . . . . 6 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> i e. ( 1 ... N ) )
11		simprr 796	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> j e. ( 1 ... ( N - 1 ) ) )
12	8, 11	sseldd 3604	. . . . . 6 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> j e. ( 1 ... N ) )
13		ovexd 6680	. . . . . 6 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> ( ( ( P o. `' S ) ` i ) U ( ( Q o. `' T ) ` j ) ) e. _V )
14		madjusmdetlem3.w	. . . . . . 7 |- W = ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( ( P o. `' S ) ` i ) U ( ( Q o. `' T ) ` j ) ) )
15	14	ovmpt4g 6783	. . . . . 6 |- ( ( i e. ( 1 ... N ) /\ j e. ( 1 ... N ) /\ ( ( ( P o. `' S ) ` i ) U ( ( Q o. `' T ) ` j ) ) e. _V ) -> ( i W j ) = ( ( ( P o. `' S ) ` i ) U ( ( Q o. `' T ) ` j ) ) )
16	10, 12, 13, 15	syl3anc 1326	. . . . 5 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> ( i W j ) = ( ( ( P o. `' S ) ` i ) U ( ( Q o. `' T ) ` j ) ) )
17	9, 11	ovresd 6801	. . . . 5 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> ( i ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) j ) = ( i W j ) )
18		eqid 2622	. . . . . . 7 |- ( I (subMat1 ` U ) J ) = ( I (subMat1 ` U ) J )
19	1	adantr 481	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> N e. NN )
20		madjusmdet.i	. . . . . . . 8 |- ( ph -> I e. ( 1 ... N ) )
21	20	adantr 481	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> I e. ( 1 ... N ) )
22		madjusmdet.j	. . . . . . . 8 |- ( ph -> J e. ( 1 ... N ) )
23	22	adantr 481	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> J e. ( 1 ... N ) )
24		madjusmdetlem3.u	. . . . . . . . 9 |- ( ph -> U e. B )
25		madjusmdet.a	. . . . . . . . . 10 |- A = ( ( 1 ... N ) Mat R )
26		eqid 2622	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
27		madjusmdet.b	. . . . . . . . . 10 |- B = ( Base ` A )
28	25, 26, 27	matbas2i 20228	. . . . . . . . 9 |- ( U e. B -> U e. ( ( Base ` R ) ^m ( ( 1 ... N ) X. ( 1 ... N ) ) ) )
29	24, 28	syl 17	. . . . . . . 8 |- ( ph -> U e. ( ( Base ` R ) ^m ( ( 1 ... N ) X. ( 1 ... N ) ) ) )
30	29	adantr 481	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> U e. ( ( Base ` R ) ^m ( ( 1 ... N ) X. ( 1 ... N ) ) ) )
31		fz1ssnn 12372	. . . . . . . 8 |- ( 1 ... N ) C_ NN
32	31, 10	sseldi 3601	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> i e. NN )
33	31, 12	sseldi 3601	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> j e. NN )
34		eqidd 2623	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> if ( i < I , i , ( i + 1 ) ) = if ( i < I , i , ( i + 1 ) ) )
35		eqidd 2623	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> if ( j < J , j , ( j + 1 ) ) = if ( j < J , j , ( j + 1 ) ) )
36	18, 19, 19, 21, 23, 30, 32, 33, 34, 35	smatlem 29863	. . . . . 6 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> ( i ( I (subMat1 ` U ) J ) j ) = ( if ( i < I , i , ( i + 1 ) ) U if ( j < J , j , ( j + 1 ) ) ) )
37		madjusmdet.d	. . . . . . . . 9 |- D = ( ( 1 ... N ) maDet R )
38		madjusmdet.k	. . . . . . . . 9 |- K = ( ( 1 ... N ) maAdju R )
39		madjusmdet.t	. . . . . . . . 9 |- .x. = ( .r ` R )
40		madjusmdet.z	. . . . . . . . 9 |- Z = ( ZRHom ` R )
41		madjusmdet.e	. . . . . . . . 9 |- E = ( ( 1 ... ( N - 1 ) ) maDet R )
42		madjusmdet.r	. . . . . . . . 9 |- ( ph -> R e. CRing )
43		madjusmdet.m	. . . . . . . . 9 |- ( ph -> M e. B )
44		madjusmdetlem2.p	. . . . . . . . 9 |- P = ( i e. ( 1 ... N ) |-> if ( i = 1 , I , if ( i <_ I , ( i - 1 ) , i ) ) )
45		madjusmdetlem2.s	. . . . . . . . 9 |- S = ( i e. ( 1 ... N ) |-> if ( i = 1 , N , if ( i <_ N , ( i - 1 ) , i ) ) )
46	27, 25, 37, 38, 39, 40, 41, 1, 42, 20, 20, 43, 44, 45	madjusmdetlem2 29894	. . . . . . . 8 |- ( ( ph /\ i e. ( 1 ... ( N - 1 ) ) ) -> if ( i < I , i , ( i + 1 ) ) = ( ( P o. `' S ) ` i ) )
47	9, 46	syldan 487	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> if ( i < I , i , ( i + 1 ) ) = ( ( P o. `' S ) ` i ) )
48		madjusmdetlem4.q	. . . . . . . . 9 |- Q = ( j e. ( 1 ... N ) |-> if ( j = 1 , J , if ( j <_ J , ( j - 1 ) , j ) ) )
49		madjusmdetlem4.t	. . . . . . . . 9 |- T = ( j e. ( 1 ... N ) |-> if ( j = 1 , N , if ( j <_ N , ( j - 1 ) , j ) ) )
50	27, 25, 37, 38, 39, 40, 41, 1, 42, 22, 22, 43, 48, 49	madjusmdetlem2 29894	. . . . . . . 8 |- ( ( ph /\ j e. ( 1 ... ( N - 1 ) ) ) -> if ( j < J , j , ( j + 1 ) ) = ( ( Q o. `' T ) ` j ) )
51	11, 50	syldan 487	. . . . . . 7 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> if ( j < J , j , ( j + 1 ) ) = ( ( Q o. `' T ) ` j ) )
52	47, 51	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> ( if ( i < I , i , ( i + 1 ) ) U if ( j < J , j , ( j + 1 ) ) ) = ( ( ( P o. `' S ) ` i ) U ( ( Q o. `' T ) ` j ) ) )
53	36, 52	eqtrd 2656	. . . . 5 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> ( i ( I (subMat1 ` U ) J ) j ) = ( ( ( P o. `' S ) ` i ) U ( ( Q o. `' T ) ` j ) ) )
54	16, 17, 53	3eqtr4rd 2667	. . . 4 |- ( ( ph /\ ( i e. ( 1 ... ( N - 1 ) ) /\ j e. ( 1 ... ( N - 1 ) ) ) ) -> ( i ( I (subMat1 ` U ) J ) j ) = ( i ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) j ) )
55	54	ralrimivva 2971	. . 3 |- ( ph -> A. i e. ( 1 ... ( N - 1 ) ) A. j e. ( 1 ... ( N - 1 ) ) ( i ( I (subMat1 ` U ) J ) j ) = ( i ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) j ) )
56		eqid 2622	. . . . 5 |- ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) ) = ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) )
57	25, 27, 56, 18, 1, 20, 22, 24	smatcl 29868	. . . 4 |- ( ph -> ( I (subMat1 ` U ) J ) e. ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) ) )
58		fzfid 12772	. . . . . . . 8 |- ( ph -> ( 1 ... N ) e. Fin )
59		eqid 2622	. . . . . . . . . . . . . 14 |- ( 1 ... N ) = ( 1 ... N )
60		eqid 2622	. . . . . . . . . . . . . 14 |- ( SymGrp ` ( 1 ... N ) ) = ( SymGrp ` ( 1 ... N ) )
61		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` ( SymGrp ` ( 1 ... N ) ) ) = ( Base ` ( SymGrp ` ( 1 ... N ) ) )
62	59, 44, 60, 61	fzto1st 29853	. . . . . . . . . . . . 13 |- ( I e. ( 1 ... N ) -> P e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
63	20, 62	syl 17	. . . . . . . . . . . 12 |- ( ph -> P e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
64		eluzfz2 12349	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 1 ) -> N e. ( 1 ... N ) )
65	3, 64	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> N e. ( 1 ... N ) )
66	59, 45, 60, 61	fzto1st 29853	. . . . . . . . . . . . . . 15 |- ( N e. ( 1 ... N ) -> S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
67	65, 66	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
68		eqid 2622	. . . . . . . . . . . . . . 15 |- ( invg ` ( SymGrp ` ( 1 ... N ) ) ) = ( invg ` ( SymGrp ` ( 1 ... N ) ) )
69	60, 61, 68	symginv 17822	. . . . . . . . . . . . . 14 |- ( S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` S ) = `' S )
70	67, 69	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` S ) = `' S )
71	60	symggrp 17820	. . . . . . . . . . . . . . 15 |- ( ( 1 ... N ) e. Fin -> ( SymGrp ` ( 1 ... N ) ) e. Grp )
72	58, 71	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( SymGrp ` ( 1 ... N ) ) e. Grp )
73	61, 68	grpinvcl 17467	. . . . . . . . . . . . . 14 |- ( ( ( SymGrp ` ( 1 ... N ) ) e. Grp /\ S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
74	72, 67, 73	syl2anc 693	. . . . . . . . . . . . 13 |- ( ph -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
75	70, 74	eqeltrrd 2702	. . . . . . . . . . . 12 |- ( ph -> `' S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
76		eqid 2622	. . . . . . . . . . . . . 14 |- ( +g ` ( SymGrp ` ( 1 ... N ) ) ) = ( +g ` ( SymGrp ` ( 1 ... N ) ) )
77	60, 61, 76	symgov 17810	. . . . . . . . . . . . 13 |- ( ( P e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( P ( +g ` ( SymGrp ` ( 1 ... N ) ) ) `' S ) = ( P o. `' S ) )
78	60, 61, 76	symgcl 17811	. . . . . . . . . . . . 13 |- ( ( P e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( P ( +g ` ( SymGrp ` ( 1 ... N ) ) ) `' S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
79	77, 78	eqeltrrd 2702	. . . . . . . . . . . 12 |- ( ( P e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' S e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( P o. `' S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
80	63, 75, 79	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( P o. `' S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
81	80	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( P o. `' S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
82		simp2 1062	. . . . . . . . . 10 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> i e. ( 1 ... N ) )
83	60, 61	symgfv 17807	. . . . . . . . . 10 |- ( ( ( P o. `' S ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( P o. `' S ) ` i ) e. ( 1 ... N ) )
84	81, 82, 83	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( ( P o. `' S ) ` i ) e. ( 1 ... N ) )
85	59, 48, 60, 61	fzto1st 29853	. . . . . . . . . . . . 13 |- ( J e. ( 1 ... N ) -> Q e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
86	22, 85	syl 17	. . . . . . . . . . . 12 |- ( ph -> Q e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
87	59, 49, 60, 61	fzto1st 29853	. . . . . . . . . . . . . . 15 |- ( N e. ( 1 ... N ) -> T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
88	65, 87	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
89	60, 61, 68	symginv 17822	. . . . . . . . . . . . . 14 |- ( T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` T ) = `' T )
90	88, 89	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` T ) = `' T )
91	61, 68	grpinvcl 17467	. . . . . . . . . . . . . 14 |- ( ( ( SymGrp ` ( 1 ... N ) ) e. Grp /\ T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
92	72, 88, 91	syl2anc 693	. . . . . . . . . . . . 13 |- ( ph -> ( ( invg ` ( SymGrp ` ( 1 ... N ) ) ) ` T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
93	90, 92	eqeltrrd 2702	. . . . . . . . . . . 12 |- ( ph -> `' T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
94	60, 61, 76	symgov 17810	. . . . . . . . . . . . 13 |- ( ( Q e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( Q ( +g ` ( SymGrp ` ( 1 ... N ) ) ) `' T ) = ( Q o. `' T ) )
95	60, 61, 76	symgcl 17811	. . . . . . . . . . . . 13 |- ( ( Q e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( Q ( +g ` ( SymGrp ` ( 1 ... N ) ) ) `' T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
96	94, 95	eqeltrrd 2702	. . . . . . . . . . . 12 |- ( ( Q e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ `' T e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) ) -> ( Q o. `' T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
97	86, 93, 96	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( Q o. `' T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
98	97	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( Q o. `' T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) )
99		simp3 1063	. . . . . . . . . 10 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> j e. ( 1 ... N ) )
100	60, 61	symgfv 17807	. . . . . . . . . 10 |- ( ( ( Q o. `' T ) e. ( Base ` ( SymGrp ` ( 1 ... N ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( Q o. `' T ) ` j ) e. ( 1 ... N ) )
101	98, 99, 100	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( ( Q o. `' T ) ` j ) e. ( 1 ... N ) )
102	24	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> U e. B )
103	25, 26, 27, 84, 101, 102	matecld 20232	. . . . . . . 8 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( ( ( P o. `' S ) ` i ) U ( ( Q o. `' T ) ` j ) ) e. ( Base ` R ) )
104	25, 26, 27, 58, 42, 103	matbas2d 20229	. . . . . . 7 |- ( ph -> ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( ( P o. `' S ) ` i ) U ( ( Q o. `' T ) ` j ) ) ) e. B )
105	14, 104	syl5eqel 2705	. . . . . 6 |- ( ph -> W e. B )
106	25, 27	submatres 29872	. . . . . 6 |- ( ( N e. NN /\ W e. B ) -> ( N (subMat1 ` W ) N ) = ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) )
107	1, 105, 106	syl2anc 693	. . . . 5 |- ( ph -> ( N (subMat1 ` W ) N ) = ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) )
108		eqid 2622	. . . . . 6 |- ( N (subMat1 ` W ) N ) = ( N (subMat1 ` W ) N )
109	25, 27, 56, 108, 1, 65, 65, 105	smatcl 29868	. . . . 5 |- ( ph -> ( N (subMat1 ` W ) N ) e. ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) ) )
110	107, 109	eqeltrrd 2702	. . . 4 |- ( ph -> ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) e. ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) ) )
111		eqid 2622	. . . . 5 |- ( ( 1 ... ( N - 1 ) ) Mat R ) = ( ( 1 ... ( N - 1 ) ) Mat R )
112	111, 56	eqmat 20230	. . . 4 |- ( ( ( I (subMat1 ` U ) J ) e. ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) ) /\ ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) e. ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) ) ) -> ( ( I (subMat1 ` U ) J ) = ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) <-> A. i e. ( 1 ... ( N - 1 ) ) A. j e. ( 1 ... ( N - 1 ) ) ( i ( I (subMat1 ` U ) J ) j ) = ( i ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) j ) ) )
113	57, 110, 112	syl2anc 693	. . 3 |- ( ph -> ( ( I (subMat1 ` U ) J ) = ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) <-> A. i e. ( 1 ... ( N - 1 ) ) A. j e. ( 1 ... ( N - 1 ) ) ( i ( I (subMat1 ` U ) J ) j ) = ( i ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) j ) ) )
114	55, 113	mpbird 247	. 2 |- ( ph -> ( I (subMat1 ` U ) J ) = ( W |` ( ( 1 ... ( N - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) )
115	114, 107	eqtr4d 2659	1 |- ( ph -> ( I (subMat1 ` U ) J ) = ( N (subMat1 ` W ) N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Fincfn 7955   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   ZZ>=cuz 11687   ...cfz 12326   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   Grpcgrp 17422   invgcminusg 17423   SymGrpcsymg 17797   CRingccrg 18548   ZRHomczrh 19848   Mat cmat 20213   maDet cmdat 20390   maAdju cmadu 20438  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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  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-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  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-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  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-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-symg 17798  df-pmtr 17862  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091  df-mat 20214  df-subma 20383  df-smat 29860

This theorem is referenced by:  madjusmdetlem4  29896