Theorem madjusmdetlem1 29893

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 )
madjusmdetlem1.g	|- G = ( Base ` ( SymGrp ` ( 1 ... N ) ) )
madjusmdetlem1.s	|- S = (pmSgn ` ( 1 ... N ) )
madjusmdetlem1.u	|- U = ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J )
madjusmdetlem1.w	|- W = ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( P ` i ) U ( Q ` j ) ) )
madjusmdetlem1.p	|- ( ph -> P e. G )
madjusmdetlem1.q	|- ( ph -> Q e. G )
madjusmdetlem1.1	|- ( ph -> ( P ` N ) = I )
madjusmdetlem1.2	|- ( ph -> ( Q ` N ) = J )
madjusmdetlem1.3	|- ( ph -> ( I (subMat1 ` U ) J ) = ( N (subMat1 ` W ) N ) )

Assertion

Ref	Expression
madjusmdetlem1	|- ( ph -> ( J ( K ` M ) I ) = ( ( Z ` ( ( S ` P ) x. ( S ` Q ) ) ) .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   i, G, j   i, W, j   U, i, j

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

Proof of Theorem madjusmdetlem1

Step	Hyp	Ref	Expression
1		madjusmdet.m	. . . 4 |- ( ph -> M e. B )
2		madjusmdet.j	. . . 4 |- ( ph -> J e. ( 1 ... N ) )
3		madjusmdet.i	. . . 4 |- ( ph -> I e. ( 1 ... N ) )
4		madjusmdet.a	. . . . 5 |- A = ( ( 1 ... N ) Mat R )
5		madjusmdet.b	. . . . 5 |- B = ( Base ` A )
6		madjusmdet.d	. . . . 5 |- D = ( ( 1 ... N ) maDet R )
7		madjusmdet.k	. . . . 5 |- K = ( ( 1 ... N ) maAdju R )
8	4, 5, 6, 7	maducoevalmin1 20458	. . . 4 |- ( ( M e. B /\ J e. ( 1 ... N ) /\ I e. ( 1 ... N ) ) -> ( J ( K ` M ) I ) = ( D ` ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ) )
9	1, 2, 3, 8	syl3anc 1326	. . 3 |- ( ph -> ( J ( K ` M ) I ) = ( D ` ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ) )
10		madjusmdetlem1.u	. . . 4 |- U = ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J )
11	10	fveq2i 6194	. . 3 |- ( D ` U ) = ( D ` ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) )
12	9, 11	syl6eqr 2674	. 2 |- ( ph -> ( J ( K ` M ) I ) = ( D ` U ) )
13		madjusmdetlem1.g	. . 3 |- G = ( Base ` ( SymGrp ` ( 1 ... N ) ) )
14		madjusmdetlem1.s	. . 3 |- S = (pmSgn ` ( 1 ... N ) )
15		madjusmdet.z	. . 3 |- Z = ( ZRHom ` R )
16		madjusmdet.t	. . 3 |- .x. = ( .r ` R )
17		madjusmdetlem1.w	. . 3 |- W = ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( P ` i ) U ( Q ` j ) ) )
18		madjusmdet.r	. . 3 |- ( ph -> R e. CRing )
19		fzfid 12772	. . 3 |- ( ph -> ( 1 ... N ) e. Fin )
20		crngring 18558	. . . . . 6 |- ( R e. CRing -> R e. Ring )
21	18, 20	syl 17	. . . . 5 |- ( ph -> R e. Ring )
22	4, 5	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 )
23	21, 1, 3, 2, 22	syl22anc 1327	. . . 4 |- ( ph -> ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) e. B )
24	10, 23	syl5eqel 2705	. . 3 |- ( ph -> U e. B )
25		madjusmdetlem1.p	. . 3 |- ( ph -> P e. G )
26		madjusmdetlem1.q	. . 3 |- ( ph -> Q e. G )
27	4, 5, 6, 13, 14, 15, 16, 17, 18, 19, 24, 25, 26	mdetpmtr12 29891	. 2 |- ( ph -> ( D ` U ) = ( ( Z ` ( ( S ` P ) x. ( S ` Q ) ) ) .x. ( D ` W ) ) )
28		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> i = N )
29	28	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> ( P ` i ) = ( P ` N ) )
30		madjusmdetlem1.1	. . . . . . . . . . . . . . 15 |- ( ph -> ( P ` N ) = I )
31	30	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( P ` N ) = I )
32	31	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> ( P ` N ) = I )
33	29, 32	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> ( P ` i ) = I )
34		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> j = N )
35	34	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> ( Q ` j ) = ( Q ` N ) )
36		madjusmdetlem1.2	. . . . . . . . . . . . . . 15 |- ( ph -> ( Q ` N ) = J )
37	36	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( Q ` N ) = J )
38	37	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> ( Q ` N ) = J )
39	35, 38	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> ( Q ` j ) = J )
40	33, 39	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) = ( I ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) J ) )
41	1	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> M e. B )
42	41	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> M e. B )
43	3	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> I e. ( 1 ... N ) )
44	43	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> I e. ( 1 ... N ) )
45	2	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> J e. ( 1 ... N ) )
46	45	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> J e. ( 1 ... N ) )
47		eqid 2622	. . . . . . . . . . . . 13 |- ( ( 1 ... N ) minMatR1 R ) = ( ( 1 ... N ) minMatR1 R )
48		eqid 2622	. . . . . . . . . . . . 13 |- ( 1r ` R ) = ( 1r ` R )
49		eqid 2622	. . . . . . . . . . . . 13 |- ( 0g ` R ) = ( 0g ` R )
50	4, 5, 47, 48, 49	minmar1eval 20455	. . . . . . . . . . . 12 |- ( ( M e. B /\ ( I e. ( 1 ... N ) /\ J e. ( 1 ... N ) ) /\ ( I e. ( 1 ... N ) /\ J e. ( 1 ... N ) ) ) -> ( I ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) J ) = if ( I = I , if ( J = J , ( 1r ` R ) , ( 0g ` R ) ) , ( I M J ) ) )
51	42, 44, 46, 44, 46, 50	syl122anc 1335	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> ( I ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) J ) = if ( I = I , if ( J = J , ( 1r ` R ) , ( 0g ` R ) ) , ( I M J ) ) )
52		eqid 2622	. . . . . . . . . . . . . 14 |- I = I
53	52	iftruei 4093	. . . . . . . . . . . . 13 |- if ( I = I , if ( J = J , ( 1r ` R ) , ( 0g ` R ) ) , ( I M J ) ) = if ( J = J , ( 1r ` R ) , ( 0g ` R ) )
54		eqid 2622	. . . . . . . . . . . . . 14 |- J = J
55	54	iftruei 4093	. . . . . . . . . . . . 13 |- if ( J = J , ( 1r ` R ) , ( 0g ` R ) ) = ( 1r ` R )
56	53, 55	eqtri 2644	. . . . . . . . . . . 12 |- if ( I = I , if ( J = J , ( 1r ` R ) , ( 0g ` R ) ) , ( I M J ) ) = ( 1r ` R )
57	56	a1i 11	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> if ( I = I , if ( J = J , ( 1r ` R ) , ( 0g ` R ) ) , ( I M J ) ) = ( 1r ` R ) )
58	40, 51, 57	3eqtrrd 2661	. . . . . . . . . 10 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ j = N ) -> ( 1r ` R ) = ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) )
59		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> i = N )
60	59	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> ( P ` i ) = ( P ` N ) )
61	31	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> ( P ` N ) = I )
62	60, 61	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> ( P ` i ) = I )
63	62	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) = ( I ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) )
64	41	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> M e. B )
65	43	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> I e. ( 1 ... N ) )
66	45	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> J e. ( 1 ... N ) )
67	26	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> Q e. G )
68		simp3 1063	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> j e. ( 1 ... N ) )
69		eqid 2622	. . . . . . . . . . . . . . 15 |- ( SymGrp ` ( 1 ... N ) ) = ( SymGrp ` ( 1 ... N ) )
70	69, 13	symgfv 17807	. . . . . . . . . . . . . 14 |- ( ( Q e. G /\ j e. ( 1 ... N ) ) -> ( Q ` j ) e. ( 1 ... N ) )
71	67, 68, 70	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( Q ` j ) e. ( 1 ... N ) )
72	71	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> ( Q ` j ) e. ( 1 ... N ) )
73	4, 5, 47, 48, 49	minmar1eval 20455	. . . . . . . . . . . 12 |- ( ( M e. B /\ ( I e. ( 1 ... N ) /\ J e. ( 1 ... N ) ) /\ ( I e. ( 1 ... N ) /\ ( Q ` j ) e. ( 1 ... N ) ) ) -> ( I ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) = if ( I = I , if ( ( Q ` j ) = J , ( 1r ` R ) , ( 0g ` R ) ) , ( I M ( Q ` j ) ) ) )
74	64, 65, 66, 65, 72, 73	syl122anc 1335	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> ( I ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) = if ( I = I , if ( ( Q ` j ) = J , ( 1r ` R ) , ( 0g ` R ) ) , ( I M ( Q ` j ) ) ) )
75	52	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> I = I )
76	75	iftrued 4094	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> if ( I = I , if ( ( Q ` j ) = J , ( 1r ` R ) , ( 0g ` R ) ) , ( I M ( Q ` j ) ) ) = if ( ( Q ` j ) = J , ( 1r ` R ) , ( 0g ` R ) ) )
77		simpr 477	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ ( Q ` j ) = J ) -> ( Q ` j ) = J )
78	77	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ ( Q ` j ) = J ) -> ( `' Q ` ( Q ` j ) ) = ( `' Q ` J ) )
79	69, 13	symgbasf1o 17803	. . . . . . . . . . . . . . . . . . . 20 |- ( Q e. G -> Q : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
80	67, 79	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> Q : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
81	80	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ ( Q ` j ) = J ) -> Q : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
82	68	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ ( Q ` j ) = J ) -> j e. ( 1 ... N ) )
83		f1ocnvfv1 6532	. . . . . . . . . . . . . . . . . 18 |- ( ( Q : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( `' Q ` ( Q ` j ) ) = j )
84	81, 82, 83	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ ( Q ` j ) = J ) -> ( `' Q ` ( Q ` j ) ) = j )
85	36	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( `' Q ` ( Q ` N ) ) = ( `' Q ` J ) )
86	26, 79	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> Q : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
87		madjusmdet.n	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> N e. NN )
88		nnuz 11723	. . . . . . . . . . . . . . . . . . . . . . 23 |- NN = ( ZZ>= ` 1 )
89	87, 88	syl6eleq 2711	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> N e. ( ZZ>= ` 1 ) )
90		eluzfz2 12349	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 1 ) -> N e. ( 1 ... N ) )
91	89, 90	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> N e. ( 1 ... N ) )
92		f1ocnvfv1 6532	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( Q : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) /\ N e. ( 1 ... N ) ) -> ( `' Q ` ( Q ` N ) ) = N )
93	86, 91, 92	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( `' Q ` ( Q ` N ) ) = N )
94	85, 93	eqtr3d 2658	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( `' Q ` J ) = N )
95	94	3ad2ant1 1082	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( `' Q ` J ) = N )
96	95	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ ( Q ` j ) = J ) -> ( `' Q ` J ) = N )
97	78, 84, 96	3eqtr3d 2664	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ ( Q ` j ) = J ) -> j = N )
98	97	ex 450	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) -> ( ( Q ` j ) = J -> j = N ) )
99	98	con3d 148	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) -> ( -. j = N -> -. ( Q ` j ) = J ) )
100	99	imp 445	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> -. ( Q ` j ) = J )
101	100	iffalsed 4097	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> if ( ( Q ` j ) = J , ( 1r ` R ) , ( 0g ` R ) ) = ( 0g ` R ) )
102	76, 101	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> if ( I = I , if ( ( Q ` j ) = J , ( 1r ` R ) , ( 0g ` R ) ) , ( I M ( Q ` j ) ) ) = ( 0g ` R ) )
103	63, 74, 102	3eqtrrd 2661	. . . . . . . . . 10 |- ( ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) /\ -. j = N ) -> ( 0g ` R ) = ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) )
104	58, 103	ifeqda 4121	. . . . . . . . 9 |- ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ i = N ) -> if ( j = N , ( 1r ` R ) , ( 0g ` R ) ) = ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) )
105		simp2 1062	. . . . . . . . . . 11 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> i e. ( 1 ... N ) )
106	105	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ -. i = N ) -> i e. ( 1 ... N ) )
107	68	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ -. i = N ) -> j e. ( 1 ... N ) )
108		ovexd 6680	. . . . . . . . . 10 |- ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ -. i = N ) -> ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) e. _V )
109	10	oveqi 6663	. . . . . . . . . . . . . 14 |- ( ( P ` i ) U ( Q ` j ) ) = ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) )
110	109	a1i 11	. . . . . . . . . . . . 13 |- ( ( i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( ( P ` i ) U ( Q ` j ) ) = ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) )
111	110	mpt2eq3ia 6720	. . . . . . . . . . . 12 |- ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( P ` i ) U ( Q ` j ) ) ) = ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) )
112	17, 111	eqtri 2644	. . . . . . . . . . 11 |- W = ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) )
113	112	ovmpt4g 6783	. . . . . . . . . 10 |- ( ( i e. ( 1 ... N ) /\ j e. ( 1 ... N ) /\ ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) e. _V ) -> ( i W j ) = ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) )
114	106, 107, 108, 113	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) /\ -. i = N ) -> ( i W j ) = ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) )
115	104, 114	ifeqda 4121	. . . . . . . 8 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> if ( i = N , if ( j = N , ( 1r ` R ) , ( 0g ` R ) ) , ( i W j ) ) = ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) )
116	115	mpt2eq3dva 6719	. . . . . . 7 |- ( ph -> ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> if ( i = N , if ( j = N , ( 1r ` R ) , ( 0g ` R ) ) , ( i W j ) ) ) = ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) ) )
117		eqid 2622	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
118	25	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> P e. G )
119	69, 13	symgfv 17807	. . . . . . . . . . . 12 |- ( ( P e. G /\ i e. ( 1 ... N ) ) -> ( P ` i ) e. ( 1 ... N ) )
120	118, 105, 119	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( P ` i ) e. ( 1 ... N ) )
121	24	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> U e. B )
122	4, 117, 5, 120, 71, 121	matecld 20232	. . . . . . . . . 10 |- ( ( ph /\ i e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( ( P ` i ) U ( Q ` j ) ) e. ( Base ` R ) )
123	4, 117, 5, 19, 18, 122	matbas2d 20229	. . . . . . . . 9 |- ( ph -> ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( P ` i ) U ( Q ` j ) ) ) e. B )
124	17, 123	syl5eqel 2705	. . . . . . . 8 |- ( ph -> W e. B )
125	117, 48	ringidcl 18568	. . . . . . . . 9 |- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) )
126	21, 125	syl 17	. . . . . . . 8 |- ( ph -> ( 1r ` R ) e. ( Base ` R ) )
127		eqid 2622	. . . . . . . . 9 |- ( ( 1 ... N ) matRRep R ) = ( ( 1 ... N ) matRRep R )
128	4, 5, 127, 49	marrepval 20368	. . . . . . . 8 |- ( ( ( W e. B /\ ( 1r ` R ) e. ( Base ` R ) ) /\ ( N e. ( 1 ... N ) /\ N e. ( 1 ... N ) ) ) -> ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) = ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> if ( i = N , if ( j = N , ( 1r ` R ) , ( 0g ` R ) ) , ( i W j ) ) ) )
129	124, 126, 91, 91, 128	syl22anc 1327	. . . . . . 7 |- ( ph -> ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) = ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> if ( i = N , if ( j = N , ( 1r ` R ) , ( 0g ` R ) ) , ( i W j ) ) ) )
130	112	a1i 11	. . . . . . 7 |- ( ph -> W = ( i e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( P ` i ) ( I ( ( ( 1 ... N ) minMatR1 R ) ` M ) J ) ( Q ` j ) ) ) )
131	116, 129, 130	3eqtr4d 2666	. . . . . 6 |- ( ph -> ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) = W )
132	131	fveq2d 6195	. . . . 5 |- ( ph -> ( D ` ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) ) = ( D ` W ) )
133		eqid 2622	. . . . . . . . . . . 12 |- ( ( 1 ... N ) subMat R ) = ( ( 1 ... N ) subMat R )
134	4, 133, 5	submaval 20387	. . . . . . . . . . 11 |- ( ( W e. B /\ N e. ( 1 ... N ) /\ N e. ( 1 ... N ) ) -> ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) = ( i e. ( ( 1 ... N ) \ { N } ) , j e. ( ( 1 ... N ) \ { N } ) |-> ( i W j ) ) )
135	124, 91, 91, 134	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) = ( i e. ( ( 1 ... N ) \ { N } ) , j e. ( ( 1 ... N ) \ { N } ) |-> ( i W j ) ) )
136		fzdif2 29551	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 1 ) -> ( ( 1 ... N ) \ { N } ) = ( 1 ... ( N - 1 ) ) )
137	89, 136	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( 1 ... N ) \ { N } ) = ( 1 ... ( N - 1 ) ) )
138		mpt2eq12 6715	. . . . . . . . . . 11 |- ( ( ( ( 1 ... N ) \ { N } ) = ( 1 ... ( N - 1 ) ) /\ ( ( 1 ... N ) \ { N } ) = ( 1 ... ( N - 1 ) ) ) -> ( i e. ( ( 1 ... N ) \ { N } ) , j e. ( ( 1 ... N ) \ { N } ) |-> ( i W j ) ) = ( i e. ( 1 ... ( N - 1 ) ) , j e. ( 1 ... ( N - 1 ) ) |-> ( i W j ) ) )
139	137, 137, 138	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( i e. ( ( 1 ... N ) \ { N } ) , j e. ( ( 1 ... N ) \ { N } ) |-> ( i W j ) ) = ( i e. ( 1 ... ( N - 1 ) ) , j e. ( 1 ... ( N - 1 ) ) |-> ( i W j ) ) )
140	135, 139	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) = ( i e. ( 1 ... ( N - 1 ) ) , j e. ( 1 ... ( N - 1 ) ) |-> ( i W j ) ) )
141		difssd 3738	. . . . . . . . . . 11 |- ( ph -> ( ( 1 ... N ) \ { N } ) C_ ( 1 ... N ) )
142	137, 141	eqsstr3d 3640	. . . . . . . . . 10 |- ( ph -> ( 1 ... ( N - 1 ) ) C_ ( 1 ... N ) )
143	4, 5	submabas 20384	. . . . . . . . . 10 |- ( ( W e. B /\ ( 1 ... ( N - 1 ) ) C_ ( 1 ... N ) ) -> ( i e. ( 1 ... ( N - 1 ) ) , j e. ( 1 ... ( N - 1 ) ) |-> ( i W j ) ) e. ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) ) )
144	124, 142, 143	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( i e. ( 1 ... ( N - 1 ) ) , j e. ( 1 ... ( N - 1 ) ) |-> ( i W j ) ) e. ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) ) )
145	140, 144	eqeltrd 2701	. . . . . . . 8 |- ( ph -> ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) e. ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) ) )
146		madjusmdet.e	. . . . . . . . 9 |- E = ( ( 1 ... ( N - 1 ) ) maDet R )
147		eqid 2622	. . . . . . . . 9 |- ( ( 1 ... ( N - 1 ) ) Mat R ) = ( ( 1 ... ( N - 1 ) ) Mat R )
148		eqid 2622	. . . . . . . . 9 |- ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) ) = ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) )
149	146, 147, 148, 117	mdetcl 20402	. . . . . . . 8 |- ( ( R e. CRing /\ ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) e. ( Base ` ( ( 1 ... ( N - 1 ) ) Mat R ) ) ) -> ( E ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) e. ( Base ` R ) )
150	18, 145, 149	syl2anc 693	. . . . . . 7 |- ( ph -> ( E ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) e. ( Base ` R ) )
151	117, 16, 48	ringlidm 18571	. . . . . . 7 |- ( ( R e. Ring /\ ( E ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) e. ( Base ` R ) ) -> ( ( 1r ` R ) .x. ( E ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) ) = ( E ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) )
152	21, 150, 151	syl2anc 693	. . . . . 6 |- ( ph -> ( ( 1r ` R ) .x. ( E ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) ) = ( E ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) )
153	4	fveq2i 6194	. . . . . . . . . . 11 |- ( Base ` A ) = ( Base ` ( ( 1 ... N ) Mat R ) )
154	5, 153	eqtri 2644	. . . . . . . . . 10 |- B = ( Base ` ( ( 1 ... N ) Mat R ) )
155	124, 154	syl6eleq 2711	. . . . . . . . 9 |- ( ph -> W e. ( Base ` ( ( 1 ... N ) Mat R ) ) )
156		smadiadetr 20481	. . . . . . . . 9 |- ( ( ( R e. CRing /\ W e. ( Base ` ( ( 1 ... N ) Mat R ) ) ) /\ ( N e. ( 1 ... N ) /\ ( 1r ` R ) e. ( Base ` R ) ) ) -> ( ( ( 1 ... N ) maDet R ) ` ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) ) = ( ( 1r ` R ) ( .r ` R ) ( ( ( ( 1 ... N ) \ { N } ) maDet R ) ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) ) )
157	18, 155, 91, 126, 156	syl22anc 1327	. . . . . . . 8 |- ( ph -> ( ( ( 1 ... N ) maDet R ) ` ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) ) = ( ( 1r ` R ) ( .r ` R ) ( ( ( ( 1 ... N ) \ { N } ) maDet R ) ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) ) )
158	6	fveq1i 6192	. . . . . . . . 9 |- ( D ` ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) ) = ( ( ( 1 ... N ) maDet R ) ` ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) )
159	16	oveqi 6663	. . . . . . . . 9 |- ( ( 1r ` R ) .x. ( ( ( ( 1 ... N ) \ { N } ) maDet R ) ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) ) = ( ( 1r ` R ) ( .r ` R ) ( ( ( ( 1 ... N ) \ { N } ) maDet R ) ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) )
160	158, 159	eqeq12i 2636	. . . . . . . 8 |- ( ( D ` ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) ) = ( ( 1r ` R ) .x. ( ( ( ( 1 ... N ) \ { N } ) maDet R ) ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) ) <-> ( ( ( 1 ... N ) maDet R ) ` ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) ) = ( ( 1r ` R ) ( .r ` R ) ( ( ( ( 1 ... N ) \ { N } ) maDet R ) ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) ) )
161	157, 160	sylibr 224	. . . . . . 7 |- ( ph -> ( D ` ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) ) = ( ( 1r ` R ) .x. ( ( ( ( 1 ... N ) \ { N } ) maDet R ) ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) ) )
162	137	oveq1d 6665	. . . . . . . . . 10 |- ( ph -> ( ( ( 1 ... N ) \ { N } ) maDet R ) = ( ( 1 ... ( N - 1 ) ) maDet R ) )
163	162, 146	syl6eqr 2674	. . . . . . . . 9 |- ( ph -> ( ( ( 1 ... N ) \ { N } ) maDet R ) = E )
164	163	fveq1d 6193	. . . . . . . 8 |- ( ph -> ( ( ( ( 1 ... N ) \ { N } ) maDet R ) ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) = ( E ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) )
165	164	oveq2d 6666	. . . . . . 7 |- ( ph -> ( ( 1r ` R ) .x. ( ( ( ( 1 ... N ) \ { N } ) maDet R ) ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) ) = ( ( 1r ` R ) .x. ( E ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) ) )
166	161, 165	eqtrd 2656	. . . . . 6 |- ( ph -> ( D ` ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) ) = ( ( 1r ` R ) .x. ( E ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) ) )
167	4, 5	submat1n 29871	. . . . . . . 8 |- ( ( N e. NN /\ W e. B ) -> ( N (subMat1 ` W ) N ) = ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) )
168	87, 124, 167	syl2anc 693	. . . . . . 7 |- ( ph -> ( N (subMat1 ` W ) N ) = ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) )
169	168	fveq2d 6195	. . . . . 6 |- ( ph -> ( E ` ( N (subMat1 ` W ) N ) ) = ( E ` ( N ( ( ( 1 ... N ) subMat R ) ` W ) N ) ) )
170	152, 166, 169	3eqtr4d 2666	. . . . 5 |- ( ph -> ( D ` ( N ( W ( ( 1 ... N ) matRRep R ) ( 1r ` R ) ) N ) ) = ( E ` ( N (subMat1 ` W ) N ) ) )
171	132, 170	eqtr3d 2658	. . . 4 |- ( ph -> ( D ` W ) = ( E ` ( N (subMat1 ` W ) N ) ) )
172	4, 5, 87, 3, 2, 21, 1, 10	submatminr1 29876	. . . . . 6 |- ( ph -> ( I (subMat1 ` M ) J ) = ( I (subMat1 ` U ) J ) )
173		madjusmdetlem1.3	. . . . . 6 |- ( ph -> ( I (subMat1 ` U ) J ) = ( N (subMat1 ` W ) N ) )
174	172, 173	eqtrd 2656	. . . . 5 |- ( ph -> ( I (subMat1 ` M ) J ) = ( N (subMat1 ` W ) N ) )
175	174	fveq2d 6195	. . . 4 |- ( ph -> ( E ` ( I (subMat1 ` M ) J ) ) = ( E ` ( N (subMat1 ` W ) N ) ) )
176	171, 175	eqtr4d 2659	. . 3 |- ( ph -> ( D ` W ) = ( E ` ( I (subMat1 ` M ) J ) ) )
177	176	oveq2d 6666	. 2 |- ( ph -> ( ( Z ` ( ( S ` P ) x. ( S ` Q ) ) ) .x. ( D ` W ) ) = ( ( Z ` ( ( S ` P ) x. ( S ` Q ) ) ) .x. ( E ` ( I (subMat1 ` M ) J ) ) ) )
178	12, 27, 177	3eqtrd 2660	1 |- ( ph -> ( J ( K ` M ) I ) = ( ( Z ` ( ( S ` P ) x. ( S ` Q ) ) ) .x. ( E ` ( I (subMat1 ` M ) J ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   {csn 4177   `'ccnv 5113   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1c1 9937   x. cmul 9941   - cmin 10266   NNcn 11020   ZZ>=cuz 11687   ...cfz 12326   Basecbs 15857   .rcmulr 15942   0gc0g 16100   SymGrpcsymg 17797  pmSgncpsgn 17909   1rcur 18501   Ringcrg 18547   CRingccrg 18548   ZRHomczrh 19848   Mat cmat 20213   matRRep cmarrep 20362   subMat csubma 20382   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:  madjusmdetlem4  29896