Theorem m1detdiag 20403

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 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		eqid 2622	. . . 4 |- ( .r ` R ) = ( .r ` R )
8		eqid 2622	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
9	1, 2, 3, 4, 5, 6, 7, 8	mdetleib 20393	. . 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 1084	. 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 6191	. . . . . . . . 9 |- ( N = { I } -> ( SymGrp ` N ) = ( SymGrp ` { I } ) )
12	11	fveq2d 6195	. . . . . . . 8 |- ( N = { I } -> ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` { I } ) ) )
13	12	adantr 481	. . . . . . 7 |- ( ( N = { I } /\ I e. V ) -> ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` { I } ) ) )
14	13	3ad2ant2 1083	. . . . . 6 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` { I } ) ) )
15		simp2r 1088	. . . . . . 7 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> I e. V )
16		eqid 2622	. . . . . . . 8 |- ( SymGrp ` { I } ) = ( SymGrp ` { I } )
17		eqid 2622	. . . . . . . 8 |- ( Base ` ( SymGrp ` { I } ) ) = ( Base ` ( SymGrp ` { I } ) )
18		eqid 2622	. . . . . . . 8 |- { I } = { I }
19	16, 17, 18	symg1bas 17816	. . . . . . 7 |- ( I e. V -> ( Base ` ( SymGrp ` { I } ) ) = { { <. I , I >. } } )
20	15, 19	syl 17	. . . . . 6 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` ( SymGrp ` { I } ) ) = { { <. I , I >. } } )
21	14, 20	eqtrd 2656	. . . . 5 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` ( SymGrp ` N ) ) = { { <. I , I >. } } )
22	21	mpteq1d 4738	. . . 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 4908	. . . . . 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 6678	. . . . 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 6191	. . . . . . . 8 |- ( p = { <. I , I >. } -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` p ) = ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) )
27		fveq1 6190	. . . . . . . . . . 11 |- ( p = { <. I , I >. } -> ( p ` x ) = ( { <. I , I >. } ` x ) )
28	27	oveq1d 6665	. . . . . . . . . 10 |- ( p = { <. I , I >. } -> ( ( p ` x ) M x ) = ( ( { <. I , I >. } ` x ) M x ) )
29	28	mpteq2dv 4745	. . . . . . . . 9 |- ( p = { <. I , I >. } -> ( x e. N |-> ( ( p ` x ) M x ) ) = ( x e. N |-> ( ( { <. I , I >. } ` x ) M x ) ) )
30	29	oveq2d 6666	. . . . . . . 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 6668	. . . . . . 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 6434	. . . . . 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 2628	. . . . 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 694	. . . 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 2622	. . . . . . . . . . . . 13 |- ( SymGrp ` N ) = ( SymGrp ` N )
36		eqid 2622	. . . . . . . . . . . . 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 17921	. . . . . . . . . . . 12 |- (pmSgn ` N ) Fn { b e. ( Base ` ( SymGrp ` N ) ) | dom ( b \ _I ) e. Fin }
38	19	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( N = { I } /\ I e. V ) -> ( Base ` ( SymGrp ` { I } ) ) = { { <. I , I >. } } )
39	13, 38	eqtrd 2656	. . . . . . . . . . . . . . . 16 |- ( ( N = { I } /\ I e. V ) -> ( Base ` ( SymGrp ` N ) ) = { { <. I , I >. } } )
40	39	3ad2ant2 1083	. . . . . . . . . . . . . . 15 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` ( SymGrp ` N ) ) = { { <. I , I >. } } )
41		rabeq 3192	. . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . 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 3721	. . . . . . . . . . . . . . . . . 18 |- ( b = { <. I , I >. } -> ( b \ _I ) = ( { <. I , I >. } \ _I ) )
44	43	dmeqd 5326	. . . . . . . . . . . . . . . . 17 |- ( b = { <. I , I >. } -> dom ( b \ _I ) = dom ( { <. I , I >. } \ _I ) )
45	44	eleq1d 2686	. . . . . . . . . . . . . . . 16 |- ( b = { <. I , I >. } -> ( dom ( b \ _I ) e. Fin <-> dom ( { <. I , I >. } \ _I ) e. Fin ) )
46	45	rabsnif 4258	. . . . . . . . . . . . . . 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 5458	. . . . . . . . . . . . . . . . . . . 20 |- ( _I |` { I } ) = ( { I } X. { I } )
49		xpsng 6406	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( I e. V /\ I e. V ) -> ( { I } X. { I } ) = { <. I , I >. } )
50	49	anidms 677	. . . . . . . . . . . . . . . . . . . 20 |- ( I e. V -> ( { I } X. { I } ) = { <. I , I >. } )
51	48, 50	syl5req 2669	. . . . . . . . . . . . . . . . . . 19 |- ( I e. V -> { <. I , I >. } = ( _I |` { I } ) )
52		fnsng 5938	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( I e. V /\ I e. V ) -> { <. I , I >. } Fn { I } )
53	52	anidms 677	. . . . . . . . . . . . . . . . . . . 20 |- ( I e. V -> { <. I , I >. } Fn { I } )
54		fnnfpeq0 6444	. . . . . . . . . . . . . . . . . . . 20 |- ( { <. I , I >. } Fn { I } -> ( dom ( { <. I , I >. } \ _I ) = (/) <-> { <. I , I >. } = ( _I |` { I } ) ) )
55	53, 54	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( I e. V -> ( dom ( { <. I , I >. } \ _I ) = (/) <-> { <. I , I >. } = ( _I |` { I } ) ) )
56	51, 55	mpbird 247	. . . . . . . . . . . . . . . . . 18 |- ( I e. V -> dom ( { <. I , I >. } \ _I ) = (/) )
57		0fin 8188	. . . . . . . . . . . . . . . . . 18 |- (/) e. Fin
58	56, 57	syl6eqel 2709	. . . . . . . . . . . . . . . . 17 |- ( I e. V -> dom ( { <. I , I >. } \ _I ) e. Fin )
59	58	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( N = { I } /\ I e. V ) -> dom ( { <. I , I >. } \ _I ) e. Fin )
60	59	3ad2ant2 1083	. . . . . . . . . . . . . . 15 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> dom ( { <. I , I >. } \ _I ) e. Fin )
61	60	iftrued 4094	. . . . . . . . . . . . . 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 2661	. . . . . . . . . . . . 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 5982	. . . . . . . . . . . 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 248	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (pmSgn ` N ) Fn { { <. I , I >. } } )
65	23	snid 4208	. . . . . . . . . . 11 |- { <. I , I >. } e. { { <. I , I >. } }
66		fvco2 6273	. . . . . . . . . . 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 694	. . . . . . . . . 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 6191	. . . . . . . . . . . . . . 15 |- ( N = { I } -> (pmSgn ` N ) = (pmSgn ` { I } ) )
69	68	adantr 481	. . . . . . . . . . . . . 14 |- ( ( N = { I } /\ I e. V ) -> (pmSgn ` N ) = (pmSgn ` { I } ) )
70	69	3ad2ant2 1083	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (pmSgn ` N ) = (pmSgn ` { I } ) )
71	70	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (pmSgn ` N ) ` { <. I , I >. } ) = ( (pmSgn ` { I } ) ` { <. I , I >. } ) )
72		snidg 4206	. . . . . . . . . . . . . . . . . 18 |- ( { <. I , I >. } e. _V -> { <. I , I >. } e. { { <. I , I >. } } )
73	23, 72	mp1i 13	. . . . . . . . . . . . . . . . 17 |- ( I e. V -> { <. I , I >. } e. { { <. I , I >. } } )
74	73, 19	eleqtrrd 2704	. . . . . . . . . . . . . . . 16 |- ( I e. V -> { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) )
75	74	ancli 574	. . . . . . . . . . . . . . 15 |- ( I e. V -> ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) )
76	75	adantl 482	. . . . . . . . . . . . . 14 |- ( ( N = { I } /\ I e. V ) -> ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) )
77	76	3ad2ant2 1083	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) )
78		eqid 2622	. . . . . . . . . . . . . 14 |- (pmSgn ` { I } ) = (pmSgn ` { I } )
79	18, 16, 17, 78	psgnsn 17940	. . . . . . . . . . . . 13 |- ( ( I e. V /\ { <. I , I >. } e. ( Base ` ( SymGrp ` { I } ) ) ) -> ( (pmSgn ` { I } ) ` { <. I , I >. } ) = 1 )
80	77, 79	syl 17	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (pmSgn ` { I } ) ` { <. I , I >. } ) = 1 )
81	71, 80	eqtrd 2656	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (pmSgn ` N ) ` { <. I , I >. } ) = 1 )
82	81	fveq2d 6195	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ZRHom ` R ) ` ( (pmSgn ` N ) ` { <. I , I >. } ) ) = ( ( ZRHom ` R ) ` 1 ) )
83		crngring 18558	. . . . . . . . . . . 12 |- ( R e. CRing -> R e. Ring )
84	83	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> R e. Ring )
85		eqid 2622	. . . . . . . . . . . 12 |- ( 1r ` R ) = ( 1r ` R )
86	5, 85	zrh1 19861	. . . . . . . . . . 11 |- ( R e. Ring -> ( ( ZRHom ` R ) ` 1 ) = ( 1r ` R ) )
87	84, 86	syl 17	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ZRHom ` R ) ` 1 ) = ( 1r ` R ) )
88	67, 82, 87	3eqtrd 2660	. . . . . . . . 9 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( ( ZRHom ` R ) o. (pmSgn ` N ) ) ` { <. I , I >. } ) = ( 1r ` R ) )
89		simp2l 1087	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> N = { I } )
90	89	mpteq1d 4738	. . . . . . . . . . 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 6666	. . . . . . . . . 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 18553	. . . . . . . . . . . . 13 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
93	83, 92	syl 17	. . . . . . . . . . . 12 |- ( R e. CRing -> (mulGrp ` R ) e. Mnd )
94	93	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (mulGrp ` R ) e. Mnd )
95		snidg 4206	. . . . . . . . . . . . . . . . 17 |- ( I e. V -> I e. { I } )
96	95	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( N = { I } /\ I e. V ) -> I e. { I } )
97		eleq2 2690	. . . . . . . . . . . . . . . . 17 |- ( N = { I } -> ( I e. N <-> I e. { I } ) )
98	97	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( N = { I } /\ I e. V ) -> ( I e. N <-> I e. { I } ) )
99	96, 98	mpbird 247	. . . . . . . . . . . . . . 15 |- ( ( N = { I } /\ I e. V ) -> I e. N )
100	3	eleq2i 2693	. . . . . . . . . . . . . . . 16 |- ( M e. B <-> M e. ( Base ` A ) )
101	100	biimpi 206	. . . . . . . . . . . . . . 15 |- ( M e. B -> M e. ( Base ` A ) )
102		simpl 473	. . . . . . . . . . . . . . . 16 |- ( ( I e. N /\ M e. ( Base ` A ) ) -> I e. N )
103		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( I e. N /\ M e. ( Base ` A ) ) -> M e. ( Base ` A ) )
104	102, 102, 103	3jca 1242	. . . . . . . . . . . . . . 15 |- ( ( I e. N /\ M e. ( Base ` A ) ) -> ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) )
105	99, 101, 104	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) )
106	105	3adant1 1079	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) )
107		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` R ) = ( Base ` R )
108	2, 107	matecl 20231	. . . . . . . . . . . . 13 |- ( ( I e. N /\ I e. N /\ M e. ( Base ` A ) ) -> ( I M I ) e. ( Base ` R ) )
109	106, 108	syl 17	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I M I ) e. ( Base ` R ) )
110	8, 107	mgpbas 18495	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` (mulGrp ` R ) )
111	109, 110	syl6eleq 2711	. . . . . . . . . . 11 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I M I ) e. ( Base ` (mulGrp ` R ) ) )
112		eqid 2622	. . . . . . . . . . . 12 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
113		fveq2 6191	. . . . . . . . . . . . . 14 |- ( x = I -> ( { <. I , I >. } ` x ) = ( { <. I , I >. } ` I ) )
114		eqvisset 3211	. . . . . . . . . . . . . . 15 |- ( x = I -> I e. _V )
115		fvsng 6447	. . . . . . . . . . . . . . 15 |- ( ( I e. _V /\ I e. _V ) -> ( { <. I , I >. } ` I ) = I )
116	114, 114, 115	syl2anc 693	. . . . . . . . . . . . . 14 |- ( x = I -> ( { <. I , I >. } ` I ) = I )
117	113, 116	eqtrd 2656	. . . . . . . . . . . . 13 |- ( x = I -> ( { <. I , I >. } ` x ) = I )
118		id 22	. . . . . . . . . . . . 13 |- ( x = I -> x = I )
119	117, 118	oveq12d 6668	. . . . . . . . . . . 12 |- ( x = I -> ( ( { <. I , I >. } ` x ) M x ) = ( I M I ) )
120	112, 119	gsumsn 18354	. . . . . . . . . . 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 1326	. . . . . . . . . 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 2656	. . . . . . . . 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 6668	. . . . . . . 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 1083	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> I e. N )
125	101	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> M e. ( Base ` A ) )
126	124, 124, 125, 108	syl3anc 1326	. . . . . . . . 9 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I M I ) e. ( Base ` R ) )
127	107, 7, 85	ringlidm 18571	. . . . . . . . 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 693	. . . . . . . 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 2656	. . . . . . 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 4409	. . . . . 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 4189	. . . . 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 6678	. . . . . 6 |- ( I M I ) e. _V
133		eqidd 2623	. . . . . . 7 |- ( y = { <. I , I >. } -> ( I M I ) = ( I M I ) )
134	133	fmptsng 6434	. . . . . 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 694	. . . . 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 2656	. . . 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 2660	. . 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 6666	. 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 18556	. . . . 5 |- ( R e. Ring -> R e. Mnd )
140	83, 139	syl 17	. . . 4 |- ( R e. CRing -> R e. Mnd )
141	140	3ad2ant1 1082	. . 3 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> R e. Mnd )
142	107, 133	gsumsn 18354	. . 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 1326	. 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 2660	1 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( D ` M ) = ( I M I ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   (/)c0 3915   ifcif 4086   {csn 4177   <.cop 4183   |-> cmpt 4729   _I cid 5023   X. cxp 5112   dom cdm 5114   |` cres 5116   o. ccom 5118   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   Fincfn 7955   1c1 9937   Basecbs 15857   .rcmulr 15942   gsum_g cgsu 16101   Mndcmnd 17294   SymGrpcsymg 17797  pmSgncpsgn 17909  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   CRingccrg 18548   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-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-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-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:  chpmat1d  20641