Theorem matunit

Description: A matrix is a unit in the ring of matrices iff its determinant is a unit in the underlying ring. (Contributed by Stefan O'Rear, 17-Jul-2018)

	Ref	Expression
Hypotheses	matunit.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
	matunit.d	⊢ 𝐷 = ( 𝑁 maDet 𝑅 )
	matunit.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
	matunit.u	⊢ 𝑈 = ( Unit ‘ 𝐴 )
	matunit.v	⊢ 𝑉 = ( Unit ‘ 𝑅 )
Assertion	matunit	⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑀 ∈ 𝑈 ↔ ( 𝐷 ‘ 𝑀 ) ∈ 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		matunit.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2		matunit.d	⊢ 𝐷 = ( 𝑁 maDet 𝑅 )
3		matunit.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
4		matunit.u	⊢ 𝑈 = ( Unit ‘ 𝐴 )
5		matunit.v	⊢ 𝑉 = ( Unit ‘ 𝑅 )
6		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
7		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
8		eqid	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
9		eqid	⊢ ( invr ‘ 𝑅 ) = ( invr ‘ 𝑅 )
10		crngring	⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring )
11	10	ad2antrr	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → 𝑅 ∈ Ring )
12	2 1 3 6	mdetcl	⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝐷 ‘ 𝑀 ) ∈ ( Base ‘ 𝑅 ) )
13	12	adantr	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( 𝐷 ‘ 𝑀 ) ∈ ( Base ‘ 𝑅 ) )
14	2 1 3 6	mdetf	⊢ ( 𝑅 ∈ CRing → 𝐷 : 𝐵 ⟶ ( Base ‘ 𝑅 ) )
15	14	ad2antrr	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → 𝐷 : 𝐵 ⟶ ( Base ‘ 𝑅 ) )
16	1 3	matrcl	⊢ ( 𝑀 ∈ 𝐵 → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) )
17	16	simpld	⊢ ( 𝑀 ∈ 𝐵 → 𝑁 ∈ Fin )
18	17	ad2antlr	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → 𝑁 ∈ Fin )
19	1	matring	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐴 ∈ Ring )
20	18 11 19	syl2anc	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → 𝐴 ∈ Ring )
21		eqid	⊢ ( invr ‘ 𝐴 ) = ( invr ‘ 𝐴 )
22	4 21 3	ringinvcl	⊢ ( ( 𝐴 ∈ Ring ∧ 𝑀 ∈ 𝑈 ) → ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ∈ 𝐵 )
23	20 22	sylancom	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ∈ 𝐵 )
24	15 23	ffvelcdmd	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( 𝐷 ‘ ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) ∈ ( Base ‘ 𝑅 ) )
25		eqid	⊢ ( .r ‘ 𝐴 ) = ( .r ‘ 𝐴 )
26		eqid	⊢ ( 1r ‘ 𝐴 ) = ( 1r ‘ 𝐴 )
27	4 21 25 26	unitrinv	⊢ ( ( 𝐴 ∈ Ring ∧ 𝑀 ∈ 𝑈 ) → ( 𝑀 ( .r ‘ 𝐴 ) ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) = ( 1r ‘ 𝐴 ) )
28	20 27	sylancom	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( 𝑀 ( .r ‘ 𝐴 ) ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) = ( 1r ‘ 𝐴 ) )
29	28	fveq2d	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( 𝐷 ‘ ( 𝑀 ( .r ‘ 𝐴 ) ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) ) = ( 𝐷 ‘ ( 1r ‘ 𝐴 ) ) )
30		simpll	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → 𝑅 ∈ CRing )
31		simplr	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → 𝑀 ∈ 𝐵 )
32	1 3 2 7 25	mdetmul	⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ∈ 𝐵 ) → ( 𝐷 ‘ ( 𝑀 ( .r ‘ 𝐴 ) ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) ) = ( ( 𝐷 ‘ 𝑀 ) ( .r ‘ 𝑅 ) ( 𝐷 ‘ ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) ) )
33	30 31 23 32	syl3anc	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( 𝐷 ‘ ( 𝑀 ( .r ‘ 𝐴 ) ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) ) = ( ( 𝐷 ‘ 𝑀 ) ( .r ‘ 𝑅 ) ( 𝐷 ‘ ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) ) )
34	2 1 26 8	mdet1	⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) → ( 𝐷 ‘ ( 1r ‘ 𝐴 ) ) = ( 1r ‘ 𝑅 ) )
35	30 18 34	syl2anc	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( 𝐷 ‘ ( 1r ‘ 𝐴 ) ) = ( 1r ‘ 𝑅 ) )
36	29 33 35	3eqtr3d	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( ( 𝐷 ‘ 𝑀 ) ( .r ‘ 𝑅 ) ( 𝐷 ‘ ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) ) = ( 1r ‘ 𝑅 ) )
37	4 21 25 26	unitlinv	⊢ ( ( 𝐴 ∈ Ring ∧ 𝑀 ∈ 𝑈 ) → ( ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ( .r ‘ 𝐴 ) 𝑀 ) = ( 1r ‘ 𝐴 ) )
38	20 37	sylancom	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ( .r ‘ 𝐴 ) 𝑀 ) = ( 1r ‘ 𝐴 ) )
39	38	fveq2d	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( 𝐷 ‘ ( ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ( .r ‘ 𝐴 ) 𝑀 ) ) = ( 𝐷 ‘ ( 1r ‘ 𝐴 ) ) )
40	1 3 2 7 25	mdetmul	⊢ ( ( 𝑅 ∈ CRing ∧ ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ∈ 𝐵 ∧ 𝑀 ∈ 𝐵 ) → ( 𝐷 ‘ ( ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ( .r ‘ 𝐴 ) 𝑀 ) ) = ( ( 𝐷 ‘ ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) ( .r ‘ 𝑅 ) ( 𝐷 ‘ 𝑀 ) ) )
41	30 23 31 40	syl3anc	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( 𝐷 ‘ ( ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ( .r ‘ 𝐴 ) 𝑀 ) ) = ( ( 𝐷 ‘ ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) ( .r ‘ 𝑅 ) ( 𝐷 ‘ 𝑀 ) ) )
42	39 41 35	3eqtr3d	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( ( 𝐷 ‘ ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) ( .r ‘ 𝑅 ) ( 𝐷 ‘ 𝑀 ) ) = ( 1r ‘ 𝑅 ) )
43	6 7 8 5 9 11 13 24 36 42	invrvald	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( ( 𝐷 ‘ 𝑀 ) ∈ 𝑉 ∧ ( ( invr ‘ 𝑅 ) ‘ ( 𝐷 ‘ 𝑀 ) ) = ( 𝐷 ‘ ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) ) ) )
44	43	simpld	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ 𝑈 ) → ( 𝐷 ‘ 𝑀 ) ∈ 𝑉 )
45		eqid	⊢ ( 𝑁 maAdju 𝑅 ) = ( 𝑁 maAdju 𝑅 )
46		eqid	⊢ ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ 𝐴 )
47	1 45 2 3 4 5 9 21 46	matinv	⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ ( 𝐷 ‘ 𝑀 ) ∈ 𝑉 ) → ( 𝑀 ∈ 𝑈 ∧ ( ( invr ‘ 𝐴 ) ‘ 𝑀 ) = ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐷 ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝐴 ) ( ( 𝑁 maAdju 𝑅 ) ‘ 𝑀 ) ) ) )
48	47	simpld	⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ ( 𝐷 ‘ 𝑀 ) ∈ 𝑉 ) → 𝑀 ∈ 𝑈 )
49	48	3expa	⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝐷 ‘ 𝑀 ) ∈ 𝑉 ) → 𝑀 ∈ 𝑈 )
50	44 49	impbida	⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑀 ∈ 𝑈 ↔ ( 𝐷 ‘ 𝑀 ) ∈ 𝑉 ) )