Theorem m1detdiag 30823

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 2438	. . . 4 |- ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` N ) )
5		eqid 2438	. . . 4 |- ( ZRHom ` R ) = ( ZRHom ` R )
6		eqid 2438	. . . 4 |- (pmSgn ` N ) = (pmSgn ` N )
7		eqid 2438	. . . 4 |- ( .r ` R ) = ( .r ` R )
8		eqid 2438	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
9	1, 2, 3, 4, 5, 6, 7, 8	mdetleib 18373	. . 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 1011	. 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 5686	. . . . . . . . 9 |- ( N = { I } -> ( SymGrp ` N ) = ( SymGrp ` { I } ) )
12	11	fveq2d 5690	. . . . . . . 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 1010	. . . . . 6 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` { I } ) ) )
15		simp2r 1015	. . . . . . 7 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> I e. V )
16		eqid 2438	. . . . . . . 8 |- ( SymGrp ` { I } ) = ( SymGrp ` { I } )
17		eqid 2438	. . . . . . . 8 |- ( Base ` ( SymGrp ` { I } ) ) = ( Base ` ( SymGrp ` { I } ) )
18		eqid 2438	. . . . . . . 8 |- { I } = { I }
19	16, 17, 18	symg1bas 15892	. . . . . . 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 2470	. . . . 5 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` ( SymGrp ` N ) ) = { { <. I , I >. } } )
22	21	mpteq1d 4368	. . . 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 4528	. . . . . 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 6111	. . . . 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 5686	. . . . . . . 8 |- ( p = { <. I , I >. } -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) )
27		fveq1 5685	. . . . . . . . . . 11 |- ( p = { <. I , I >. } -> ( p ` x ) = ( { <. I , I >. } ` x ) )
28	27	oveq1d 6101	. . . . . . . . . 10 |- ( p = { <. I , I >. } -> ( ( p ` x ) M x ) = ( ( { <. I , I >. } ` x ) M x ) )
29	28	mpteq2dv 4374	. . . . . . . . 9 |- ( p = { <. I , I >. } -> ( x e. N |-> ( ( p ` x ) M x ) ) = ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) )
30	29	oveq2d 6102	. . . . . . . 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 6104	. . . . . . 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 5895	. . . . . 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 2443	. . . . 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 2438	. . . . . . . . . . . . 13 |- ( SymGrp ` N ) = ( SymGrp ` N )
36		eqid 2438	. . . . . . . . . . . . 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 15998	. . . . . . . . . . . 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 2470	. . . . . . . . . . . . . . . 16 |- ( ( N = { I } /\ I e. V ) -> ( Base ` ( SymGrp ` N ) ) = { { <. I , I >. } } )
40	39	3ad2ant2 1010	. . . . . . . . . . . . . . 15 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` ( SymGrp ` N ) ) = { { <. I , I >. } } )
41		rabeq 2961	. . . . . . . . . . . . . . 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 3462	. . . . . . . . . . . . . . . . . 18 |- ( b = { <. I , I >. } -> ( b \ _I ) = ( { <. I , I >. } \ _I ) )
44	43	dmeqd 5037	. . . . . . . . . . . . . . . . 17 |- ( b = { <. I , I >. } -> dom ( b \ _I ) = dom ( { <. I , I >. } \ _I ) )
45	44	eleq1d 2504	. . . . . . . . . . . . . . . 16 |- ( b = { <. I , I >. } -> ( dom ( b \ _I ) e. Fin <-> dom ( { <. I , I >. } \ _I ) e. Fin ) )
46	45	rabsnif 3939	. . . . . . . . . . . . . . 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 5157	. . . . . . . . . . . . . . . . . . . 20 |- ( _I |` { I } ) = ( { I } X. { I } )
49		xpsng 5879	. . . . . . . . . . . . . . . . . . . . 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 2483	. . . . . . . . . . . . . . . . . . 19 |- ( I e. V -> { <. I , I >. } = ( _I |` { I } ) )
52		fnsng 5460	. . . . . . . . . . . . . . . . . . . . 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 5904	. . . . . . . . . . . . . . . . . . . 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 7532	. . . . . . . . . . . . . . . . . 18 |- (/) e. Fin
58	56, 57	syl6eqel 2526	. . . . . . . . . . . . . . . . 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 1010	. . . . . . . . . . . . . . 15 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> dom ( { <. I , I >. } \ _I ) e. Fin )
61		iftrue 3792	. . . . . . . . . . . . . . 15 |- ( dom ( { <. I , I >. } \ _I ) e. Fin -> if ( dom ( { <. I , I >. } \ _I ) e. Fin , { { <. I , I >. } } , (/) ) = { { <. I , I >. } } )
62	60, 61	syl 16	. . . . . . . . . . . . . 14 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> if ( dom ( { <. I , I >. } \ _I ) e. Fin , { { <. I , I >. } } , (/) ) = { { <. I , I >. } } )
63	42, 47, 62	3eqtrrd 2475	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> { { <. I , I >. } } = { b e. ( Base ` ( SymGrp ` N ) ) | dom ( b \ _I ) e. Fin } )
64	63	fneq2d 5497	. . . . . . . . . . . 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 } ) )
65	37, 64	mpbiri 233	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (pmSgn ` N ) Fn { { <. I , I >. } } )
66	23	snid 3900	. . . . . . . . . . 11 |- { <. I , I >. } e. { { <. I , I >. } }
67		fvco2 5761	. . . . . . . . . . 11 |- ( ( (pmSgn ` N ) Fn { { <. I , I >. } } /\ { <. I , I >. } e. { { <. I , I >. } } ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) = ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. I , I >. } ) ) )
68	65, 66, 67	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 >. } ) ) )
69		fveq2 5686	. . . . . . . . . . . . . . 15 |- ( N = { I } -> (pmSgn ` N ) = (pmSgn ` { I } ) )
70	69	adantr 465	. . . . . . . . . . . . . 14 |- ( ( N = { I } /\ I e. V ) -> (pmSgn ` N ) = (pmSgn ` { I } ) )
71	70	3ad2ant2 1010	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (pmSgn ` N ) = (pmSgn ` { I } ) )
72	71	fveq1d 5688	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (pmSgn ` N ) ` { <. I , I >. } ) = ( (pmSgn ` { I } ) ` { <. I , I >. } ) )
73		snidg 3898	. . . . . . . . . . . . . . . . . 18 |- ( { <. I , I >. } e. _V -> { <. I , I >. } e. { { <. I , I >. } } )
74	23, 73	mp1i 12	. . . . . . . . . . . . . . . . 17 |- ( I e. V -> { <. I , I >. } e. { { <. I , I >. } } )
75	74, 19	eleqtrrd 2515	. . . . . . . . . . . . . . . 16 |- ( I e. V -> { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) )
76	75	ancli 551	. . . . . . . . . . . . . . 15 |- ( I e. V -> ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) )
77	76	adantl 466	. . . . . . . . . . . . . 14 |- ( ( N = { I } /\ I e. V ) -> ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) )
78	77	3ad2ant2 1010	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) )
79		eqid 2438	. . . . . . . . . . . . . 14 |- (pmSgn ` { I } ) = (pmSgn ` { I } )
80	18, 16, 17, 79	psgnsn 30722	. . . . . . . . . . . . 13 |- ( ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) -> ( (pmSgn ` { I } ) ` { <. I , I >. } ) = 1 )
81	78, 80	syl 16	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (pmSgn ` { I } ) ` { <. I , I >. } ) = 1 )
82	72, 81	eqtrd 2470	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (pmSgn ` N ) ` { <. I , I >. } ) = 1 )
83	82	fveq2d 5690	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. I , I >. } ) ) = ( ( ZRHom ` R ) ` 1 ) )
84		crngrng 16643	. . . . . . . . . . . 12 |- ( R e. CRing -> R e. Ring )
85	84	3ad2ant1 1009	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> R e. Ring )
86		eqid 2438	. . . . . . . . . . . 12 |- ( 1r ` R ) = ( 1r ` R )
87	5, 86	zrh1 17919	. . . . . . . . . . 11 |- ( R e. Ring -> ( ( ZRHom ` R ) ` 1 ) = ( 1r ` R ) )
88	85, 87	syl 16	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ZRHom ` R ) ` 1 ) = ( 1r ` R ) )
89	68, 83, 88	3eqtrd 2474	. . . . . . . . 9 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) = ( 1r ` R ) )
90		simp2l 1014	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> N = { I } )
91	90	mpteq1d 4368	. . . . . . . . . . 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 ) ) )
92	91	oveq2d 6102	. . . . . . . . . 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 ) ) ) )
93	8	rngmgp 16639	. . . . . . . . . . . . 13 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
94	84, 93	syl 16	. . . . . . . . . . . 12 |- ( R e. CRing -> (mulGrp ` R ) e. Mnd )
95	94	3ad2ant1 1009	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (mulGrp ` R ) e. Mnd )
96		snidg 3898	. . . . . . . . . . . . . . . . 17 |- ( I e. V -> I e. { I } )
97	96	adantl 466	. . . . . . . . . . . . . . . 16 |- ( ( N = { I } /\ I e. V ) -> I e. { I } )
98		eleq2 2499	. . . . . . . . . . . . . . . . 17 |- ( N = { I } -> ( I e. N <-> I e. { I } ) )
99	98	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( N = { I } /\ I e. V ) -> ( I e. N <-> I e. { I } ) )
100	97, 99	mpbird 232	. . . . . . . . . . . . . . 15 |- ( ( N = { I } /\ I e. V ) -> I e. N )
101	3	eleq2i 2502	. . . . . . . . . . . . . . . 16 |- ( M e. B <-> M e. ( Base ` A ) )
102	101	biimpi 194	. . . . . . . . . . . . . . 15 |- ( M e. B -> M e. ( Base ` A ) )
103		simpl 457	. . . . . . . . . . . . . . . 16 |- ( ( I e. N /\ M e. ( Base ` A ) ) -> I e. N )
104		simpr 461	. . . . . . . . . . . . . . . 16 |- ( ( I e. N /\ M e. ( Base ` A ) ) -> M e. ( Base ` A ) )
105	103, 103, 104	3jca 1168	. . . . . . . . . . . . . . 15 |- ( ( I e. N /\ M e. ( Base ` A ) ) -> ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) )
106	100, 102, 105	syl2an 477	. . . . . . . . . . . . . 14 |- ( ( ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) )
107	106	3adant1 1006	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) )
108		eqid 2438	. . . . . . . . . . . . . 14 |- ( Base ` R ) = ( Base ` R )
109	2, 108	matecl 18301	. . . . . . . . . . . . 13 |- ( ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) -> ( I M I ) e. ( Base ` R ) )
110	107, 109	syl 16	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I M I ) e. ( Base ` R ) )
111	8, 108	mgpbas 16585	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` (mulGrp ` R ) )
112	110, 111	syl6eleq 2528	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I M I ) e. ( Base ` (mulGrp ` R ) ) )
113		eqid 2438	. . . . . . . . . . . 12 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
114		fveq2 5686	. . . . . . . . . . . . . 14 |- ( x = I -> ( { <. I , I >. } ` x ) = ( { <. I , I >. } ` I ) )
115		eqvisset 2975	. . . . . . . . . . . . . . 15 |- ( x = I -> I e. _V )
116		fvsng 5907	. . . . . . . . . . . . . . 15 |- ( ( I e. _V /\ I e. _V ) -> ( { <. I , I >. } ` I ) = I )
117	115, 115, 116	syl2anc 661	. . . . . . . . . . . . . 14 |- ( x = I -> ( { <. I , I >. } ` I ) = I )
118	114, 117	eqtrd 2470	. . . . . . . . . . . . 13 |- ( x = I -> ( { <. I , I >. } ` x ) = I )
119		id 22	. . . . . . . . . . . . 13 |- ( x = I -> x = I )
120	118, 119	oveq12d 6104	. . . . . . . . . . . 12 |- ( x = I -> ( ( { <. I , I >. } ` x ) M x ) = ( I M I ) )
121	113, 120	gsumsn 16439	. . . . . . . . . . 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 ) )
122	95, 15, 112, 121	syl3anc 1218	. . . . . . . . . 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 ) )
123	92, 122	eqtrd 2470	. . . . . . . . 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 ) )
124	89, 123	oveq12d 6104	. . . . . . . 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 ) ) )
125	100	3ad2ant2 1010	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> I e. N )
126	102	3ad2ant3 1011	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> M e. ( Base ` A ) )
127	125, 125, 126, 109	syl3anc 1218	. . . . . . . . 9 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I M I ) e. ( Base ` R ) )
128	108, 7, 86	rnglidm 16656	. . . . . . . . 9 |- ( ( R e. Ring /\ ( I M I ) e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) ( I M I ) ) = ( I M I ) )
129	85, 127, 128	syl2anc 661	. . . . . . . 8 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( 1r ` R ) ( .r ` R ) ( I M I ) ) = ( I M I ) )
130	124, 129	eqtrd 2470	. . . . . . 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 ) )
131	130	opeq2d 4061	. . . . . 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 ) >. )
132	131	sneqd 3884	. . . . 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 ) >. } )
133		ovex 6111	. . . . . 6 |- ( I M I ) e. _V
134		eqidd 2439	. . . . . . 7 |- ( y = { <. I , I >. } -> ( I M I ) = ( I M I ) )
135	134	fmptsng 5895	. . . . . 6 |- ( ( { <. I , I >. } e. _V /\ ( I M I ) e. _V ) -> { <. { <. I , I >. } , ( I M I ) >. } = ( y e. { { <. I , I >. } } |-> ( I M I ) ) )
136	24, 133, 135	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 ) ) )
137	132, 136	eqtrd 2470	. . . 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 ) ) )
138	22, 34, 137	3eqtrd 2474	. . 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 ) ) )
139	138	oveq2d 6102	. 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 ) ) ) )
140		rngmnd 16642	. . . . 5 |- ( R e. Ring -> R e. Mnd )
141	84, 140	syl 16	. . . 4 |- ( R e. CRing -> R e. Mnd )
142	141	3ad2ant1 1009	. . 3 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> R e. Mnd )
143	108, 134	gsumsn 16439	. . 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 ) )
144	142, 24, 127, 143	syl3anc 1218	. 2 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( R gsumg ( y e. { { <. I , I >. } } |-> ( I M I ) ) ) = ( I M I ) )
145	10, 139, 144	3eqtrd 2474	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 965   = wceq 1369   e. wcel 1756   {crab 2714   _Vcvv 2967   \ cdif 3320   (/)c0 3632   ifcif 3786   {csn 3872   <.cop 3878   e. cmpt 4345   _I cid 4626   X. cxp 4833   dom cdm 4835   |` cres 4837   o. ccom 4839   Fn wfn 5408   ` cfv 5413  (class class class)co 6086   Fincfn 7302   1c1 9275   Basecbs 14166   .rcmulr 14231   gsum_g cgsu 14371   Mndcmnd 15401   SymGrpcsymg 15873  pmSgncpsgn 15986  mulGrpcmgp 16579   1rcur 16591   Ringcrg 16633   CRingccrg 16634   ZRHomczrh 17906   Mat cmat 18255   maDet cmdat 18370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-ot 3881  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-word 12221  df-concat 12223  df-s1 12224  df-substr 12225  df-splice 12226  df-reverse 12227  df-s2 12467  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-0g 14372  df-gsum 14373  df-prds 14378  df-pws 14380  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-grp 15536  df-minusg 15537  df-mulg 15539  df-subg 15669  df-ghm 15736  df-gim 15778  df-cntz 15826  df-oppg 15852  df-symg 15874  df-pmtr 15939  df-psgn 15988  df-cmn 16270  df-mgp 16580  df-ur 16592  df-rng 16635  df-cring 16636  df-rnghom 16794  df-subrg 16841  df-sra 17230  df-rgmod 17231  df-cnfld 17794  df-zring 17859  df-zrh 17910  df-dsmm 18132  df-frlm 18147  df-mat 18257  df-mdet 18371

This theorem is referenced by: (None)