Theorem m1detdiag 18972

Description: The determinant of a 1-dimensional matrix equals its (single) entry. (Contributed by AV, 6-Aug-2019.)

Hypotheses

Ref	Expression
mdetdiag.d	|- D = ( N maDet R )
mdetdiag.a	|- A = ( N Mat R )
mdetdiag.b	|- B = ( Base ` A )

Assertion

Ref	Expression
m1detdiag	|- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( D ` M ) = ( I M I ) )

Proof of Theorem m1detdiag

Dummy variables b p x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdetdiag.d	. . . 4 |- D = ( N maDet R )
2		mdetdiag.a	. . . 4 |- A = ( N Mat R )
3		mdetdiag.b	. . . 4 |- B = ( Base ` A )
4		eqid 2443	. . . 4 |- ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` N ) )
5		eqid 2443	. . . 4 |- ( ZRHom ` R ) = ( ZRHom ` R )
6		eqid 2443	. . . 4 |- (pmSgn ` N ) = (pmSgn ` N )
7		eqid 2443	. . . 4 |- ( .r ` R ) = ( .r ` R )
8		eqid 2443	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
9	1, 2, 3, 4, 5, 6, 7, 8	mdetleib 18962	. . 3 |- ( M e. B -> ( D ` M ) = ( R gsumg ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( p ` x ) M x ) ) ) ) ) ) )
10	9	3ad2ant3 1020	. 2 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( D ` M ) = ( R gsumg ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( p ` x ) M x ) ) ) ) ) ) )
11		fveq2 5856	. . . . . . . . 9 |- ( N = { I } -> ( SymGrp ` N ) = ( SymGrp ` { I } ) )
12	11	fveq2d 5860	. . . . . . . 8 |- ( N = { I } -> ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` { I } ) ) )
13	12	adantr 465	. . . . . . 7 |- ( ( N = { I } /\ I e. V ) -> ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` { I } ) ) )
14	13	3ad2ant2 1019	. . . . . 6 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` { I } ) ) )
15		simp2r 1024	. . . . . . 7 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> I e. V )
16		eqid 2443	. . . . . . . 8 |- ( SymGrp ` { I } ) = ( SymGrp ` { I } )
17		eqid 2443	. . . . . . . 8 |- ( Base ` ( SymGrp ` { I } ) ) = ( Base ` ( SymGrp ` { I } ) )
18		eqid 2443	. . . . . . . 8 |- { I } = { I }
19	16, 17, 18	symg1bas 16295	. . . . . . 7 |- ( I e. V -> ( Base ` ( SymGrp ` { I } ) ) = { { <. I , I >. } } )
20	15, 19	syl 16	. . . . . 6 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` ( SymGrp ` { I } ) ) = { { <. I , I >. } } )
21	14, 20	eqtrd 2484	. . . . 5 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` ( SymGrp ` N ) ) = { { <. I , I >. } } )
22	21	mpteq1d 4518	. . . 4 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( p ` x ) M x ) ) ) ) ) = ( p e. { { <. I , I >. } } |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( p ` x ) M x ) ) ) ) ) )
23		snex 4678	. . . . . 6 |- { <. I , I >. } e. _V
24	23	a1i 11	. . . . 5 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> { <. I , I >. } e. _V )
25		ovex 6309	. . . . 5 |- ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) e. _V
26		fveq2 5856	. . . . . . . 8 |- ( p = { <. I , I >. } -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) )
27		fveq1 5855	. . . . . . . . . . 11 |- ( p = { <. I , I >. } -> ( p ` x ) = ( { <. I , I >. } ` x ) )
28	27	oveq1d 6296	. . . . . . . . . 10 |- ( p = { <. I , I >. } -> ( ( p ` x ) M x ) = ( ( { <. I , I >. } ` x ) M x ) )
29	28	mpteq2dv 4524	. . . . . . . . 9 |- ( p = { <. I , I >. } -> ( x e. N |-> ( ( p ` x ) M x ) ) = ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) )
30	29	oveq2d 6297	. . . . . . . 8 |- ( p = { <. I , I >. } -> ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( p ` x ) M x ) ) ) = ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) )
31	26, 30	oveq12d 6299	. . . . . . 7 |- ( p = { <. I , I >. } -> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( p ` x ) M x ) ) ) ) = ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) )
32	31	fmptsng 6077	. . . . . 6 |- ( ( { <. I , I >. } e. _V /\ ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) e. _V ) -> { <. { <. I , I >. } , ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) >. } = ( p e. { { <. I , I >. } } |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( p ` x ) M x ) ) ) ) ) )
33	32	eqcomd 2451	. . . . 5 |- ( ( { <. I , I >. } e. _V /\ ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) e. _V ) -> ( p e. { { <. I , I >. } } |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( p ` x ) M x ) ) ) ) ) = { <. { <. I , I >. } , ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) >. } )
34	24, 25, 33	sylancl 662	. . . 4 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( p e. { { <. I , I >. } } |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( p ` x ) M x ) ) ) ) ) = { <. { <. I , I >. } , ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) >. } )
35		eqid 2443	. . . . . . . . . . . . 13 |- ( SymGrp ` N ) = ( SymGrp ` N )
36		eqid 2443	. . . . . . . . . . . . 13 |- { b e. ( Base ` ( SymGrp ` N ) ) | dom ( b \ _I ) e. Fin } = { b e. ( Base ` ( SymGrp ` N ) ) | dom ( b \ _I ) e. Fin }
37	35, 4, 36, 6	psgnfn 16400	. . . . . . . . . . . 12 |- (pmSgn ` N ) Fn { b e. ( Base ` ( SymGrp ` N ) ) | dom ( b \ _I ) e. Fin }
38	19	adantl 466	. . . . . . . . . . . . . . . . 17 |- ( ( N = { I } /\ I e. V ) -> ( Base ` ( SymGrp ` { I } ) ) = { { <. I , I >. } } )
39	13, 38	eqtrd 2484	. . . . . . . . . . . . . . . 16 |- ( ( N = { I } /\ I e. V ) -> ( Base ` ( SymGrp ` N ) ) = { { <. I , I >. } } )
40	39	3ad2ant2 1019	. . . . . . . . . . . . . . 15 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` ( SymGrp ` N ) ) = { { <. I , I >. } } )
41		rabeq 3089	. . . . . . . . . . . . . . 15 |- ( ( Base ` ( SymGrp ` N ) ) = { { <. I , I >. } } -> { b e. ( Base ` ( SymGrp ` N ) ) | dom ( b \ _I ) e. Fin } = { b e. { { <. I , I >. } } | dom ( b \ _I ) e. Fin } )
42	40, 41	syl 16	. . . . . . . . . . . . . 14 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> { b e. ( Base ` ( SymGrp ` N ) ) | dom ( b \ _I ) e. Fin } = { b e. { { <. I , I >. } } | dom ( b \ _I ) e. Fin } )
43		difeq1 3600	. . . . . . . . . . . . . . . . . 18 |- ( b = { <. I , I >. } -> ( b \ _I ) = ( { <. I , I >. } \ _I ) )
44	43	dmeqd 5195	. . . . . . . . . . . . . . . . 17 |- ( b = { <. I , I >. } -> dom ( b \ _I ) = dom ( { <. I , I >. } \ _I ) )
45	44	eleq1d 2512	. . . . . . . . . . . . . . . 16 |- ( b = { <. I , I >. } -> ( dom ( b \ _I ) e. Fin <-> dom ( { <. I , I >. } \ _I ) e. Fin ) )
46	45	rabsnif 4084	. . . . . . . . . . . . . . 15 |- { b e. { { <. I , I >. } } | dom ( b \ _I ) e. Fin } = if ( dom ( { <. I , I >. } \ _I ) e. Fin , { { <. I , I >. } } , (/) )
47	46	a1i 11	. . . . . . . . . . . . . 14 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> { b e. { { <. I , I >. } } | dom ( b \ _I ) e. Fin } = if ( dom ( { <. I , I >. } \ _I ) e. Fin , { { <. I , I >. } } , (/) ) )
48		restidsing 5320	. . . . . . . . . . . . . . . . . . . 20 |- ( _I |` { I } ) = ( { I } X. { I } )
49		xpsng 6057	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( I e. V /\ I e. V ) -> ( { I } X. { I } ) = { <. I , I >. } )
50	49	anidms 645	. . . . . . . . . . . . . . . . . . . 20 |- ( I e. V -> ( { I } X. { I } ) = { <. I , I >. } )
51	48, 50	syl5req 2497	. . . . . . . . . . . . . . . . . . 19 |- ( I e. V -> { <. I , I >. } = ( _I |` { I } ) )
52		fnsng 5625	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( I e. V /\ I e. V ) -> { <. I , I >. } Fn { I } )
53	52	anidms 645	. . . . . . . . . . . . . . . . . . . 20 |- ( I e. V -> { <. I , I >. } Fn { I } )
54		fnnfpeq0 6087	. . . . . . . . . . . . . . . . . . . 20 |- ( { <. I , I >. } Fn { I } -> ( dom ( { <. I , I >. } \ _I ) = (/) <-> { <. I , I >. } = ( _I |` { I } ) ) )
55	53, 54	syl 16	. . . . . . . . . . . . . . . . . . 19 |- ( I e. V -> ( dom ( { <. I , I >. } \ _I ) = (/) <-> { <. I , I >. } = ( _I |` { I } ) ) )
56	51, 55	mpbird 232	. . . . . . . . . . . . . . . . . 18 |- ( I e. V -> dom ( { <. I , I >. } \ _I ) = (/) )
57		0fin 7749	. . . . . . . . . . . . . . . . . 18 |- (/) e. Fin
58	56, 57	syl6eqel 2539	. . . . . . . . . . . . . . . . 17 |- ( I e. V -> dom ( { <. I , I >. } \ _I ) e. Fin )
59	58	adantl 466	. . . . . . . . . . . . . . . 16 |- ( ( N = { I } /\ I e. V ) -> dom ( { <. I , I >. } \ _I ) e. Fin )
60	59	3ad2ant2 1019	. . . . . . . . . . . . . . 15 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> dom ( { <. I , I >. } \ _I ) e. Fin )
61	60	iftrued 3934	. . . . . . . . . . . . . 14 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> if ( dom ( { <. I , I >. } \ _I ) e. Fin , { { <. I , I >. } } , (/) ) = { { <. I , I >. } } )
62	42, 47, 61	3eqtrrd 2489	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> { { <. I , I >. } } = { b e. ( Base ` ( SymGrp ` N ) ) | dom ( b \ _I ) e. Fin } )
63	62	fneq2d 5662	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (pmSgn ` N ) Fn { { <. I , I >. } } <-> (pmSgn ` N ) Fn { b e. ( Base ` ( SymGrp ` N ) ) | dom ( b \ _I ) e. Fin } ) )
64	37, 63	mpbiri 233	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (pmSgn ` N ) Fn { { <. I , I >. } } )
65	23	snid 4042	. . . . . . . . . . 11 |- { <. I , I >. } e. { { <. I , I >. } }
66		fvco2 5933	. . . . . . . . . . 11 |- ( ( (pmSgn ` N ) Fn { { <. I , I >. } } /\ { <. I , I >. } e. { { <. I , I >. } } ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) = ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. I , I >. } ) ) )
67	64, 65, 66	sylancl 662	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) = ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. I , I >. } ) ) )
68		fveq2 5856	. . . . . . . . . . . . . . 15 |- ( N = { I } -> (pmSgn ` N ) = (pmSgn ` { I } ) )
69	68	adantr 465	. . . . . . . . . . . . . 14 |- ( ( N = { I } /\ I e. V ) -> (pmSgn ` N ) = (pmSgn ` { I } ) )
70	69	3ad2ant2 1019	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (pmSgn ` N ) = (pmSgn ` { I } ) )
71	70	fveq1d 5858	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (pmSgn ` N ) ` { <. I , I >. } ) = ( (pmSgn ` { I } ) ` { <. I , I >. } ) )
72		snidg 4040	. . . . . . . . . . . . . . . . . 18 |- ( { <. I , I >. } e. _V -> { <. I , I >. } e. { { <. I , I >. } } )
73	23, 72	mp1i 12	. . . . . . . . . . . . . . . . 17 |- ( I e. V -> { <. I , I >. } e. { { <. I , I >. } } )
74	73, 19	eleqtrrd 2534	. . . . . . . . . . . . . . . 16 |- ( I e. V -> { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) )
75	74	ancli 551	. . . . . . . . . . . . . . 15 |- ( I e. V -> ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) )
76	75	adantl 466	. . . . . . . . . . . . . 14 |- ( ( N = { I } /\ I e. V ) -> ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) )
77	76	3ad2ant2 1019	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) )
78		eqid 2443	. . . . . . . . . . . . . 14 |- (pmSgn ` { I } ) = (pmSgn ` { I } )
79	18, 16, 17, 78	psgnsn 16419	. . . . . . . . . . . . 13 |- ( ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) -> ( (pmSgn ` { I } ) ` { <. I , I >. } ) = 1 )
80	77, 79	syl 16	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (pmSgn ` { I } ) ` { <. I , I >. } ) = 1 )
81	71, 80	eqtrd 2484	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (pmSgn ` N ) ` { <. I , I >. } ) = 1 )
82	81	fveq2d 5860	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. I , I >. } ) ) = ( ( ZRHom ` R ) ` 1 ) )
83		crngring 17083	. . . . . . . . . . . 12 |- ( R e. CRing -> R e. Ring )
84	83	3ad2ant1 1018	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> R e. Ring )
85		eqid 2443	. . . . . . . . . . . 12 |- ( 1r ` R ) = ( 1r ` R )
86	5, 85	zrh1 18423	. . . . . . . . . . 11 |- ( R e. Ring -> ( ( ZRHom ` R ) ` 1 ) = ( 1r ` R ) )
87	84, 86	syl 16	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ZRHom ` R ) ` 1 ) = ( 1r ` R ) )
88	67, 82, 87	3eqtrd 2488	. . . . . . . . 9 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) = ( 1r ` R ) )
89		simp2l 1023	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> N = { I } )
90	89	mpteq1d 4518	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) = ( x e. { I } |-> ( ( { <. I , I >. } ` x ) M x ) ) )
91	90	oveq2d 6297	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) = ( (mulGrp ` R ) gsumg ( x e. { I } |-> ( ( { <. I , I >. } ` x ) M x ) ) ) )
92	8	ringmgp 17078	. . . . . . . . . . . . 13 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
93	83, 92	syl 16	. . . . . . . . . . . 12 |- ( R e. CRing -> (mulGrp ` R ) e. Mnd )
94	93	3ad2ant1 1018	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (mulGrp ` R ) e. Mnd )
95		snidg 4040	. . . . . . . . . . . . . . . . 17 |- ( I e. V -> I e. { I } )
96	95	adantl 466	. . . . . . . . . . . . . . . 16 |- ( ( N = { I } /\ I e. V ) -> I e. { I } )
97		eleq2 2516	. . . . . . . . . . . . . . . . 17 |- ( N = { I } -> ( I e. N <-> I e. { I } ) )
98	97	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( N = { I } /\ I e. V ) -> ( I e. N <-> I e. { I } ) )
99	96, 98	mpbird 232	. . . . . . . . . . . . . . 15 |- ( ( N = { I } /\ I e. V ) -> I e. N )
100	3	eleq2i 2521	. . . . . . . . . . . . . . . 16 |- ( M e. B <-> M e. ( Base ` A ) )
101	100	biimpi 194	. . . . . . . . . . . . . . 15 |- ( M e. B -> M e. ( Base ` A ) )
102		simpl 457	. . . . . . . . . . . . . . . 16 |- ( ( I e. N /\ M e. ( Base ` A ) ) -> I e. N )
103		simpr 461	. . . . . . . . . . . . . . . 16 |- ( ( I e. N /\ M e. ( Base ` A ) ) -> M e. ( Base ` A ) )
104	102, 102, 103	3jca 1177	. . . . . . . . . . . . . . 15 |- ( ( I e. N /\ M e. ( Base ` A ) ) -> ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) )
105	99, 101, 104	syl2an 477	. . . . . . . . . . . . . 14 |- ( ( ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) )
106	105	3adant1 1015	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) )
107		eqid 2443	. . . . . . . . . . . . . 14 |- ( Base ` R ) = ( Base ` R )
108	2, 107	matecl 18800	. . . . . . . . . . . . 13 |- ( ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) -> ( I M I ) e. ( Base ` R ) )
109	106, 108	syl 16	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I M I ) e. ( Base ` R ) )
110	8, 107	mgpbas 17021	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` (mulGrp ` R ) )
111	109, 110	syl6eleq 2541	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I M I ) e. ( Base ` (mulGrp ` R ) ) )
112		eqid 2443	. . . . . . . . . . . 12 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
113		fveq2 5856	. . . . . . . . . . . . . 14 |- ( x = I -> ( { <. I , I >. } ` x ) = ( { <. I , I >. } ` I ) )
114		eqvisset 3103	. . . . . . . . . . . . . . 15 |- ( x = I -> I e. _V )
115		fvsng 6090	. . . . . . . . . . . . . . 15 |- ( ( I e. _V /\ I e. _V ) -> ( { <. I , I >. } ` I ) = I )
116	114, 114, 115	syl2anc 661	. . . . . . . . . . . . . 14 |- ( x = I -> ( { <. I , I >. } ` I ) = I )
117	113, 116	eqtrd 2484	. . . . . . . . . . . . 13 |- ( x = I -> ( { <. I , I >. } ` x ) = I )
118		id 22	. . . . . . . . . . . . 13 |- ( x = I -> x = I )
119	117, 118	oveq12d 6299	. . . . . . . . . . . 12 |- ( x = I -> ( ( { <. I , I >. } ` x ) M x ) = ( I M I ) )
120	112, 119	gsumsn 16855	. . . . . . . . . . 11 |- ( ( (mulGrp ` R ) e. Mnd /\ I e. V /\ ( I M I ) e. ( Base ` (mulGrp ` R ) ) ) -> ( (mulGrp ` R ) gsumg ( x e. { I } |-> ( ( { <. I , I >. } ` x ) M x ) ) ) = ( I M I ) )
121	94, 15, 111, 120	syl3anc 1229	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( x e. { I } |-> ( ( { <. I , I >. } ` x ) M x ) ) ) = ( I M I ) )
122	91, 121	eqtrd 2484	. . . . . . . . 9 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) = ( I M I ) )
123	88, 122	oveq12d 6299	. . . . . . . 8 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) = ( ( 1r ` R ) ( .r ` R ) ( I M I ) ) )
124	99	3ad2ant2 1019	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> I e. N )
125	101	3ad2ant3 1020	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> M e. ( Base ` A ) )
126	124, 124, 125, 108	syl3anc 1229	. . . . . . . . 9 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I M I ) e. ( Base ` R ) )
127	107, 7, 85	ringlidm 17096	. . . . . . . . 9 |- ( ( R e. Ring /\ ( I M I ) e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) ( I M I ) ) = ( I M I ) )
128	84, 126, 127	syl2anc 661	. . . . . . . 8 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( 1r ` R ) ( .r ` R ) ( I M I ) ) = ( I M I ) )
129	123, 128	eqtrd 2484	. . . . . . 7 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) = ( I M I ) )
130	129	opeq2d 4209	. . . . . 6 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> <. { <. I , I >. } , ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) >. = <. { <. I , I >. } , ( I M I ) >. )
131	130	sneqd 4026	. . . . 5 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> { <. { <. I , I >. } , ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) >. } = { <. { <. I , I >. } , ( I M I ) >. } )
132		ovex 6309	. . . . . 6 |- ( I M I ) e. _V
133		eqidd 2444	. . . . . . 7 |- ( y = { <. I , I >. } -> ( I M I ) = ( I M I ) )
134	133	fmptsng 6077	. . . . . 6 |- ( ( { <. I , I >. } e. _V /\ ( I M I ) e. _V ) -> { <. { <. I , I >. } , ( I M I ) >. } = ( y e. { { <. I , I >. } } |-> ( I M I ) ) )
135	24, 132, 134	sylancl 662	. . . . 5 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> { <. { <. I , I >. } , ( I M I ) >. } = ( y e. { { <. I , I >. } } |-> ( I M I ) ) )
136	131, 135	eqtrd 2484	. . . 4 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> { <. { <. I , I >. } , ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) ) ) >. } = ( y e. { { <. I , I >. } } |-> ( I M I ) ) )
137	22, 34, 136	3eqtrd 2488	. . 3 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( p ` x ) M x ) ) ) ) ) = ( y e. { { <. I , I >. } } |-> ( I M I ) ) )
138	137	oveq2d 6297	. 2 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( R gsumg ( p e. ( Base ` ( SymGrp ` N ) ) |-> ( ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) ( .r ` R ) ( (mulGrp ` R ) gsumg ( x e. N |-> ( ( p ` x ) M x ) ) ) ) ) ) = ( R gsumg ( y e. { { <. I , I >. } } |-> ( I M I ) ) ) )
139		ringmnd 17081	. . . . 5 |- ( R e. Ring -> R e. Mnd )
140	83, 139	syl 16	. . . 4 |- ( R e. CRing -> R e. Mnd )
141	140	3ad2ant1 1018	. . 3 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> R e. Mnd )
142	107, 133	gsumsn 16855	. . 3 |- ( ( R e. Mnd /\ { <. I , I >. } e. _V /\ ( I M I ) e. ( Base ` R ) ) -> ( R gsumg ( y e. { { <. I , I >. } } |-> ( I M I ) ) ) = ( I M I ) )
143	141, 24, 126, 142	syl3anc 1229	. 2 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( R gsumg ( y e. { { <. I , I >. } } |-> ( I M I ) ) ) = ( I M I ) )
144	10, 138, 143	3eqtrd 2488	1 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( D ` M ) = ( I M I ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   {crab 2797   _Vcvv 3095   \ cdif 3458   (/)c0 3770   ifcif 3926   {csn 4014   <.cop 4020   |-> cmpt 4495   _I cid 4780   X. cxp 4987   dom cdm 4989   |` cres 4991   o. ccom 4993   Fn wfn 5573   ` cfv 5578  (class class class)co 6281   Fincfn 7518   1c1 9496   Basecbs 14509   .rcmulr 14575   gsum_g cgsu 14715   Mndcmnd 15793   SymGrpcsymg 16276  pmSgncpsgn 16388  mulGrpcmgp 17015   1rcur 17027   Ringcrg 17072   CRingccrg 17073   ZRHomczrh 18410   Mat cmat 18782   maDet cmdat 18959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-addf 9574  ax-mulf 9575

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-xor 1365  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-rp 11230  df-fz 11682  df-fzo 11804  df-seq 12087  df-exp 12146  df-hash 12385  df-word 12521  df-concat 12523  df-s1 12524  df-substr 12525  df-splice 12526  df-reverse 12527  df-s2 12792  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-0g 14716  df-gsum 14717  df-prds 14722  df-pws 14724  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-submnd 15841  df-grp 15931  df-minusg 15932  df-mulg 15934  df-subg 16072  df-ghm 16139  df-gim 16181  df-cntz 16229  df-oppg 16255  df-symg 16277  df-pmtr 16341  df-psgn 16390  df-cmn 16674  df-mgp 17016  df-ur 17028  df-ring 17074  df-cring 17075  df-rnghom 17238  df-subrg 17301  df-sra 17692  df-rgmod 17693  df-cnfld 18295  df-zring 18363  df-zrh 18414  df-dsmm 18636  df-frlm 18651  df-mat 18783  df-mdet 18960

This theorem is referenced by:  chpmat1d  19210