Theorem m2detleib 20437

Description: Leibniz' Formula for 2x2-matrices. (Contributed by AV, 21-Dec-2018.) (Revised by AV, 26-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.)

Hypotheses

Ref	Expression
m2detleib.n	|- N = { 1 , 2 }
m2detleib.d	|- D = ( N maDet R )
m2detleib.a	|- A = ( N Mat R )
m2detleib.b	|- B = ( Base ` A )
m2detleib.m	|- .- = ( -g ` R )
m2detleib.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
m2detleib	|- ( ( R e. Ring /\ M e. B ) -> ( D ` M ) = ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) .- ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) )

Proof of Theorem m2detleib

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

Step	Hyp	Ref	Expression
1		m2detleib.d	. . . 4 |- D = ( N maDet R )
2		m2detleib.a	. . . 4 |- A = ( N Mat R )
3		m2detleib.b	. . . 4 |- B = ( Base ` A )
4		eqid 2622	. . . 4 |- ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` N ) )
5		eqid 2622	. . . 4 |- ( ZRHom ` R ) = ( ZRHom ` R )
6		eqid 2622	. . . 4 |- (pmSgn ` N ) = (pmSgn ` N )
7		m2detleib.t	. . . 4 |- .x. = ( .r ` R )
8		eqid 2622	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
9	1, 2, 3, 4, 5, 6, 7, 8	mdetleib1 20397	. . 3 |- ( M e. B -> ( D ` M ) = ( R gsumg ( k e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) )
10	9	adantl 482	. 2 |- ( ( R e. Ring /\ M e. B ) -> ( D ` M ) = ( R gsumg ( k e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) )
11		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
12		eqid 2622	. . 3 |- ( +g ` R ) = ( +g ` R )
13		ringcmn 18581	. . . 4 |- ( R e. Ring -> R e. CMnd )
14	13	adantr 481	. . 3 |- ( ( R e. Ring /\ M e. B ) -> R e. CMnd )
15		m2detleib.n	. . . . . 6 |- N = { 1 , 2 }
16		prfi 8235	. . . . . 6 |- { 1 , 2 } e. Fin
17	15, 16	eqeltri 2697	. . . . 5 |- N e. Fin
18		eqid 2622	. . . . . 6 |- ( SymGrp ` N ) = ( SymGrp ` N )
19	18, 4	symgbasfi 17806	. . . . 5 |- ( N e. Fin -> ( Base ` ( SymGrp ` N ) ) e. Fin )
20	17, 19	ax-mp 5	. . . 4 |- ( Base ` ( SymGrp ` N ) ) e. Fin
21	20	a1i 11	. . 3 |- ( ( R e. Ring /\ M e. B ) -> ( Base ` ( SymGrp ` N ) ) e. Fin )
22		simpl 473	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> R e. Ring )
23	22	adantr 481	. . . 4 |- ( ( ( R e. Ring /\ M e. B ) /\ k e. ( Base ` ( SymGrp ` N ) ) ) -> R e. Ring )
24	4, 6, 5	zrhpsgnelbas 19940	. . . . . 6 |- ( ( R e. Ring /\ N e. Fin /\ k e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) e. ( Base ` R ) )
25	17, 24	mp3an2 1412	. . . . 5 |- ( ( R e. Ring /\ k e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) e. ( Base ` R ) )
26	25	adantlr 751	. . . 4 |- ( ( ( R e. Ring /\ M e. B ) /\ k e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) e. ( Base ` R ) )
27		simpr 477	. . . . 5 |- ( ( ( R e. Ring /\ M e. B ) /\ k e. ( Base ` ( SymGrp ` N ) ) ) -> k e. ( Base ` ( SymGrp ` N ) ) )
28		simpr 477	. . . . . 6 |- ( ( R e. Ring /\ M e. B ) -> M e. B )
29	28	adantr 481	. . . . 5 |- ( ( ( R e. Ring /\ M e. B ) /\ k e. ( Base ` ( SymGrp ` N ) ) ) -> M e. B )
30	15, 4, 2, 3, 8	m2detleiblem2 20434	. . . . 5 |- ( ( R e. Ring /\ k e. ( Base ` ( SymGrp ` N ) ) /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) e. ( Base ` R ) )
31	23, 27, 29, 30	syl3anc 1326	. . . 4 |- ( ( ( R e. Ring /\ M e. B ) /\ k e. ( Base ` ( SymGrp ` N ) ) ) -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) e. ( Base ` R ) )
32	11, 7	ringcl 18561	. . . 4 |- ( ( R e. Ring /\ ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) e. ( Base ` R ) /\ ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) e. ( Base ` R ) ) -> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) e. ( Base ` R ) )
33	23, 26, 31, 32	syl3anc 1326	. . 3 |- ( ( ( R e. Ring /\ M e. B ) /\ k e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) e. ( Base ` R ) )
34		opex 4932	. . . . . . . 8 |- <. 1 , 1 >. e. _V
35		opex 4932	. . . . . . . 8 |- <. 2 , 2 >. e. _V
36	34, 35	pm3.2i 471	. . . . . . 7 |- ( <. 1 , 1 >. e. _V /\ <. 2 , 2 >. e. _V )
37		opex 4932	. . . . . . . 8 |- <. 1 , 2 >. e. _V
38		opex 4932	. . . . . . . 8 |- <. 2 , 1 >. e. _V
39	37, 38	pm3.2i 471	. . . . . . 7 |- ( <. 1 , 2 >. e. _V /\ <. 2 , 1 >. e. _V )
40	36, 39	pm3.2i 471	. . . . . 6 |- ( ( <. 1 , 1 >. e. _V /\ <. 2 , 2 >. e. _V ) /\ ( <. 1 , 2 >. e. _V /\ <. 2 , 1 >. e. _V ) )
41		1ne2 11240	. . . . . . . . . 10 |- 1 =/= 2
42	41	olci 406	. . . . . . . . 9 |- ( 1 =/= 1 \/ 1 =/= 2 )
43		1ex 10035	. . . . . . . . . 10 |- 1 e. _V
44	43, 43	opthne 4951	. . . . . . . . 9 |- ( <. 1 , 1 >. =/= <. 1 , 2 >. <-> ( 1 =/= 1 \/ 1 =/= 2 ) )
45	42, 44	mpbir 221	. . . . . . . 8 |- <. 1 , 1 >. =/= <. 1 , 2 >.
46	41	orci 405	. . . . . . . . 9 |- ( 1 =/= 2 \/ 1 =/= 1 )
47	43, 43	opthne 4951	. . . . . . . . 9 |- ( <. 1 , 1 >. =/= <. 2 , 1 >. <-> ( 1 =/= 2 \/ 1 =/= 1 ) )
48	46, 47	mpbir 221	. . . . . . . 8 |- <. 1 , 1 >. =/= <. 2 , 1 >.
49	45, 48	pm3.2i 471	. . . . . . 7 |- ( <. 1 , 1 >. =/= <. 1 , 2 >. /\ <. 1 , 1 >. =/= <. 2 , 1 >. )
50	49	orci 405	. . . . . 6 |- ( ( <. 1 , 1 >. =/= <. 1 , 2 >. /\ <. 1 , 1 >. =/= <. 2 , 1 >. ) \/ ( <. 2 , 2 >. =/= <. 1 , 2 >. /\ <. 2 , 2 >. =/= <. 2 , 1 >. ) )
51	40, 50	pm3.2i 471	. . . . 5 |- ( ( ( <. 1 , 1 >. e. _V /\ <. 2 , 2 >. e. _V ) /\ ( <. 1 , 2 >. e. _V /\ <. 2 , 1 >. e. _V ) ) /\ ( ( <. 1 , 1 >. =/= <. 1 , 2 >. /\ <. 1 , 1 >. =/= <. 2 , 1 >. ) \/ ( <. 2 , 2 >. =/= <. 1 , 2 >. /\ <. 2 , 2 >. =/= <. 2 , 1 >. ) ) )
52	51	a1i 11	. . . 4 |- ( ( R e. Ring /\ M e. B ) -> ( ( ( <. 1 , 1 >. e. _V /\ <. 2 , 2 >. e. _V ) /\ ( <. 1 , 2 >. e. _V /\ <. 2 , 1 >. e. _V ) ) /\ ( ( <. 1 , 1 >. =/= <. 1 , 2 >. /\ <. 1 , 1 >. =/= <. 2 , 1 >. ) \/ ( <. 2 , 2 >. =/= <. 1 , 2 >. /\ <. 2 , 2 >. =/= <. 2 , 1 >. ) ) ) )
53		prneimg 4388	. . . . 5 |- ( ( ( <. 1 , 1 >. e. _V /\ <. 2 , 2 >. e. _V ) /\ ( <. 1 , 2 >. e. _V /\ <. 2 , 1 >. e. _V ) ) -> ( ( ( <. 1 , 1 >. =/= <. 1 , 2 >. /\ <. 1 , 1 >. =/= <. 2 , 1 >. ) \/ ( <. 2 , 2 >. =/= <. 1 , 2 >. /\ <. 2 , 2 >. =/= <. 2 , 1 >. ) ) -> { <. 1 , 1 >. , <. 2 , 2 >. } =/= { <. 1 , 2 >. , <. 2 , 1 >. } ) )
54	53	imp 445	. . . 4 |- ( ( ( ( <. 1 , 1 >. e. _V /\ <. 2 , 2 >. e. _V ) /\ ( <. 1 , 2 >. e. _V /\ <. 2 , 1 >. e. _V ) ) /\ ( ( <. 1 , 1 >. =/= <. 1 , 2 >. /\ <. 1 , 1 >. =/= <. 2 , 1 >. ) \/ ( <. 2 , 2 >. =/= <. 1 , 2 >. /\ <. 2 , 2 >. =/= <. 2 , 1 >. ) ) ) -> { <. 1 , 1 >. , <. 2 , 2 >. } =/= { <. 1 , 2 >. , <. 2 , 1 >. } )
55		disjsn2 4247	. . . 4 |- ( { <. 1 , 1 >. , <. 2 , 2 >. } =/= { <. 1 , 2 >. , <. 2 , 1 >. } -> ( { { <. 1 , 1 >. , <. 2 , 2 >. } } i^i { { <. 1 , 2 >. , <. 2 , 1 >. } } ) = (/) )
56	52, 54, 55	3syl 18	. . 3 |- ( ( R e. Ring /\ M e. B ) -> ( { { <. 1 , 1 >. , <. 2 , 2 >. } } i^i { { <. 1 , 2 >. , <. 2 , 1 >. } } ) = (/) )
57		2nn 11185	. . . . . 6 |- 2 e. NN
58	18, 4, 15	symg2bas 17818	. . . . . 6 |- ( ( 1 e. _V /\ 2 e. NN ) -> ( Base ` ( SymGrp ` N ) ) = { { <. 1 , 1 >. , <. 2 , 2 >. } , { <. 1 , 2 >. , <. 2 , 1 >. } } )
59	43, 57, 58	mp2an 708	. . . . 5 |- ( Base ` ( SymGrp ` N ) ) = { { <. 1 , 1 >. , <. 2 , 2 >. } , { <. 1 , 2 >. , <. 2 , 1 >. } }
60		df-pr 4180	. . . . 5 |- { { <. 1 , 1 >. , <. 2 , 2 >. } , { <. 1 , 2 >. , <. 2 , 1 >. } } = ( { { <. 1 , 1 >. , <. 2 , 2 >. } } u. { { <. 1 , 2 >. , <. 2 , 1 >. } } )
61	59, 60	eqtri 2644	. . . 4 |- ( Base ` ( SymGrp ` N ) ) = ( { { <. 1 , 1 >. , <. 2 , 2 >. } } u. { { <. 1 , 2 >. , <. 2 , 1 >. } } )
62	61	a1i 11	. . 3 |- ( ( R e. Ring /\ M e. B ) -> ( Base ` ( SymGrp ` N ) ) = ( { { <. 1 , 1 >. , <. 2 , 2 >. } } u. { { <. 1 , 2 >. , <. 2 , 1 >. } } ) )
63	11, 12, 14, 21, 33, 56, 62	gsummptfidmsplit 18330	. 2 |- ( ( R e. Ring /\ M e. B ) -> ( R gsumg ( k e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) = ( ( R gsumg ( k e. { { <. 1 , 1 >. , <. 2 , 2 >. } } |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) ( +g ` R ) ( R gsumg ( k e. { { <. 1 , 2 >. , <. 2 , 1 >. } } |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) ) )
64		ringmnd 18556	. . . . . 6 |- ( R e. Ring -> R e. Mnd )
65	64	adantr 481	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> R e. Mnd )
66		prex 4909	. . . . . 6 |- { <. 1 , 1 >. , <. 2 , 2 >. } e. _V
67	66	a1i 11	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> { <. 1 , 1 >. , <. 2 , 2 >. } e. _V )
68	66	prid1 4297	. . . . . . . . 9 |- { <. 1 , 1 >. , <. 2 , 2 >. } e. { { <. 1 , 1 >. , <. 2 , 2 >. } , { <. 1 , 2 >. , <. 2 , 1 >. } }
69	68, 59	eleqtrri 2700	. . . . . . . 8 |- { <. 1 , 1 >. , <. 2 , 2 >. } e. ( Base ` ( SymGrp ` N ) )
70	69	a1i 11	. . . . . . 7 |- ( M e. B -> { <. 1 , 1 >. , <. 2 , 2 >. } e. ( Base ` ( SymGrp ` N ) ) )
71	4, 6, 5	zrhpsgnelbas 19940	. . . . . . . 8 |- ( ( R e. Ring /\ N e. Fin /\ { <. 1 , 1 >. , <. 2 , 2 >. } e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) e. ( Base ` R ) )
72	17, 71	mp3an2 1412	. . . . . . 7 |- ( ( R e. Ring /\ { <. 1 , 1 >. , <. 2 , 2 >. } e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) e. ( Base ` R ) )
73	70, 72	sylan2 491	. . . . . 6 |- ( ( R e. Ring /\ M e. B ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) e. ( Base ` R ) )
74	15, 4, 2, 3, 8	m2detleiblem2 20434	. . . . . . 7 |- ( ( R e. Ring /\ { <. 1 , 1 >. , <. 2 , 2 >. } e. ( Base ` ( SymGrp ` N ) ) /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) e. ( Base ` R ) )
75	69, 74	mp3an2 1412	. . . . . 6 |- ( ( R e. Ring /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) e. ( Base ` R ) )
76	11, 7	ringcl 18561	. . . . . 6 |- ( ( R e. Ring /\ ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) e. ( Base ` R ) /\ ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) e. ( Base ` R ) ) -> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) ) e. ( Base ` R ) )
77	22, 73, 75, 76	syl3anc 1326	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) ) e. ( Base ` R ) )
78		fveq2 6191	. . . . . . . 8 |- ( k = { <. 1 , 1 >. , <. 2 , 2 >. } -> ( (pmSgn ` N ) ` k ) = ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) )
79	78	fveq2d 6195	. . . . . . 7 |- ( k = { <. 1 , 1 >. , <. 2 , 2 >. } -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) = ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) )
80		fveq1 6190	. . . . . . . . . 10 |- ( k = { <. 1 , 1 >. , <. 2 , 2 >. } -> ( k ` n ) = ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) )
81	80	oveq1d 6665	. . . . . . . . 9 |- ( k = { <. 1 , 1 >. , <. 2 , 2 >. } -> ( ( k ` n ) M n ) = ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) )
82	81	mpteq2dv 4745	. . . . . . . 8 |- ( k = { <. 1 , 1 >. , <. 2 , 2 >. } -> ( n e. N |-> ( ( k ` n ) M n ) ) = ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) )
83	82	oveq2d 6666	. . . . . . 7 |- ( k = { <. 1 , 1 >. , <. 2 , 2 >. } -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) = ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) )
84	79, 83	oveq12d 6668	. . . . . 6 |- ( k = { <. 1 , 1 >. , <. 2 , 2 >. } -> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) = ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) ) )
85	11, 84	gsumsn 18354	. . . . 5 |- ( ( R e. Mnd /\ { <. 1 , 1 >. , <. 2 , 2 >. } e. _V /\ ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) ) e. ( Base ` R ) ) -> ( R gsumg ( k e. { { <. 1 , 1 >. , <. 2 , 2 >. } } |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) = ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) ) )
86	65, 67, 77, 85	syl3anc 1326	. . . 4 |- ( ( R e. Ring /\ M e. B ) -> ( R gsumg ( k e. { { <. 1 , 1 >. , <. 2 , 2 >. } } |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) = ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) ) )
87		prex 4909	. . . . . 6 |- { <. 1 , 2 >. , <. 2 , 1 >. } e. _V
88	87	a1i 11	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> { <. 1 , 2 >. , <. 2 , 1 >. } e. _V )
89	87	prid2 4298	. . . . . . . . 9 |- { <. 1 , 2 >. , <. 2 , 1 >. } e. { { <. 1 , 1 >. , <. 2 , 2 >. } , { <. 1 , 2 >. , <. 2 , 1 >. } }
90	89, 59	eleqtrri 2700	. . . . . . . 8 |- { <. 1 , 2 >. , <. 2 , 1 >. } e. ( Base ` ( SymGrp ` N ) )
91	90	a1i 11	. . . . . . 7 |- ( M e. B -> { <. 1 , 2 >. , <. 2 , 1 >. } e. ( Base ` ( SymGrp ` N ) ) )
92	4, 6, 5	zrhpsgnelbas 19940	. . . . . . . 8 |- ( ( R e. Ring /\ N e. Fin /\ { <. 1 , 2 >. , <. 2 , 1 >. } e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) e. ( Base ` R ) )
93	17, 92	mp3an2 1412	. . . . . . 7 |- ( ( R e. Ring /\ { <. 1 , 2 >. , <. 2 , 1 >. } e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) e. ( Base ` R ) )
94	91, 93	sylan2 491	. . . . . 6 |- ( ( R e. Ring /\ M e. B ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) e. ( Base ` R ) )
95	15, 4, 2, 3, 8	m2detleiblem2 20434	. . . . . . 7 |- ( ( R e. Ring /\ { <. 1 , 2 >. , <. 2 , 1 >. } e. ( Base ` ( SymGrp ` N ) ) /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) e. ( Base ` R ) )
96	90, 95	mp3an2 1412	. . . . . 6 |- ( ( R e. Ring /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) e. ( Base ` R ) )
97	11, 7	ringcl 18561	. . . . . 6 |- ( ( R e. Ring /\ ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) e. ( Base ` R ) /\ ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) e. ( Base ` R ) ) -> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) ) e. ( Base ` R ) )
98	22, 94, 96, 97	syl3anc 1326	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) ) e. ( Base ` R ) )
99		fveq2 6191	. . . . . . . 8 |- ( k = { <. 1 , 2 >. , <. 2 , 1 >. } -> ( (pmSgn ` N ) ` k ) = ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) )
100	99	fveq2d 6195	. . . . . . 7 |- ( k = { <. 1 , 2 >. , <. 2 , 1 >. } -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) = ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) )
101		fveq1 6190	. . . . . . . . . 10 |- ( k = { <. 1 , 2 >. , <. 2 , 1 >. } -> ( k ` n ) = ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) )
102	101	oveq1d 6665	. . . . . . . . 9 |- ( k = { <. 1 , 2 >. , <. 2 , 1 >. } -> ( ( k ` n ) M n ) = ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) )
103	102	mpteq2dv 4745	. . . . . . . 8 |- ( k = { <. 1 , 2 >. , <. 2 , 1 >. } -> ( n e. N |-> ( ( k ` n ) M n ) ) = ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) )
104	103	oveq2d 6666	. . . . . . 7 |- ( k = { <. 1 , 2 >. , <. 2 , 1 >. } -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) = ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) )
105	100, 104	oveq12d 6668	. . . . . 6 |- ( k = { <. 1 , 2 >. , <. 2 , 1 >. } -> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) = ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) ) )
106	11, 105	gsumsn 18354	. . . . 5 |- ( ( R e. Mnd /\ { <. 1 , 2 >. , <. 2 , 1 >. } e. _V /\ ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) ) e. ( Base ` R ) ) -> ( R gsumg ( k e. { { <. 1 , 2 >. , <. 2 , 1 >. } } |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) = ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) ) )
107	65, 88, 98, 106	syl3anc 1326	. . . 4 |- ( ( R e. Ring /\ M e. B ) -> ( R gsumg ( k e. { { <. 1 , 2 >. , <. 2 , 1 >. } } |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) = ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) ) )
108	86, 107	oveq12d 6668	. . 3 |- ( ( R e. Ring /\ M e. B ) -> ( ( R gsumg ( k e. { { <. 1 , 1 >. , <. 2 , 2 >. } } |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) ( +g ` R ) ( R gsumg ( k e. { { <. 1 , 2 >. , <. 2 , 1 >. } } |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) ) = ( ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) ) ( +g ` R ) ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) ) ) )
109		eqidd 2623	. . . . . . 7 |- ( M e. B -> { <. 1 , 1 >. , <. 2 , 2 >. } = { <. 1 , 1 >. , <. 2 , 2 >. } )
110		eqid 2622	. . . . . . . 8 |- ( 1r ` R ) = ( 1r ` R )
111	15, 4, 5, 6, 110	m2detleiblem5 20431	. . . . . . 7 |- ( ( R e. Ring /\ { <. 1 , 1 >. , <. 2 , 2 >. } = { <. 1 , 1 >. , <. 2 , 2 >. } ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) = ( 1r ` R ) )
112	109, 111	sylan2 491	. . . . . 6 |- ( ( R e. Ring /\ M e. B ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) = ( 1r ` R ) )
113		eqidd 2623	. . . . . . 7 |- ( ( R e. Ring /\ M e. B ) -> { <. 1 , 1 >. , <. 2 , 2 >. } = { <. 1 , 1 >. , <. 2 , 2 >. } )
114	8, 7	mgpplusg 18493	. . . . . . . 8 |- .x. = ( +g ` (mulGrp ` R ) )
115	15, 4, 2, 3, 8, 114	m2detleiblem3 20435	. . . . . . 7 |- ( ( R e. Ring /\ { <. 1 , 1 >. , <. 2 , 2 >. } = { <. 1 , 1 >. , <. 2 , 2 >. } /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) = ( ( 1 M 1 ) .x. ( 2 M 2 ) ) )
116	22, 113, 28, 115	syl3anc 1326	. . . . . 6 |- ( ( R e. Ring /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) = ( ( 1 M 1 ) .x. ( 2 M 2 ) ) )
117	112, 116	oveq12d 6668	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) ) = ( ( 1r ` R ) .x. ( ( 1 M 1 ) .x. ( 2 M 2 ) ) ) )
118	43	prid1 4297	. . . . . . . . . 10 |- 1 e. { 1 , 2 }
119	118, 15	eleqtrri 2700	. . . . . . . . 9 |- 1 e. N
120	119	a1i 11	. . . . . . . 8 |- ( ( R e. Ring /\ M e. B ) -> 1 e. N )
121	3	eleq2i 2693	. . . . . . . . . 10 |- ( M e. B <-> M e. ( Base ` A ) )
122	121	biimpi 206	. . . . . . . . 9 |- ( M e. B -> M e. ( Base ` A ) )
123	122	adantl 482	. . . . . . . 8 |- ( ( R e. Ring /\ M e. B ) -> M e. ( Base ` A ) )
124	2, 11	matecl 20231	. . . . . . . 8 |- ( ( 1 e. N /\ 1 e. N /\ M e. ( Base ` A ) ) -> ( 1 M 1 ) e. ( Base ` R ) )
125	120, 120, 123, 124	syl3anc 1326	. . . . . . 7 |- ( ( R e. Ring /\ M e. B ) -> ( 1 M 1 ) e. ( Base ` R ) )
126		prid2g 4296	. . . . . . . . . . 11 |- ( 2 e. NN -> 2 e. { 1 , 2 } )
127	57, 126	ax-mp 5	. . . . . . . . . 10 |- 2 e. { 1 , 2 }
128	127, 15	eleqtrri 2700	. . . . . . . . 9 |- 2 e. N
129	128	a1i 11	. . . . . . . 8 |- ( ( R e. Ring /\ M e. B ) -> 2 e. N )
130	2, 11	matecl 20231	. . . . . . . 8 |- ( ( 2 e. N /\ 2 e. N /\ M e. ( Base ` A ) ) -> ( 2 M 2 ) e. ( Base ` R ) )
131	129, 129, 123, 130	syl3anc 1326	. . . . . . 7 |- ( ( R e. Ring /\ M e. B ) -> ( 2 M 2 ) e. ( Base ` R ) )
132	11, 7	ringcl 18561	. . . . . . 7 |- ( ( R e. Ring /\ ( 1 M 1 ) e. ( Base ` R ) /\ ( 2 M 2 ) e. ( Base ` R ) ) -> ( ( 1 M 1 ) .x. ( 2 M 2 ) ) e. ( Base ` R ) )
133	22, 125, 131, 132	syl3anc 1326	. . . . . 6 |- ( ( R e. Ring /\ M e. B ) -> ( ( 1 M 1 ) .x. ( 2 M 2 ) ) e. ( Base ` R ) )
134	11, 7, 110	ringlidm 18571	. . . . . 6 |- ( ( R e. Ring /\ ( ( 1 M 1 ) .x. ( 2 M 2 ) ) e. ( Base ` R ) ) -> ( ( 1r ` R ) .x. ( ( 1 M 1 ) .x. ( 2 M 2 ) ) ) = ( ( 1 M 1 ) .x. ( 2 M 2 ) ) )
135	133, 134	syldan 487	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> ( ( 1r ` R ) .x. ( ( 1 M 1 ) .x. ( 2 M 2 ) ) ) = ( ( 1 M 1 ) .x. ( 2 M 2 ) ) )
136	117, 135	eqtrd 2656	. . . 4 |- ( ( R e. Ring /\ M e. B ) -> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) ) = ( ( 1 M 1 ) .x. ( 2 M 2 ) ) )
137		eqidd 2623	. . . . . 6 |- ( M e. B -> { <. 1 , 2 >. , <. 2 , 1 >. } = { <. 1 , 2 >. , <. 2 , 1 >. } )
138		eqid 2622	. . . . . . 7 |- ( invg ` R ) = ( invg ` R )
139	15, 4, 5, 6, 110, 138	m2detleiblem6 20432	. . . . . 6 |- ( ( R e. Ring /\ { <. 1 , 2 >. , <. 2 , 1 >. } = { <. 1 , 2 >. , <. 2 , 1 >. } ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) = ( ( invg ` R ) ` ( 1r ` R ) ) )
140	137, 139	sylan2 491	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) = ( ( invg ` R ) ` ( 1r ` R ) ) )
141		eqidd 2623	. . . . . 6 |- ( ( R e. Ring /\ M e. B ) -> { <. 1 , 2 >. , <. 2 , 1 >. } = { <. 1 , 2 >. , <. 2 , 1 >. } )
142	15, 4, 2, 3, 8, 114	m2detleiblem4 20436	. . . . . 6 |- ( ( R e. Ring /\ { <. 1 , 2 >. , <. 2 , 1 >. } = { <. 1 , 2 >. , <. 2 , 1 >. } /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) = ( ( 2 M 1 ) .x. ( 1 M 2 ) ) )
143	22, 141, 28, 142	syl3anc 1326	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) = ( ( 2 M 1 ) .x. ( 1 M 2 ) ) )
144	140, 143	oveq12d 6668	. . . 4 |- ( ( R e. Ring /\ M e. B ) -> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) ) = ( ( ( invg ` R ) ` ( 1r ` R ) ) .x. ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) )
145	136, 144	oveq12d 6668	. . 3 |- ( ( R e. Ring /\ M e. B ) -> ( ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 1 >. , <. 2 , 2 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 1 >. , <. 2 , 2 >. } ` n ) M n ) ) ) ) ( +g ` R ) ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. 1 , 2 >. , <. 2 , 1 >. } ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( { <. 1 , 2 >. , <. 2 , 1 >. } ` n ) M n ) ) ) ) ) = ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) ( +g ` R ) ( ( ( invg ` R ) ` ( 1r ` R ) ) .x. ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) ) )
146	2, 11	matecl 20231	. . . . . 6 |- ( ( 2 e. N /\ 1 e. N /\ M e. ( Base ` A ) ) -> ( 2 M 1 ) e. ( Base ` R ) )
147	129, 120, 123, 146	syl3anc 1326	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> ( 2 M 1 ) e. ( Base ` R ) )
148	2, 11	matecl 20231	. . . . . 6 |- ( ( 1 e. N /\ 2 e. N /\ M e. ( Base ` A ) ) -> ( 1 M 2 ) e. ( Base ` R ) )
149	120, 129, 123, 148	syl3anc 1326	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> ( 1 M 2 ) e. ( Base ` R ) )
150	11, 7	ringcl 18561	. . . . 5 |- ( ( R e. Ring /\ ( 2 M 1 ) e. ( Base ` R ) /\ ( 1 M 2 ) e. ( Base ` R ) ) -> ( ( 2 M 1 ) .x. ( 1 M 2 ) ) e. ( Base ` R ) )
151	22, 147, 149, 150	syl3anc 1326	. . . 4 |- ( ( R e. Ring /\ M e. B ) -> ( ( 2 M 1 ) .x. ( 1 M 2 ) ) e. ( Base ` R ) )
152		m2detleib.m	. . . . 5 |- .- = ( -g ` R )
153	15, 4, 5, 6, 110, 138, 7, 152	m2detleiblem7 20433	. . . 4 |- ( ( R e. Ring /\ ( ( 1 M 1 ) .x. ( 2 M 2 ) ) e. ( Base ` R ) /\ ( ( 2 M 1 ) .x. ( 1 M 2 ) ) e. ( Base ` R ) ) -> ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) ( +g ` R ) ( ( ( invg ` R ) ` ( 1r ` R ) ) .x. ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) ) = ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) .- ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) )
154	22, 133, 151, 153	syl3anc 1326	. . 3 |- ( ( R e. Ring /\ M e. B ) -> ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) ( +g ` R ) ( ( ( invg ` R ) ` ( 1r ` R ) ) .x. ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) ) = ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) .- ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) )
155	108, 145, 154	3eqtrd 2660	. 2 |- ( ( R e. Ring /\ M e. B ) -> ( ( R gsumg ( k e. { { <. 1 , 1 >. , <. 2 , 2 >. } } |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) ( +g ` R ) ( R gsumg ( k e. { { <. 1 , 2 >. , <. 2 , 1 >. } } |-> ( ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` k ) ) .x. ( (mulGrp ` R ) gsumg ( n e. N |-> ( ( k ` n ) M n ) ) ) ) ) ) ) = ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) .- ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) )
156	10, 63, 155	3eqtrd 2660	1 |- ( ( R e. Ring /\ M e. B ) -> ( D ` M ) = ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) .- ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   1c1 9937   NNcn 11020   2c2 11070   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   gsum_g cgsu 16101   Mndcmnd 17294   invgcminusg 17423   -gcsg 17424   SymGrpcsymg 17797  pmSgncpsgn 17909  CMndccmn 18193  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   ZRHomczrh 19848   Mat cmat 20213   maDet cmdat 20390

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-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-cda 8990  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-fac 13061  df-bc 13090  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-sbg 17427  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-rnghom 18715  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-mdet 20391

This theorem is referenced by:  lmat22det  29888